{ "schema": "apstats-misconception-map/v1", "reviewed": false, "provenance": "codex-draft-2026-09-11", "rubrics": { "WS-U1L10-reflect1": { "elements": [ { "id": "percentile-definition", "description": "Explains that percentile is the percent of values less than or equal to a given value" }, { "id": "zscore-definition", "description": "Explains that a z-score tells how many standard deviations a value is from the mean" }, { "id": "sign-meaning", "description": "States that a positive z-score means above the mean and a negative z-score means below the mean" }, { "id": "two-tools", "description": "Makes clear that percentile and z-score are two different ways to describe relative position" }, { "id": "context", "description": "Uses Flint water-sample context such as lead levels, water samples, or parts per billion" }, { "id": "example-values", "description": "May reference examples such as 20 parts per billion at the 91.5th percentile and 0.88 standard deviations above the mean, or 2 parts per billion at the 39.4th percentile and 0.37 standard deviations below the mean" }, { "id": "any-distribution", "description": "May note that percentiles and z-scores can be used for any distribution and do not require a normal distribution" } ], "commonMistakes": [ "Forgetting that percentile includes values less than or equal to the given value", "Describing a z-score as a percent instead of a number of standard deviations", "Mixing up what positive and negative z-scores mean", "Describing percentile and z-score as if they were the same thing" ], "worksheet": "u1_lesson10_live.html" }, "WS-U1L10-reflect2": { "elements": [ { "id": "table-a-left", "description": "States that Table A gives the area to the left of a z-score" }, { "id": "area-right", "description": "Explains that area to the right is found by subtracting the left-tail area from 1" }, { "id": "area-between", "description": "Explains that area between two values is found by calculating two z-scores and subtracting two left-tail areas" }, { "id": "backward-process", "description": "Explains that backward problems use the target area to find a z-score first and then solve for the original value with the mean and standard deviation" }, { "id": "zscore-language", "description": "Uses z-score language such as area to the left, z-score, mean, or standard deviation" }, { "id": "context", "description": "Uses normal-distribution context such as the blood-pressure examples from class" }, { "id": "example-values", "description": "May mention examples like 0.0668 for above 125, 0.13 between 120 and 129, or z = 1.28 and x = 122.8 for the highest 10%" } ], "commonMistakes": [ "Forgetting that Table A gives area to the left, not area to the right", "Using one minus when the problem is asking for area between two values", "Working backward by searching for the area in the z-score column instead of in the body of the table", "Failing to mention the role of the mean and standard deviation when solving for the cutoff value" ], "worksheet": "u1_lesson10_live.html" }, "WS-U1L10-exitTicket": { "elements": [ { "id": "shape-parameters", "description": "States that the distribution is normal, symmetric, and mound-shaped and identifies the mean as 16 ounces and the standard deviation as 0.5 ounce" }, { "id": "within-one-sd", "description": "Uses the empirical rule to state that about 68% of loaves are between 15.5 and 16.5 ounces" }, { "id": "above-two-sd", "description": "Uses the empirical rule to state that about 2.5% of loaves are above 17 ounces" }, { "id": "backward-left-area", "description": "Explains that the heaviest 10% means 90% is to the left before using Table A" }, { "id": "backward-zscore", "description": "Uses a z-score of about 1.28 and solves for a cutoff of about 16.64 ounces or about 16.6 ounces" }, { "id": "context", "description": "Uses loaf-weight context throughout the response" }, { "id": "notation", "description": "May refer to the parameters with symbols such as mu and sigma or mention the z-score formula" } ], "commonMistakes": [ "Forgetting that 15.5 to 16.5 is exactly within one standard deviation of the mean", "Saying the percent above 17 ounces is 5% instead of 2.5%", "Using 10% as the left-tail area instead of converting it to 90% to the left", "Failing to solve back to an actual loaf weight for the cutoff" ], "worksheet": "u1_lesson10_live.html" }, "WS-U1L1-reflect1": { "elements": [ { "id": "needs-context", "description": "Explains that a number by itself does not mean much unless we know what is being measured or described" }, { "id": "units-or-variable", "description": "Identifies the context as lead level in water samples or mentions units such as parts per billion or percent of samples" }, { "id": "links-to-question", "description": "Connects the number to the real question of whether Flint's water was safe to drink or whether the threshold was exceeded" }, { "id": "threshold-idea", "description": "May mention that 15 ppb and 11.3% matter because they can be compared to a cutoff or threshold" } ], "commonMistakes": [ "Saying the number speaks for itself without needing context", "Mentioning the number but not what it measures", "Treating the number as meaningful without saying what it measures", "Ignoring the role of units or percent in the interpretation" ], "worksheet": "u1_lesson1_live.html" }, "WS-U1L1-reflect2": { "elements": [ { "id": "changed-percent", "description": "Explains that the percent above 15 ppb changed from 11.3% (8 of 71) to 8.7% (6 of 69)" }, { "id": "changed-conclusion", "description": "States that the conclusion changed from not safe to safe when the two samples were removed" }, { "id": "responsible-use", "description": "Explains that leaving out relevant data can change results and that statistics should be used honestly and carefully" }, { "id": "real-consequences", "description": "May note that these choices can affect real people and real decisions" } ], "commonMistakes": [ "Stating that the omitted points did not affect the conclusion", "Mentioning only that two points were removed without saying how the conclusion changed", "Ignoring that the conclusion changed from unsafe to safe", "Failing to address why data should be handled responsibly" ], "worksheet": "u1_lesson1_live.html" }, "WS-U1L1-exitTicket": { "elements": [ { "id": "question-in-context", "description": "Identifies a sensible question in context, such as whether too many students are getting too little sleep before the exam" }, { "id": "variable", "description": "Describes the variable as the number of hours of sleep the night before the exam" }, { "id": "variation", "description": "Explains that the data vary because different students got different amounts of sleep" }, { "id": "context-for-30", "description": "Explains that 9 out of 30, or 30%, is meaningful because it describes the proportion of sampled students who got fewer than 6 hours of sleep" }, { "id": "statistics-process", "description": "Describes how statistics can help by collecting data, analyzing the data, and interpreting the results to answer the principal's question" } ], "commonMistakes": [ "Giving a number without explaining what it represents", "Failing to identify the variable clearly", "Not explaining what variation means in the data", "Leaving out the role of collecting, analyzing, and interpreting data" ], "worksheet": "u1_lesson1_live.html" }, "WS-U1L2-reflect1": { "elements": [ { "id": "individuals-are-properties", "description": "Identifies the individuals as the properties or homes for sale in Charleston" }, { "id": "variable-definition", "description": "Explains that a variable is a characteristic recorded for each individual that can change from one property to another" }, { "id": "gives-example-variable", "description": "Gives at least one example of a variable from the Charleston data set, such as price, year built, pool, or zip code" }, { "id": "rows-vs-columns", "description": "May note that individuals appear in rows while variables appear in columns" } ], "commonMistakes": [ "Calling the houses variables instead of individuals", "Naming a column heading without explaining what a variable is", "Giving an identifier such as ID as an example of a variable", "Ignoring the Charleston context entirely" ], "worksheet": "u1_lesson2_live.html" }, "WS-U1L2-reflect2": { "elements": [ { "id": "numbers-as-labels", "description": "Explains that the numbers act as labels or categories rather than measurements or counts" }, { "id": "zip-code-location", "description": "Explains that zip code represents a location or category of place" }, { "id": "grade-level-categories", "description": "Explains that grade level really stands for labels such as freshman, sophomore, junior, or senior" }, { "id": "not-average", "description": "May mention that taking an average would not be meaningful for these values" } ], "commonMistakes": [ "Assuming that any variable written with numbers must be quantitative", "Explaining zip code as a count or measurement", "Ignoring the grade-level example", "Failing to distinguish labels from measured or counted quantities" ], "worksheet": "u1_lesson2_live.html" }, "WS-U1L2-exitTicket": { "elements": [ { "id": "individuals", "description": "Identifies the individuals as the 25 students who play a spring sport" }, { "id": "two-variables", "description": "Names two actual variables from the spreadsheet in context, such as sport played, resting heart rate, or number of practices missed" }, { "id": "sport-played-categorical", "description": "Classifies sport played as categorical because it uses category names or labels" }, { "id": "heart-rate-quantitative", "description": "Classifies resting heart rate as quantitative because it is a numerical measured quantity in beats per minute" }, { "id": "id-not-variable", "description": "Explains that student ID is an identifier used to label each student, not a characteristic that varies in the lesson sense" } ], "commonMistakes": [ "Calling student ID a variable just because it is listed in a column", "Treating sport played as quantitative", "Failing to state that resting heart rate is measured in beats per minute", "Not identifying the students as the individuals" ], "worksheet": "u1_lesson2_live.html" }, "WS-U1L3-reflect1": { "elements": [ { "id": "frequency-counts", "description": "Explains that a frequency table gives the number of individuals or cases in each category" }, { "id": "relative-frequency-proportion", "description": "Explains that a relative frequency table gives the proportion or percent in each category" }, { "id": "superpower-context", "description": "Uses the superpower survey context or a correct category example such as fly, telepathy, or invisibility" }, { "id": "divide-by-total", "description": "May mention that relative frequencies are found by dividing by the total of 50 students" } ], "commonMistakes": [ "Saying both tables only show counts", "Describing relative frequency without mentioning proportion or percent", "Ignoring the superpower survey context entirely", "Confusing a category name with a relative frequency" ], "worksheet": "u1_lesson3_live.html" }, "WS-U1L3-reflect2": { "elements": [ { "id": "use-counts-or-percentages", "description": "Explains that counts, proportions, or percentages from the table are used to test a claim" }, { "id": "specific-example", "description": "Uses a correct example from the lesson, such as fly plus telepathy equaling 50 percent or can't say drug unfamiliar being 8.5 percent" }, { "id": "claim-judgment", "description": "States whether the claim is supported or not supported and why the numbers justify that decision" }, { "id": "majority-threshold", "description": "May mention a benchmark such as majority meaning more than 50 percent or over one-third meaning above about 33 percent" } ], "commonMistakes": [ "Giving an opinion instead of using table values", "Using an example that does not match the lesson numbers", "Failing to say whether the claim is supported or not supported", "Ignoring the need to justify the claim in context" ], "worksheet": "u1_lesson3_live.html" }, "WS-U1L3-exitTicket": { "elements": [ { "id": "frequency-definition", "description": "Explains that a frequency table records the number of cases in each category" }, { "id": "chips-relative-frequency", "description": "Finds chips as 18 out of 40, which is 0.45 or 45 percent" }, { "id": "yogurt-relative-frequency", "description": "Finds yogurt as 4 out of 40, which is 0.10 or 10 percent" }, { "id": "no-majority-chips", "description": "States that the claim of a majority preferring chips is not supported because 45 percent is less than 50 percent" }, { "id": "context-justification", "description": "Uses counts or relative frequencies to justify the conclusion in the lunch-side context" } ], "commonMistakes": [ "Treating 18 students as a majority without comparing to the total of 40", "Giving counts instead of relative frequencies for chips or yogurt", "Forgetting that a majority means more than 50 percent", "Explaining the numbers without tying them to the lunch-side context" ], "worksheet": "u1_lesson3_live.html" }, "WS-U1L4-reflect1": { "elements": [ { "id": "frequency-counts", "description": "Explains that a frequency bar chart shows counts or frequencies for each category" }, { "id": "relative-frequency-proportions", "description": "Explains that a relative frequency bar chart shows proportions or percents for each category" }, { "id": "same-superpower-data", "description": "States that both graphs display the same superpower categories or same distribution" }, { "id": "superpower-example", "description": "Uses the superpower context or a correct category example such as telepathy, fly, or freeze time" }, { "id": "divide-by-total", "description": "May mention that relative frequencies come from dividing by the total of 50 students" } ], "commonMistakes": [ "Saying both graphs only show counts", "Describing relative frequency without mentioning proportion or percent", "Ignoring that both graphs come from the same superpower data set", "Giving graphing details without explaining the difference in what the heights represent" ], "worksheet": "u1_lesson4_live.html" }, "WS-U1L4-reflect2": { "elements": [ { "id": "different-group-sizes", "description": "Explains that the male and female groups have different sizes, 80 males and 125 females" }, { "id": "use-proportions", "description": "Explains that relative frequencies or proportions should be used instead of just raw counts" }, { "id": "lesson-example", "description": "Uses a correct lesson example such as freeze time being 28 and 28 but more popular among males, or telepathy being less than 10 percent of males and almost 25 percent of females" }, { "id": "comparison-conclusion", "description": "Explains how the relative frequencies change or clarify the comparison" }, { "id": "graph-or-table-link", "description": "May mention that relative frequency tables or side-by-side relative frequency bar charts make the comparison easier" } ], "commonMistakes": [ "Comparing counts only and ignoring the different totals", "Giving an example whose numbers do not support the comparison", "Saying equal counts mean equal popularity", "Not explaining how the proportion changes the conclusion" ], "worksheet": "u1_lesson4_live.html" }, "WS-U1L4-exitTicket": { "elements": [ { "id": "different-grade-sizes", "description": "Explains that relative frequencies are better because the grades are different sizes, 30 ninth graders and 20 tenth graders" }, { "id": "ninth-jersey-frequency", "description": "Finds Jersey Day for 9th grade as 6 out of 30, which is 0.20 or 20 percent" }, { "id": "tenth-jersey-frequency", "description": "Finds Jersey Day for 10th grade as 8 out of 20, which is 0.40 or 40 percent" }, { "id": "claim-not-supported", "description": "States that the claim Jersey Day was equally popular is not supported" }, { "id": "context-justification", "description": "Uses the spirit day context and counts or relative frequencies to justify that Jersey Day was more popular among 10th graders" } ], "commonMistakes": [ "Comparing the Jersey Day counts without noticing the grades have different totals", "Giving 6 and 8 instead of relative frequencies or percents", "Saying the claim is supported because both groups liked Jersey Day", "Forgetting to tie the conclusion back to the spirit day context" ], "worksheet": "u1_lesson4_live.html" }, "WS-U1L5-reflect1": { "elements": [ { "id": "discrete-definition", "description": "Explains that a discrete variable has countable values or has gaps between possible values" }, { "id": "continuous-definition", "description": "Explains that a continuous variable has infinitely many possible values or no gaps between values" }, { "id": "counting-measuring", "description": "Connects discrete to counting and continuous to measuring" }, { "id": "lesson-example", "description": "Uses a correct lesson example such as number of siblings for discrete, height for continuous, or Flint water measurements as continuous" }, { "id": "rounded-values-note", "description": "May mention that rounded data can still come from a continuous variable when the original values included decimals" } ], "commonMistakes": [ "Saying discrete just means small numbers", "Calling any decimal value discrete", "Ignoring the idea of gaps between possible values", "Giving an example that does not match the definition" ], "worksheet": "u1_lesson5_live.html" }, "WS-U1L5-reflect2": { "elements": [ { "id": "large-data-set", "description": "Explains that a histogram is more useful for a large data set" }, { "id": "shows-individual-values", "description": "Explains that a dotplot or stem-and-leaf plot shows each individual value" }, { "id": "histogram-intervals", "description": "Explains that a histogram groups data into intervals or bins" }, { "id": "lose-exact-values", "description": "States that a histogram does not show each exact individual value" }, { "id": "shape-idea", "description": "May mention that both types of displays still help show the shape of the distribution" } ], "commonMistakes": [ "Saying histograms show every exact data value", "Ignoring the role of intervals or bins in a histogram", "Not mentioning that large data sets make histograms more useful", "Comparing graph appearance without explaining what information is lost" ], "worksheet": "u1_lesson5_live.html" }, "WS-U1L5-exitTicket": { "elements": [ { "id": "discrete-classification", "description": "Classifies number of siblings as a discrete variable" }, { "id": "countable-gaps-explanation", "description": "Explains that the values are countable whole-number counts with gaps, so values like 2.5 siblings are not possible" }, { "id": "best-display-choice", "description": "Chooses a dotplot or stem-and-leaf plot as the better display for this small data set" }, { "id": "chosen-display-advantage", "description": "Explains that the chosen display shows each individual value and/or makes the shape easy to see" }, { "id": "histogram-limitation", "description": "Explains that a histogram would group the data into intervals and hide the exact values for this small set" } ], "commonMistakes": [ "Calling number of siblings continuous because the values are numbers", "Choosing a histogram without explaining the loss of exact values", "Forgetting that sibling counts cannot take values like 2.5", "Describing the graph without tying it to this small data set" ], "worksheet": "u1_lesson5_live.html" }, "WS-U1L6-reflect1": { "elements": [ { "id": "shape-description", "description": "Describes the shape as unimodal and skewed right or gives equivalent correct shape language" }, { "id": "center-description", "description": "Identifies a typical value around 3 to 4 parts per billion" }, { "id": "variability-description", "description": "Describes the spread or variability as going from 0 to 104 parts per billion or gives an equivalent spread statement" }, { "id": "unusual-features", "description": "Mentions unusual features such as the cluster from 0 to 10, the gap between 42 and 104, or high outliers" }, { "id": "context", "description": "Uses context by referring to Flint lead levels or Flint water samples" }, { "id": "single-peak-detail", "description": "May mention that the single peak is at or near 0" } ], "commonMistakes": [ "Describing only the shape and forgetting center or spread", "Leaving out context and writing only generic graph vocabulary", "Calling the Flint distribution symmetric", "Ignoring the unusual features such as the gap or outliers" ], "worksheet": "u1_lesson6_live.html" }, "WS-U1L6-reflect2": { "elements": [ { "id": "two-unusual-features", "description": "Correctly explains any two of these lesson terms: outlier, gap, and cluster" }, { "id": "flint-connection", "description": "Connects at least one unusual feature to the Flint data, such as the cluster from 0 to 10, the gap from 42 to 104, or the high outlier" }, { "id": "distribution-description-role", "description": "States that unusual features are one of the important characteristics in a full distribution description" }, { "id": "gaps-clusters-together", "description": "May mention that gaps and clusters often appear together" } ], "commonMistakes": [ "Treating a gap as a low bar rather than a region with no data", "Calling any large value an outlier without explaining why it does not fit", "Mentioning clusters without explaining that they are concentrations of values", "Forgetting to connect the explanation to the Flint example" ], "worksheet": "u1_lesson6_live.html" }, "WS-U1L6-exitTicket": { "elements": [ { "id": "shape", "description": "Describes the distribution as skewed right or right-skewed and also identifies one main peak or says unimodal" }, { "id": "center", "description": "Gives a typical value around 4 minutes or an equivalent center statement in context" }, { "id": "variability", "description": "Describes the spread as running from 1 to 18 minutes or gives an equivalent variability statement in context" }, { "id": "unusual-features", "description": "Identifies unusual features such as a cluster from about 1 to 6, a gap before 18, or the value 18 as an outlier" }, { "id": "context", "description": "Uses context by referring to minutes waiting for the late bus or waiting times for students" } ], "commonMistakes": [ "Giving only one shape word when the prompt asks for at least two appropriate terms", "Listing the minimum and maximum without describing center", "Ignoring the high value of 18 as an unusual feature", "Describing the numbers without mentioning bus wait times" ], "worksheet": "u1_lesson6_live.html" }, "WS-U1L7-reflect1": { "elements": [ { "id": "mean-interpretation", "description": "Explains that the mean describes the average Flint lead level and may reference 7.31 parts per billion" }, { "id": "median-interpretation", "description": "Explains that the median is the middle ordered value and may note that at least half the Flint lead levels are at or below 3 parts per billion and at least half are at or above 3 parts per billion" }, { "id": "iqr-interpretation", "description": "Explains that the IQR describes the spread of the middle 50% of Flint lead levels and may reference IQR 5, Q1 2, or Q3 7" }, { "id": "context", "description": "Uses context by referring to Flint lead levels or Flint water samples" }, { "id": "center-vs-variability", "description": "Distinguishes that mean and median describe center while IQR describes variability" }, { "id": "numerical-detail", "description": "Includes at least one correct numerical value from the lesson, such as mean 7.31, median 3, IQR 5, Q1 2, or Q3 7" } ], "commonMistakes": [ "Listing the statistics without explaining what they mean", "Confusing the mean with the median", "Treating the IQR as the full range of the data", "Describing a statistic without saying what it tells you about the data" ], "worksheet": "u1_lesson7_live.html" }, "WS-U1L7-reflect2": { "elements": [ { "id": "shape-or-outliers", "description": "States that the Flint distribution is skewed right or has large high outliers" }, { "id": "median-iqr-choice", "description": "States that median and IQR are the best measures for the Flint distribution" }, { "id": "resistant-language", "description": "Explains that median and IQR are resistant measures" }, { "id": "nonresistant-language", "description": "Explains that mean and standard deviation are nonresistant and are pulled by outliers" }, { "id": "context", "description": "Uses context by referring to Flint lead levels or Flint water samples" }, { "id": "removal-example", "description": "May mention that removing 104 changed the mean and standard deviation a lot but changed the median and IQR very little" } ], "commonMistakes": [ "Saying median and IQR are always better for every distribution", "Mentioning outliers without explaining resistance", "Choosing mean and standard deviation even after noting the skew or outliers", "Saying which measures to use without explaining why outliers affect the other pair" ], "worksheet": "u1_lesson7_live.html" }, "WS-U1L7-exitTicket": { "elements": [ { "id": "best-center", "description": "Chooses the median as the best measure of center and may identify it as 5 minutes" }, { "id": "best-variability", "description": "Chooses the IQR as the best measure of variability and may identify it as 2 minutes or note that the middle 50% run from 4 to 6 minutes" }, { "id": "outlier-identification", "description": "Identifies 22 minutes as an outlier using a valid lesson method such as the 1.5 IQR rule or the two-standard-deviation rule" }, { "id": "resistant-justification", "description": "Explains that median and IQR are resistant and therefore appropriate because of the skew or outlier" }, { "id": "context", "description": "Uses context by referring to minutes waiting for rides after practice or student waiting times" }, { "id": "skew-language", "description": "May describe the distribution as right-skewed or skewed right" } ], "commonMistakes": [ "Choosing mean and standard deviation even though 22 creates a strong outlier", "Calling 22 unusual without connecting it to an outlier rule", "Giving the range instead of the IQR as the preferred variability measure", "Calling 22 an outlier without showing the rule that makes it one" ], "worksheet": "u1_lesson7_live.html" }, "WS-U1L8-reflect1": { "elements": [ { "id": "five-number-summary", "description": "Identifies that the five-number summary is the minimum, Q1, median, Q3, and maximum" }, { "id": "boxplot-feature", "description": "Correctly explains at least one requested box-plot feature: the box runs from Q1 to Q3 with the median inside it, whiskers extend to the most extreme non-outliers, or outliers are plotted separately" }, { "id": "flint-context", "description": "Uses context by referring to Flint lead levels or Flint water samples" }, { "id": "numerical-details", "description": "Includes at least two correct numerical values such as 0, 2, 3, 7, 13, or 104" }, { "id": "quartile-meaning", "description": "May explain that the box plot splits the data into quartiles or that each section contains 25% of the data" } ], "commonMistakes": [ "Listing numbers without explaining what part of the box plot they represent", "Saying the whiskers always go to the minimum and maximum even when outliers exist", "Forgetting that the median is the line inside the box", "Listing five numbers without saying which part of the box plot each one becomes" ], "worksheet": "u1_lesson8_live.html" }, "WS-U1L8-reflect2": { "elements": [ { "id": "skewed-right", "description": "States that in a skewed-right distribution the mean is usually greater than the median" }, { "id": "skewed-left", "description": "States that in a skewed-left distribution the mean is usually less than the median" }, { "id": "symmetric", "description": "States that in a relatively symmetric distribution the mean and median are about equal or very close" }, { "id": "outlier-effect", "description": "Explains that outliers or a long tail pull the mean more than the median because the mean is nonresistant" }, { "id": "flint-connection", "description": "Connects the rule to the Flint box plot or Flint lead levels by noting that the Flint distribution is skewed right and has mean 7.31 above median 3" }, { "id": "median-resistant", "description": "May mention that the median is resistant compared with the mean" } ], "commonMistakes": [ "Reversing the relationships for skewed-right and skewed-left distributions", "Saying the mean and median are always exactly equal for symmetric distributions", "Ignoring the way outliers pull the mean more than the median", "Stating the rules without saying why the tail moves the mean" ], "worksheet": "u1_lesson8_live.html" }, "WS-U1L8-exitTicket": { "elements": [ { "id": "box-and-median", "description": "Explains that the box would run from Q1 = 18 to Q3 = 31 with a median line at 22" }, { "id": "whiskers-and-outlier", "description": "Explains that the whiskers would extend to 12 and 38 and that 58 would be shown as a separate outlier point" }, { "id": "middle-50", "description": "States that the middle 50% of homework times are between 18 and 31 minutes" }, { "id": "mean-vs-median", "description": "Predicts that the mean is greater than the median because the high outlier suggests a skewed-right distribution" }, { "id": "context", "description": "Uses context by referring to homework times or minutes spent on homework" }, { "id": "quartile-language", "description": "May mention quartiles or the five-number summary by name" } ], "commonMistakes": [ "Sending the right whisker to 58 even though 58 is identified as an outlier", "Saying the middle 50% runs from the minimum to the maximum", "Predicting the mean is less than the median despite the high outlier on the right", "Sending the whisker to the outlier instead of the largest non-outlier" ], "worksheet": "u1_lesson8_live.html" }, "WS-U1L9-reflect1": { "elements": [ { "id": "shape-comparison", "description": "States that the salary distributions appear similar or fairly symmetric for both corporations" }, { "id": "center-comparison", "description": "States that the medians are approximately the same for corporations A and B" }, { "id": "variability-comparison", "description": "Explains that corporation A has greater variability than corporation B, such as a larger range or IQR" }, { "id": "unusual-features", "description": "Explains that corporation A has two high outliers while corporation B has no outliers" }, { "id": "comparative-language", "description": "Uses comparative words such as similar, same, greater, or less" }, { "id": "context", "description": "Uses context by referring to yearly salaries or corporations A and B" }, { "id": "boxplot-caution", "description": "May note that box plots only show that the distributions appear symmetric because values inside quartiles are hidden" } ], "commonMistakes": [ "Describing each corporation separately without directly comparing them", "Leaving out one of the four characteristics such as variability or unusual features", "Forgetting that corporation A has greater variability than corporation B", "Ignoring context and writing only generic statements about box plots" ], "worksheet": "u1_lesson9_live.html" }, "WS-U1L9-reflect2": { "elements": [ { "id": "reason-for-a", "description": "Gives a valid reason to choose corporation A, such as the possibility of earning a higher salary because some salaries at A exceed the maximum at B" }, { "id": "a-feature", "description": "Supports the reason for corporation A with a box-plot feature such as the high outliers or the fact that at least 3 of 30 salaries at A are above the maximum at B" }, { "id": "reason-for-b", "description": "Gives a valid reason to choose corporation B, such as the higher minimum salary or less risk of never getting a raise" }, { "id": "b-feature", "description": "Supports the reason for corporation B with a box-plot feature such as the higher minimum or the absence of salaries still at the starting value" }, { "id": "context", "description": "Uses context by referring to salary, raises, or corporations A and B" }, { "id": "based-on-box-plots", "description": "May explicitly say that the reasons come from features visible on the box plots" } ], "commonMistakes": [ "Giving a preference without referring to any feature of the box plots", "Mixing up which corporation has the higher minimum salary", "Ignoring the high salaries and outliers at corporation A", "Failing to explain why the box-plot feature would matter to someone choosing a job" ], "worksheet": "u1_lesson9_live.html" }, "WS-U1L9-exitTicket": { "elements": [ { "id": "center", "description": "States that the centers are about the same because both medians are 60 minutes" }, { "id": "variability", "description": "Explains that Study Hall A has greater variability than Study Hall B, such as a larger IQR or range" }, { "id": "unusual-features", "description": "Notes that Study Hall A has a high outlier at 150 while Study Hall B has no outliers" }, { "id": "shape", "description": "States that Study Hall A appears skewed right and that Study Hall B appears fairly symmetric or less skewed based on the box plots" }, { "id": "comparative-context", "description": "Uses comparative words and refers to nightly homework times or study halls in context" }, { "id": "numerical-support", "description": "May include supporting values such as Q1 and Q3 or describe the middle 50% for each study hall" } ], "commonMistakes": [ "Saying the distributions have different centers even though both medians are 60", "Missing that Study Hall A has much greater variability than Study Hall B", "Ignoring the high outlier at 150 in Study Hall A", "Forgetting to use comparative words or to mention homework times in context" ], "worksheet": "u1_lesson9_live.html" }, "WS-U2L1-reflect1": { "elements": [ { "id": "identify-type-first", "description": "Explains that you first identify whether the variables are categorical or quantitative" }, { "id": "categorical-tools", "description": "States that two categorical variables use displays such as segmented bar graphs or mosaic plots and numerical summaries such as conditional relative frequencies from a two-way table" }, { "id": "quantitative-tools", "description": "States that two quantitative variables use a scatter plot and numerical summaries such as correlation, line of best fit, or coefficient of determination" }, { "id": "match-method-to-data", "description": "May explain that the variable type tells you which method fits the data and helps you judge whether a relationship appears to exist" } ], "commonMistakes": [ "Mixing up categorical and quantitative variables", "Saying to use a scatter plot for categorical variables", "Leaving out numerical summaries entirely", "Naming tools without explaining that variable type determines the choice" ], "worksheet": "u2_lesson1_live.html" }, "WS-U2L1-reflect2": { "elements": [ { "id": "random-variation", "description": "Explains that a visible pattern or association could happen because of random variation or chance" }, { "id": "not-automatic-proof", "description": "Explains that seeing a pattern does not automatically prove a real relationship" }, { "id": "need-analysis", "description": "Explains that we use graphs and numerical summaries to investigate the relationship more carefully before drawing conclusions" }, { "id": "uncertain-conclusions", "description": "May mention that conclusions in statistics are uncertain because variation may be random or not" } ], "commonMistakes": [ "Claiming that any pattern in a graph proves the variables are related", "Ignoring random variation or chance", "Talking only about strong relationships without mentioning uncertainty", "Confusing an apparent pattern with a guaranteed cause-and-effect conclusion" ], "worksheet": "u2_lesson1_live.html" }, "WS-U2L1-exitTicket": { "elements": [ { "id": "study-a-types", "description": "States that in Study A both variables are categorical" }, { "id": "study-a-tools", "description": "Names an appropriate graph for Study A such as a segmented bar graph or mosaic plot and a numerical summary such as conditional relative frequencies from a two-way table" }, { "id": "study-b-types", "description": "States that in Study B both variables are quantitative" }, { "id": "study-b-tools", "description": "Names an appropriate graph for Study B as a scatter plot and names two numerical summaries such as correlation, line of best fit, or coefficient of determination" }, { "id": "pattern-study-a", "description": "Describes a possible pattern for Study A such as noticeably different conditional distributions across grade levels or activity groups" }, { "id": "pattern-study-b", "description": "Describes a possible pattern for Study B such as points trending upward or downward in the scatter plot" }, { "id": "random-variation", "description": "Explains that an apparent pattern in either study could still be caused by random variation or chance and therefore does not automatically prove a real relationship" }, { "id": "careful-conclusion", "description": "May mention that we use the chosen displays and summaries to investigate before making a conclusion" } ], "commonMistakes": [ "Treating grade level and after-school activity as quantitative variables", "Using a scatter plot for Study A or a bar graph for Study B", "Listing only graphs and no numerical summaries", "Forgetting to explain that an apparent pattern might be due to chance", "Describing a pattern without connecting it to whether the variables might be related" ], "worksheet": "u2_lesson1_live.html" }, "WS-U2L2-reflect1": { "elements": [ { "id": "within-group-percent", "description": "Explains that dividing by the column total converts counts to percents or relative frequencies within each age group" }, { "id": "fair-comparison", "description": "Explains that this makes the comparison fair because the age groups can have different sizes" }, { "id": "compare-distributions", "description": "Explains that using percents lets you compare the distributions across groups rather than raw counts" }, { "id": "sum-to-100", "description": "May mention that the percents within a group add to 100%" } ], "commonMistakes": [ "Saying the column total is used only because the calculator requires it", "Comparing raw counts instead of percents", "Not mentioning that the groups can be different sizes", "Confusing a column relative frequency with a joint relative frequency" ], "worksheet": "u2_lesson2_live.html" }, "WS-U2L2-reflect2": { "elements": [ { "id": "compare-distributions", "description": "Explains that you compare the distributions for the groups" }, { "id": "not-same-means-association", "description": "States that if the distributions are not the same, the variables are associated" }, { "id": "predictive-language", "description": "May explain that if knowing one variable helps predict the other, that indicates association" }, { "id": "mosaic-widths", "description": "May mention that a mosaic plot also shows group sizes through bar widths" } ], "commonMistakes": [ "Claiming there is association whenever the groups have different sizes", "Looking only at raw counts instead of distributions", "Saying the bars must be exactly equal for the variables to be associated", "Describing correlation or slope instead of association between categorical variables" ], "worksheet": "u2_lesson2_live.html" }, "WS-U2L2-exitTicket": { "elements": [ { "id": "identify-variables", "description": "Identifies the variables as grade level and school sport participation and recognizes that both are categorical" }, { "id": "two-way-table", "description": "Explains that it is a two-way table because it summarizes counts for two categorical variables together" }, { "id": "ninth-grade-distribution", "description": "Gives the 9th-grade conditional distribution as 60% plays a sport and 40% does not" }, { "id": "eleventh-grade-distribution", "description": "Gives the 11th-grade conditional distribution as 28% plays a sport and 72% does not, or equivalent approximations" }, { "id": "association-conclusion", "description": "Explains that the segmented bars would look different because the distributions are different, so the variables are associated" }, { "id": "mosaic-advantage", "description": "States that a mosaic plot also shows the different group sizes through bar widths" } ], "commonMistakes": [ "Using raw counts instead of conditional percents", "Dividing by the grand total instead of the grade-level total", "Saying the variables are associated only because the totals 30, 40, and 50 are different", "Forgetting that a mosaic plot shows group size with bar width" ], "worksheet": "u2_lesson2_live.html" }, "WS-U2L3-reflect1": { "elements": [ { "id": "marginal-definition", "description": "Explains that a marginal relative frequency uses a row total or column total divided by the table total" }, { "id": "conditional-definition", "description": "Explains that a conditional relative frequency uses a cell count divided by a specific row total or column total" }, { "id": "overall-vs-within-group", "description": "Explains that marginal describes an overall proportion while conditional describes a proportion within a group" }, { "id": "denominator-language", "description": "May explicitly mention that the denominators are different" } ], "commonMistakes": [ "Using the table total for both statistics", "Saying marginal and conditional relative frequencies are the same thing", "Describing only the numerator and not the denominator", "Not explaining that conditional relative frequency is within a chosen row or column" ], "worksheet": "u2_lesson3_live.html" }, "WS-U2L3-reflect2": { "elements": [ { "id": "within-group-distributions", "description": "Explains that conditional relative frequencies show the distribution within each group" }, { "id": "compare-groups", "description": "Explains that these within-group distributions can be compared across groups" }, { "id": "different-means-association", "description": "States that if the conditional distributions are not the same, the variables are associated" }, { "id": "supports-graphical-conclusion", "description": "May mention that these summary statistics support the conclusion seen in a segmented bar graph" } ], "commonMistakes": [ "Using marginal relative frequencies to decide association", "Comparing raw counts instead of within-group percentages", "Saying the variables are associated only because the groups have different totals", "Not stating that the conditional distributions must be compared across groups" ], "worksheet": "u2_lesson3_live.html" }, "WS-U2L3-exitTicket": { "elements": [ { "id": "identify-variables", "description": "Identifies the variables as grade level and preferred study style and recognizes that both are categorical" }, { "id": "two-way-table", "description": "Explains that it is a two-way table because it summarizes counts for two categorical variables together" }, { "id": "joint-relative-frequency", "description": "Gives the joint relative frequency for 11th grade and prefers group as 30% or 0.30" }, { "id": "marginal-relative-frequency", "description": "Gives the marginal relative frequency for 10th grade as 30% or 0.30" }, { "id": "conditional-ninth", "description": "Gives the 9th-grade conditional distribution as 60% alone and 40% group" }, { "id": "conditional-eleventh", "description": "Gives the 11th-grade conditional distribution as 25% alone and 75% group" }, { "id": "association-conclusion", "description": "Explains that the variables are associated because the conditional distributions are different across grade levels" } ], "commonMistakes": [ "Using the wrong denominator for the joint relative frequency", "Using the grand total instead of the grade-level total for the conditional distributions", "Confusing the marginal relative frequency with a conditional relative frequency", "Concluding association from totals alone instead of comparing conditional distributions" ], "worksheet": "u2_lesson3_live.html" }, "WS-U2L4-reflect1": { "elements": [ { "id": "explanatory-role", "description": "Explains that the explanatory variable is used to explain or predict the response variable" }, { "id": "x-axis", "description": "States that the explanatory variable goes on the x-axis" }, { "id": "y-axis", "description": "States that the response variable goes on the y-axis" }, { "id": "paired-values", "description": "Explains that each point represents one observation with both an x-value and a y-value" } ], "commonMistakes": [ "Putting the response variable on the x-axis", "Saying the explanatory and response variables can go on either axis with no reason", "Defining the variables without explaining which axis each belongs on", "Not connecting explanatory variable to prediction or explanation" ], "worksheet": "u2_lesson4_live.html" }, "WS-U2L4-reflect2": { "elements": [ { "id": "direction-only", "description": "Explains that positive only describes the direction of the association" }, { "id": "form", "description": "States that a complete description should also include the form, such as linear or nonlinear" }, { "id": "strength", "description": "States that a complete description should also include the strength of the association" }, { "id": "features-context", "description": "States that a complete description should mention unusual features and the context of the variables" } ], "commonMistakes": [ "Treating positive as a complete description by itself", "Listing only one extra feature, such as linear, and omitting strength or unusual features", "Describing the graph without mentioning context", "Confusing positive with strong" ], "worksheet": "u2_lesson4_live.html" }, "WS-U2L4-exitTicket": { "elements": [ { "id": "identify-variables", "description": "Identifies the variables as hours studied and quiz score and recognizes that both are quantitative" }, { "id": "explanatory-response", "description": "States that hours studied is the explanatory variable and quiz score is the response variable" }, { "id": "scatterplot-appropriate", "description": "Explains that a scatterplot is appropriate because each student has two quantitative values" }, { "id": "positive-direction", "description": "Describes the likely direction as positive, meaning higher study time tends to go with higher quiz scores" }, { "id": "linear-form", "description": "Describes the likely form as approximately linear" }, { "id": "strong-strength", "description": "Describes the likely strength as strong or notes that the points would be fairly close to a line" }, { "id": "construction-features", "description": "Lists at least two correct scatterplot features, such as a title, labeled axes, and scale with tick marks" }, { "id": "no-obvious-unusual-features", "description": "May mention that there is no obvious unusual feature in this small data set" } ], "commonMistakes": [ "Reversing the explanatory and response variables", "Saying a scatterplot is used because the variables are categorical", "Describing only direction and not form or strength", "Listing graph features without explaining the relationship in context", "Forgetting to mention labeled axes or tick-mark scales" ], "worksheet": "u2_lesson4_live.html" }, "WS-U2L5-reflect1": { "elements": [ { "id": "sign-direction", "description": "Explains that the sign of r tells the direction of the relationship" }, { "id": "positive-negative", "description": "States that positive means y tends to increase as x increases, while negative means y tends to decrease as x increases" }, { "id": "magnitude-strength", "description": "Explains that the magnitude or distance from 0 tells the strength of the linear relationship" }, { "id": "close-values", "description": "States that values closer to 0 are weaker and values closer to 1 or -1 are stronger" }, { "id": "linear-focus", "description": "Recognizes that r is describing a linear relationship" } ], "commonMistakes": [ "Confusing sign with strength", "Saying positive means the relationship is strong", "Ignoring what values near 0 mean", "Forgetting that r describes a linear relationship" ], "worksheet": "u2_lesson5_live.html" }, "WS-U2L5-reflect2": { "elements": [ { "id": "no-causation", "description": "States that correlation does not imply causation" }, { "id": "no-form", "description": "Explains that r alone does not tell the form of the relationship" }, { "id": "no-unusual-features", "description": "Explains that r alone does not show unusual features such as outliers or clusters" }, { "id": "need-graph-context", "description": "States that you need to look at the scatterplot or consider context to make those judgments" }, { "id": "attendance-example", "description": "May mention that attendance rose while test scores stayed flat despite a strong positive correlation" } ], "commonMistakes": [ "Using strong correlation as proof of cause and effect", "Saying r alone can identify outliers or clusters", "Treating r as enough to decide whether a linear model is appropriate", "Ignoring the need to inspect the scatterplot" ], "worksheet": "u2_lesson5_live.html" }, "WS-U2L5-exitTicket": { "elements": [ { "id": "positive-direction", "description": "Interprets the sign as positive, meaning more tutoring tends to go with higher quiz scores" }, { "id": "strong-strength", "description": "Interprets the strength as strong because 0.89 is close to 1" }, { "id": "r-alone-limit", "description": "Explains that r alone does not tell form or unusual features and that you need the scatterplot for that" }, { "id": "no-causation", "description": "Explains that the correlation does not prove that tutoring caused the higher quiz scores" }, { "id": "technology-use", "description": "States that technology is usually used because the formula is cumbersome or because it quickly computes r accurately" }, { "id": "unit-free-range", "description": "May mention that r is unit-free and between -1 and 1" } ], "commonMistakes": [ "Saying 0.89 proves causation", "Describing only direction and not strength", "Claiming r alone shows the exact shape or unusual features", "Forgetting to explain why technology is commonly used" ], "worksheet": "u2_lesson5_live.html" }, "WS-U2L6-reflect1": { "elements": [ { "id": "plug-in-x", "description": "Explains that you substitute the explanatory-variable value for x in the regression equation" }, { "id": "slope-and-intercept", "description": "Explains that you multiply the slope by x and combine it with the y-intercept to get ŷ" }, { "id": "predicted-response", "description": "States that ŷ is the predicted response rather than an exact observed value" }, { "id": "decimal-possible", "description": "Explains that the result can be a decimal because it is a model prediction, not an actual count or exact data value" }, { "id": "variability-idea", "description": "May mention that data have variability, so the model is estimating rather than giving exact values" } ], "commonMistakes": [ "Mixing up the slope and y-intercept", "Treating ŷ as an actual observed value", "Forgetting to explain what x represents", "Saying the answer cannot be a decimal" ], "worksheet": "u2_lesson6_live.html" }, "WS-U2L6-reflect2": { "elements": [ { "id": "define-extrapolation", "description": "Defines extrapolation as predicting outside the interval of observed x-values" }, { "id": "trend-may-not-continue", "description": "Explains that extrapolation is dangerous because the existing trend may not continue" }, { "id": "less-reliable", "description": "States that predictions made by extrapolation are less reliable" }, { "id": "compare-to-interval", "description": "Explains that you decide by comparing the chosen x-value to the interval of x-values used to build the model" }, { "id": "example-link", "description": "May mention the 2048 example or the swine prediction at x = 0.2" } ], "commonMistakes": [ "Saying any regression prediction is reliable", "Ignoring the interval of observed x-values", "Treating extrapolation as the same as interpolation", "Failing to explain why the trend may change" ], "worksheet": "u2_lesson6_live.html" }, "WS-U2L6-exitTicket": { "elements": [ { "id": "variables-context", "description": "Identifies x as hours of after-school tutoring and ŷ as the predicted quiz score" }, { "id": "slope-context", "description": "Explains that the slope 1.8 means the predicted quiz score increases by 1.8 points for each additional hour of tutoring" }, { "id": "intercept-context", "description": "Explains that the y-intercept 61.4 is the predicted quiz score for a student with 0 hours of tutoring" }, { "id": "correct-prediction", "description": "Calculates the predicted quiz score for 8 hours as 75.8" }, { "id": "prediction-not-exact", "description": "Explains that the model gives a predicted score, so it does not have to match a student’s exact observed score" }, { "id": "extrapolation-unreliable", "description": "States that x = 15 is outside the observed interval from 2 to 10, so it is extrapolation and is not reliable or is less reliable because the trend may not continue" }, { "id": "further-less-reliable", "description": "May note that predictions become less reliable the farther they are beyond the observed x-range" } ], "commonMistakes": [ "Treating 75.8 as an exact quiz score", "Describing 15 hours as reliable even though it is outside the observed interval", "Failing to interpret the slope and intercept in context", "Using the wrong x-value in the calculation" ], "worksheet": "u2_lesson6_live.html" }, "WS-U2L7-reflect1": { "elements": [ { "id": "actual-minus-predicted", "description": "States that a residual is the actual value minus the predicted value, or y - ŷ" }, { "id": "single-point-difference", "description": "Explains that a residual tells how far one observed point is from the model's prediction" }, { "id": "positive-underestimate", "description": "Explains that a positive residual means the actual value is greater than predicted, so the model underestimated" }, { "id": "negative-overestimate", "description": "Explains that a negative residual means the actual value is less than predicted, so the model overestimated" }, { "id": "context-example", "description": "May mention the example of 84 actual items, 51.4 predicted items, and a residual of 32.6" } ], "commonMistakes": [ "Reversing the subtraction and saying predicted minus actual", "Saying a positive residual means overestimation", "Saying a negative residual means underestimation", "Failing to explain that a residual is tied to one observed data point" ], "worksheet": "u2_lesson7_live.html" }, "WS-U2L7-reflect2": { "elements": [ { "id": "apparent-randomness", "description": "States that a good residual plot shows apparent randomness" }, { "id": "centered-at-zero", "description": "States that the residuals should be scattered around zero" }, { "id": "no-clear-pattern", "description": "Explains that no clear pattern is a good sign for a linear model" }, { "id": "pattern-warning", "description": "Explains that a visible pattern or curvature is a warning sign that the linear model may not be appropriate" }, { "id": "nonlinear-next-step", "description": "May mention considering a nonlinear model or another model when a pattern appears" } ], "commonMistakes": [ "Saying a patterned residual plot is a good sign", "Ignoring whether residuals are centered around zero", "Describing the original scatterplot instead of the residual plot", "Failing to connect patterns to poor linear fit" ], "worksheet": "u2_lesson7_live.html" }, "WS-U2L7-exitTicket": { "elements": [ { "id": "correct-residual", "description": "Calculates the residual as 84 - 51.4 = 32.6" }, { "id": "context-interpretation", "description": "Interprets the residual in context by saying the actual store had 32.6 more organic items than predicted" }, { "id": "underpredicted", "description": "States that the positive residual means the model underpredicted" }, { "id": "good-fit-randomness", "description": "Explains that random scatter around zero with no clear pattern suggests a linear model is appropriate" }, { "id": "bad-fit-pattern", "description": "Describes that a visible pattern such as curvature or an up-down-up shape would make a linear model questionable" }, { "id": "residual-plot-role", "description": "May mention that residual plots are used to investigate model fit or appropriateness" } ], "commonMistakes": [ "Computing predicted minus actual instead of actual minus predicted", "Calling the positive residual overprediction", "Saying a random residual plot is bad for a linear model", "Failing to give an example of a pattern that would make linear fit doubtful" ], "worksheet": "u2_lesson7_live.html" }, "WS-U2L8-reflect1": { "elements": [ { "id": "canceling-problem", "description": "Explains that raw residuals can cancel because positive and negative values offset each other" }, { "id": "least-squares-definition", "description": "States that the LSRL minimizes the sum of squared residuals" }, { "id": "mean-point-property", "description": "States that the LSRL contains the point (x-bar, y-bar)" }, { "id": "slope-formula-property", "description": "States that the slope can be calculated with b = r(s_y / s_x)" }, { "id": "numeric-example", "description": "May mention the example where r = 0.95, s_y = 6.08, s_x = 10.2, and b = 0.57" } ], "commonMistakes": [ "Saying the LSRL minimizes the sum of raw residuals", "Ignoring that positive and negative residuals can cancel", "Forgetting that the line contains the mean point", "Confusing the slope formula with the regression equation itself" ], "worksheet": "u2_lesson8_live.html" }, "WS-U2L8-reflect2": { "elements": [ { "id": "slope-change", "description": "Explains that the slope is the predicted change in y for every one-unit increase in x" }, { "id": "attendance-context", "description": "Interprets the example slope by saying each 1 percentage point increase in attendance predicts about 0.57 more questions correct" }, { "id": "intercept-definition", "description": "Explains that the y-intercept is the predicted y-value when x = 0" }, { "id": "intercept-not-meaningful", "description": "Explains that the y-intercept may not be meaningful when x = 0 is not reasonable in context or gives an illogical prediction" }, { "id": "example-illogical-value", "description": "May mention that 0% attendance gives a prediction of -7.69 questions correct, which is not logical" } ], "commonMistakes": [ "Treating slope as an exact rather than predicted change", "Forgetting that the y-intercept corresponds to x = 0", "Claiming the y-intercept is always meaningful", "Ignoring the context of attendance and exam questions" ], "worksheet": "u2_lesson8_live.html" }, "WS-U2L8-reflect3": { "elements": [ { "id": "r-squared-meaning", "description": "States that r squared is the proportion or percent of variation in the response variable explained by the explanatory variable in the model" }, { "id": "context-interpretation", "description": "Interprets the example by saying 90.3% of the variation in exam questions correct is explained by the linear relationship with attendance" }, { "id": "r-squared-equals-r-squared", "description": "States that r squared equals the correlation squared" }, { "id": "output-slope", "description": "Explains that the slope is the coefficient of the explanatory variable in computer output" }, { "id": "output-intercept", "description": "Explains that the y-intercept is the constant in computer output" }, { "id": "strength-note", "description": "May mention that larger r squared values indicate a stronger relationship" } ], "commonMistakes": [ "Saying r squared is the percent of points on the line", "Confusing r squared with the slope", "Reading the constant as the slope", "Failing to connect r squared to explained variation" ], "worksheet": "u2_lesson8_live.html" }, "WS-U2L8-exitTicket": { "elements": [ { "id": "squared-residuals", "description": "Explains that squared residuals are used so positive and negative errors do not cancel and the LSRL minimizes their sum" }, { "id": "slope-interpretation", "description": "Interprets 0.57 as the predicted average increase in questions correct for each 1 percentage point increase in attendance" }, { "id": "intercept-definition", "description": "States that -7.69 is the predicted y-value when attendance is 0%" }, { "id": "intercept-not-meaningful", "description": "Explains that the y-intercept is not meaningful because 0% attendance is not a reasonable context and the prediction is illogical" }, { "id": "r-squared-context", "description": "Interprets 0.903 as 90.3% of the variation in exam questions correct being explained by the linear relationship with attendance" }, { "id": "mean-point", "description": "Names the point (x-bar, y-bar) as a point that must lie on the LSRL" }, { "id": "slope-formula-note", "description": "May mention that the slope could also be found using b = r(s_y / s_x)" } ], "commonMistakes": [ "Saying residuals should just be added instead of squared", "Interpreting the slope without the idea of predicted change", "Treating the y-intercept as automatically meaningful", "Describing r squared without mentioning explained variation", "Forgetting that the line contains (x-bar, y-bar)" ], "worksheet": "u2_lesson8_live.html" }, "WS-U2L9-reflect1": { "elements": [ { "id": "high-leverage-definition", "description": "Explains that a high leverage point has an unusually large or small x-value and is far from x-bar" }, { "id": "outlier-definition", "description": "Explains that an outlier has a large residual and does not follow the general trend" }, { "id": "influential-definition", "description": "Explains that an influential point is one that substantially changes the relationship when removed" }, { "id": "high-leverage-effect", "description": "States that high leverage points often affect the slope and/or y-intercept" }, { "id": "outlier-effect", "description": "States that outliers often affect the correlation and may change r squared" }, { "id": "both-type", "description": "May mention that a point can be both high leverage and an outlier" } ], "commonMistakes": [ "Confusing a high leverage point with a point that simply has a large residual", "Calling every outlier influential without explaining the removal idea", "Saying high leverage points mainly change correlation instead of slope or intercept", "Ignoring that influential points are defined by what happens when the point is removed" ], "worksheet": "u2_lesson9_live.html" }, "WS-U2L9-reflect2": { "elements": [ { "id": "nonlinear-trigger", "description": "Explains that transformation should be considered when the scatterplot or residual plot suggests a nonlinear pattern" }, { "id": "purpose-of-transform", "description": "States that the goal is to make the relationship more linear so a linear model fits better" }, { "id": "log-skew", "description": "Explains that a log transformation can reduce right skew or make large values less extreme" }, { "id": "residual-evidence", "description": "States that the transformed model is better if the residual plot looks more random and centered around zero" }, { "id": "r-squared-evidence", "description": "States that a higher r squared or an r squared closer to 1 supports the transformed model" }, { "id": "numeric-example", "description": "May mention the example where r squared improved from 46.6% to 71.1%" } ], "commonMistakes": [ "Saying data should be transformed only because r squared is not perfect", "Ignoring the residual plot as evidence", "Forgetting that the purpose is to make the relationship more linear", "Describing log transformation without mentioning reduced skew or compressed high values" ], "worksheet": "u2_lesson9_live.html" }, "WS-U2L9-exitTicket": { "elements": [ { "id": "high-leverage-identification", "description": "Identifies the point far from x-bar as the high leverage point because it has an unusual x-value" }, { "id": "outlier-identification", "description": "Identifies the point with the very large residual near x-bar as the outlier because it does not follow the trend" }, { "id": "influential-definition", "description": "Explains that a point is influential if removing it substantially changes the relationship" }, { "id": "different-effects", "description": "Explains that high leverage points often change slope and/or y-intercept, while outliers often change correlation and possibly r squared" }, { "id": "log-transform-purpose", "description": "Explains that a log transformation can reduce right skew and make the relationship more linear by making large income values less extreme" }, { "id": "better-fit-evidence", "description": "Explains that the transformed model is more appropriate because the residuals are more random around zero and r squared increased from 46.6% to 71.1%" }, { "id": "both-point-type", "description": "May mention that a point can be both high leverage and an outlier" } ], "commonMistakes": [ "Calling the point near x-bar a high leverage point just because it has a large residual", "Describing influential points without the removal idea", "Forgetting that high leverage and outlier points usually affect different parts of the model", "Explaining the transformed model only with r squared and not with residual-plot evidence", "Ignoring how log transformation changes skew in income data" ], "worksheet": "u2_lesson9_live.html" }, "WS-U3L1-reflect1": { "elements": [ { "id": "returned-planes", "description": "Explains that the data came only from planes that returned from the mission" }, { "id": "missing-planes", "description": "Notes that planes shot down were not included in the sample" }, { "id": "vulnerable-areas", "description": "Explains that missing bullet holes may point to vulnerable areas where armor is needed" }, { "id": "analysis-depends", "description": "May state that the way data are collected changes how the data should be analyzed" } ], "commonMistakes": [ "Saying the sample included all planes from the missions", "Focusing only on where bullet holes were seen without discussing the missing planes", "Not explaining why areas without bullet holes might be the vulnerable spots", "Talking about war history without connecting it to data collection" ], "worksheet": "u3_lesson1_live.html" }, "WS-U3L1-reflect2": { "elements": [ { "id": "who-sampled", "description": "Asks who was included in the sample" }, { "id": "who-missed", "description": "Asks who was left out or how unsampled individuals might differ" }, { "id": "chance-method", "description": "Asks whether the sample was collected using chance or random selection" }, { "id": "representative", "description": "Asks whether the sample is representative of the population" }, { "id": "untrustworthy", "description": "May mention that nonchance methods can lead to untrustworthy conclusions" } ], "commonMistakes": [ "Only asking whether the sample size is large enough", "Ignoring who was left out of the sample", "Not mentioning whether the sample was chosen by chance", "Assuming any collected sample automatically represents the population" ], "worksheet": "u3_lesson1_live.html" }, "WS-U3L1-exitTicket": { "elements": [ { "id": "population", "description": "Identifies the population as all students at the high school" }, { "id": "sample", "description": "Identifies the sample as the students who responded to the Instagram survey" }, { "id": "nonchance", "description": "Explains that the method is not based on chance or is a voluntary response sample" }, { "id": "difference", "description": "Explains that respondents may differ from nonrespondents, such as being more engaged, more opinionated, or more likely to follow student council social media" }, { "id": "better-method", "description": "Suggests a random or chance-based method for selecting students from the whole school" }, { "id": "trust-conclusion", "description": "States that the reported 82% may not be trustworthy or representative of the whole school" } ], "commonMistakes": [ "Confusing the sample with the population", "Treating the Instagram survey as random just because many students could see it", "Failing to explain how respondents might differ from students who did not respond", "Suggesting a better survey without using chance or random selection", "Assuming 82% must reflect the whole school" ], "worksheet": "u3_lesson1_live.html" }, "WS-U3L2-reflect1": { "elements": [ { "id": "returned-only", "description": "Explains that the sample included only planes that returned from the mission" }, { "id": "shot-down-excluded", "description": "Notes that planes shot down were left out of the sample" }, { "id": "not-representative", "description": "Explains that the sample was not representative because the missing planes may have been hit in different spots" }, { "id": "generalization-risk", "description": "States that generalizing to all hit planes is not appropriate from this sample alone" }, { "id": "tail-insight", "description": "May mention that missing bullet holes point to vulnerable areas like the tail" } ], "commonMistakes": [ "Saying the sample included all planes that were hit", "Ignoring the planes that were shot down", "Not connecting representativeness to whether generalization is appropriate", "Focusing only on where armor should go without explaining the sample problem" ], "worksheet": "u3_lesson2_live.html" }, "WS-U3L2-reflect2": { "elements": [ { "id": "observational-no-treatment", "description": "Explains that observational studies do not impose treatments" }, { "id": "observational-no-cause", "description": "States that observational studies cannot establish cause and effect" }, { "id": "experiment-treatment", "description": "Explains that experiments impose treatments or assign different conditions to subjects" }, { "id": "experiment-causal", "description": "States that a well-designed experiment can support a causal conclusion" }, { "id": "random-assignment", "description": "May mention random assignment or the resume experiment example" } ], "commonMistakes": [ "Claiming observational studies can prove cause and effect", "Failing to mention imposed treatments in experiments", "Describing an experiment as just a larger or more careful observational study", "Talking only about generalization without addressing causation" ], "worksheet": "u3_lesson2_live.html" }, "WS-U3L2-exitTicket": { "elements": [ { "id": "population", "description": "Identifies the population as all students at that high school" }, { "id": "sample", "description": "Identifies the sample as the 120 randomly chosen students from the high school" }, { "id": "observational", "description": "Identifies the study as an observational study because no treatments were imposed and existing records were used" }, { "id": "generalize-yes", "description": "Explains that generalization to the high school is appropriate because the sample was chosen at random from that population" }, { "id": "no-causation", "description": "Explains that the study cannot show that sports cause higher GPAs because it is observational" }, { "id": "confounding", "description": "May mention that other factors could explain GPA differences" } ], "commonMistakes": [ "Confusing the sample with the population", "Calling the study an experiment because two groups are being compared", "Ignoring that the sample was randomly selected from the high school", "Claiming the principal can conclude sports cause higher GPAs", "Forgetting that observational studies may have confounding variables" ], "worksheet": "u3_lesson2_live.html" }, "WS-U3L3-reflect1": { "elements": [ { "id": "both-group", "description": "Explains that both methods begin by dividing the population into groups" }, { "id": "cluster-select-groups", "description": "States that cluster sampling randomly selects whole clusters or groups" }, { "id": "cluster-all-members", "description": "Explains that all individuals in the selected clusters are included in the sample" }, { "id": "stratified-sample-within", "description": "States that stratified sampling takes a simple random sample within each stratum" }, { "id": "stratified-combine", "description": "Explains that the sampled individuals from all strata are combined into one sample" }, { "id": "hetero-homo", "description": "May mention that clusters are ideally heterogeneous while strata are ideally homogeneous" } ], "commonMistakes": [ "Saying cluster and stratified sampling are the same because both use groups", "Claiming that stratified sampling selects entire strata", "Claiming that cluster sampling samples only some people within each selected cluster", "Leaving out what happens after the groups are chosen" ], "worksheet": "u3_lesson3_live.html" }, "WS-U3L3-reflect2": { "elements": [ { "id": "segregation-pattern", "description": "Explains that incomes are similar within regions or neighborhoods and different across regions because of economic segregation" }, { "id": "sample-each-region", "description": "States that stratified sampling takes some homes from each region or stratum" }, { "id": "cluster-risk", "description": "Explains that a cluster sample could accidentally select only high-income or low-income areas" }, { "id": "better-estimate", "description": "States that stratified sampling gives a more consistent or more precise estimate because it includes a better mix of incomes" }, { "id": "harder-to-collect", "description": "Notes that stratified sampling is harder to implement because data must be collected across many regions or with a more complicated procedure" }, { "id": "low-variability", "description": "May mention lower variability or greater precision directly" } ], "commonMistakes": [ "Saying cluster sampling is automatically better because it is easier to collect", "Ignoring that incomes vary a lot between neighborhoods", "Forgetting that stratified sampling includes people from every group", "Saying stratified is better without connecting it to how incomes vary across regions" ], "worksheet": "u3_lesson3_live.html" }, "WS-U3L3-exitTicket": { "elements": [ { "id": "population", "description": "Identifies the population as all students at the high school" }, { "id": "stratified", "description": "Identifies the sampling method as a stratified random sample" }, { "id": "strata-grade", "description": "Explains that grade levels are the strata or groups based on a shared characteristic" }, { "id": "within-each-grade", "description": "States that students are randomly selected within each grade and then combined into one sample" }, { "id": "appropriate-representation", "description": "Explains that the method is appropriate because it guarantees representation from every grade and homework time may differ by grade" }, { "id": "not-cluster", "description": "Explains that this is not a cluster sample because not all students in a selected grade are surveyed" }, { "id": "disadvantage", "description": "Describes one disadvantage such as being harder to organize, more time-consuming, or more complicated to implement" } ], "commonMistakes": [ "Calling the method a cluster sample just because students are grouped by grade", "Forgetting to identify the population as the whole high school", "Not explaining that students are sampled within each grade", "Ignoring why representation from every grade could matter", "Leaving out any disadvantage or challenge of stratified sampling" ], "worksheet": "u3_lesson3_live.html" }, "WS-U3L4-reflect1": { "elements": [ { "id": "population", "description": "Identifies the population as all students at the university" }, { "id": "sample", "description": "Identifies the sample as the students who responded to the survey" }, { "id": "nonresponse", "description": "Explains that this is nonresponse bias because many selected students did not respond" }, { "id": "different-responders", "description": "Explains that respondents may differ from nonrespondents, such as students without internships being less likely to respond" }, { "id": "overestimate", "description": "States that the reported percentage with internships is likely an overestimate of the true percentage" }, { "id": "underrepresented", "description": "May note that students without internships are underrepresented in the responses" } ], "commonMistakes": [ "Calling the sample all selected students instead of the students who actually responded", "Ignoring that students without internships may respond at lower rates", "Saying the result is unbiased just because the original selection was random", "Leaving out whether the estimate is likely too high or too low" ], "worksheet": "u3_lesson4_live.html" }, "WS-U3L4-reflect2": { "elements": [ { "id": "undercoverage-definition", "description": "Defines undercoverage bias as part of the population having a reduced chance of being included" }, { "id": "volunteer-definition", "description": "Defines volunteer response bias as people choosing themselves to participate" }, { "id": "undercoverage-difference", "description": "Explains that undercoverage can leave out a subgroup whose responses differ from the rest of the population" }, { "id": "volunteer-difference", "description": "Explains that volunteer response can overrepresent people with stronger interest, stronger opinions, or more favorable attitudes" }, { "id": "direction", "description": "States that either bias can distort the estimate by causing an overestimate or underestimate because the sample is not representative" }, { "id": "example", "description": "May reference an example such as nongraduates being excluded or runners volunteering" } ], "commonMistakes": [ "Treating undercoverage and volunteer response as the same thing", "Defining volunteer response bias as simply low response rate", "Forgetting to explain how the sample differs from the population", "Leaving out that the bias changes the direction of the estimate" ], "worksheet": "u3_lesson4_live.html" }, "WS-U3L4-exitTicket": { "elements": [ { "id": "population", "description": "Identifies the population as all city residents" }, { "id": "sample", "description": "Identifies the sample as the residents who choose to respond to the survey" }, { "id": "volunteer-bias", "description": "Identifies volunteer or voluntary response bias because people choose whether to respond" }, { "id": "overestimate", "description": "Explains that support for bike lanes is likely overestimated because supporters may be more likely to respond" }, { "id": "wording-bias", "description": "Identifies question wording bias because words like \"important\" and \"safer\" are leading" }, { "id": "improvement", "description": "Suggests an improvement such as taking a random sample of residents or using neutral wording" } ], "commonMistakes": [ "Calling the sample all residents instead of only the people who respond", "Saying the survey is random because every household got the invitation", "Missing that supporters may be more likely to respond than opponents", "Ignoring the leading words in the survey question", "Suggesting an improvement without addressing either sampling or wording" ], "worksheet": "u3_lesson4_live.html" }, "WS-U3L5-reflect1": { "elements": [ { "id": "observational", "description": "States that this is an observational study and therefore cannot establish causation" }, { "id": "explanatory", "description": "Identifies the explanatory variable as whether or not a student takes notes" }, { "id": "response", "description": "Identifies the response variable as student grades or test scores" }, { "id": "confounding-variable", "description": "Identifies a possible confounding variable such as academic motivation" }, { "id": "confounding-explanation", "description": "Explains that the confounding variable affects both note-taking and grades, creating a possible false association" }, { "id": "motivation-example", "description": "May note that more motivated students may both take notes more often and earn higher grades" } ], "commonMistakes": [ "Claiming that higher grades for note-takers prove note-taking causes higher grades", "Mixing up the explanatory variable and the response variable", "Naming a confounding variable without explaining that it affects both note-taking and grades", "Ignoring that the study is observational rather than experimental" ], "worksheet": "u3_lesson5_live.html" }, "WS-U3L5-reflect2": { "elements": [ { "id": "crd-definition", "description": "Defines a completely randomized design as assigning treatments completely at random to all experimental units" }, { "id": "randomization-benefit", "description": "May explain that random assignment helps balance confounding or uncontrolled variables between treatment groups" }, { "id": "block-definition", "description": "Defines a randomized block design as first grouping similar units into blocks based on a blocking variable" }, { "id": "within-block-randomization", "description": "Explains that treatments are randomly assigned within each block" }, { "id": "blocking-purpose", "description": "Explains that blocking helps separate variability due to the blocking variable from treatment differences" }, { "id": "example", "description": "May use an example such as cancer severity or matched pairs to illustrate blocking" } ], "commonMistakes": [ "Saying that blocks are formed randomly instead of by a relevant variable", "Forgetting that treatments are still randomized within each block", "Saying that a completely randomized design removes all confounding", "Describing blocking without explaining why it helps" ], "worksheet": "u3_lesson5_live.html" }, "WS-U3L5-exitTicket": { "elements": [ { "id": "units", "description": "Identifies the experimental units as the patients with eczema" }, { "id": "factor", "description": "Identifies the explanatory variable or factor as the type of cream received" }, { "id": "treatments", "description": "Identifies the treatments as the new anti-itch cream and the placebo cream" }, { "id": "response", "description": "Identifies the response variable as the measured eczema symptoms, itching, or symptom improvement after treatment" }, { "id": "blocking", "description": "Explains that patients are blocked by severity, such as mild versus severe eczema" }, { "id": "within-block-randomization", "description": "Explains that patients are randomly assigned to treatments within each severity block" }, { "id": "control-group", "description": "Identifies the placebo group as the control group" }, { "id": "double-blind", "description": "Explains that neither the patients nor the nurses who interact with them know which treatment each patient receives" }, { "id": "placebo-purpose", "description": "May note that the placebo helps researchers judge whether any effect is due to the treatment rather than the placebo effect" } ], "commonMistakes": [ "Confusing the blocks with the treatments", "Forgetting that random assignment occurs within each severity block", "Naming severity as the response variable instead of the blocking variable", "Failing to identify the placebo group as the control group", "Saying the study is double-blind without mentioning both patients and nurses" ], "worksheet": "u3_lesson5_live.html" }, "WS-U3L6-7-reflect53": { "elements": [ { "id": "selection-def", "description": "Random selection = choosing sample from population" }, { "id": "assignment-def", "description": "Random assignment = assigning treatments to units" }, { "id": "selection-purpose", "description": "Random selection allows generalization to population" }, { "id": "assignment-purpose", "description": "Random assignment allows causal conclusions" } ], "commonMistakes": [ "Confusing the two terms (using them interchangeably)", "Only defining without explaining when/why each is used", "Saying random selection establishes causation (it doesn't)" ], "worksheet": "u3_lesson6-7_live.html" }, "WS-U3L6-7-reflect54a": { "elements": [ { "id": "identify-variable", "description": "Names a confounding variable (soil type, terrain, moisture, sunlight, drainage)" }, { "id": "explain-confounding", "description": "Explains WHY it's confounding: related to treatment AND affects response" }, { "id": "assignment-issue", "description": "Notes that different fields got different treatments without proper randomization" } ], "commonMistakes": [ "Listing variables without explaining why they're confounding", "Naming the treatment (fertilizer) as the confounding variable", "Confusing confounding with lurking variables" ], "worksheet": "u3_lesson6-7_live.html" }, "WS-U3L6-7-reflect54b": { "elements": [ { "id": "blocking-variable", "description": "Identifies blocking variable (soil type)" }, { "id": "block-creation", "description": "Groups fields by soil type into blocks" }, { "id": "random-within-blocks", "description": "Random assignment of treatments WITHIN each block" }, { "id": "response-measurement", "description": "Measure and compare yield" } ], "commonMistakes": [ "Blocking without random assignment within blocks", "Proposing more replication without blocking", "Confusing blocking with stratified sampling" ], "worksheet": "u3_lesson6-7_live.html" }, "WS-U3L6-7-reflect55": { "elements": [ { "id": "significance-meaning", "description": "Assuming no true treatment difference, p = 0.02 means there is a 2% probability of obtaining a difference at least as extreme as the observed one from chance variation alone" }, { "id": "not-by-chance", "description": "Concludes that the observed difference is statistically significant, or unlikely under a no-difference chance model" }, { "id": "causation-requirement", "description": "Can conclude causation IF random assignment was used" }, { "id": "random-assignment-check", "description": "Need to verify study used random assignment to treatments" } ], "commonMistakes": [ "Saying p = 0.02 means 2% chance the null hypothesis is true", "Claiming causation without mentioning need for random assignment", "Confusing statistical significance with practical significance" ], "worksheet": "u3_lesson6-7_live.html" }, "WS-U3L6-7-reflect56": { "elements": [ { "id": "random-selection", "description": "Random selection of experimental units from population" }, { "id": "representative-sample", "description": "OR experimental units are representative of the population" }, { "id": "population-specification", "description": "Results generalize to population from which sample was drawn" } ], "commonMistakes": [ "Confusing generalization with causation (mentioning random assignment)", "Only mentioning random selection without representative alternative", "Saying large sample size is sufficient (it's not)" ], "worksheet": "u3_lesson6-7_live.html" }, "WS-U3L6-7-exitTicket": { "elements": [ { "id": "random-assignment-role", "description": "Random assignment balances confounding variables across groups" }, { "id": "significance-role", "description": "Statistical significance indicates difference unlikely due to chance" }, { "id": "causation-connection", "description": "Together they allow causal conclusions: treatment caused the effect" }, { "id": "chain-of-logic", "description": "Shows logical connection between all three concepts" } ], "commonMistakes": [ "Treating the three concepts as independent rather than connected", "Confusing random assignment with random selection", "Claiming any significant result proves causation (without random assignment)" ], "worksheet": "u3_lesson6-7_live.html" }, "WS-U4L1-3-reflect1": { "elements": [ { "id": "humans-avoid-streaks", "description": "Humans tend to avoid long streaks/runs of the same outcome" }, { "id": "real-random-has-streaks", "description": "Truly random sequences often contain longer streaks than expected" }, { "id": "humans-alternate", "description": "Humans tend to alternate too frequently (HTHTHTHT pattern)" }, { "id": "streaks-are-normal", "description": "Acknowledges that streaks of 5, 6, 7, or 8+ are actually common in 100 flips" } ], "commonMistakes": [ "Saying random means 'equal' or 'balanced' (e.g., exactly 50-50)", "Not recognizing that streaks are normal in random data", "Focusing on total counts rather than patterns/streaks" ], "worksheet": "u4_lesson1-2-3_live.html" }, "WS-U4L1-3-reflect2": { "elements": [ { "id": "independence", "description": "Each flip is independent - past flips don't affect future flips" }, { "id": "no-due-effect", "description": "The coin is not 'due' for heads; probability remains 50%" }, { "id": "lln-long-run", "description": "Law of Large Numbers applies to LONG RUN, not next single flip" }, { "id": "gamblers-fallacy", "description": "This is called the gambler's fallacy (optional term)" } ], "commonMistakes": [ "Saying the Law of Large Numbers means things 'balance out' in the short run", "Not clearly stating that each flip remains 50-50", "Confusing long-run patterns with predictions for the next trial" ], "worksheet": "u4_lesson1-2-3_live.html" }, "WS-U4L1-3-reflect3": { "elements": [ { "id": "long-run-interpretation", "description": "If we randomly select many albums, about 20% would be jazz" }, { "id": "relative-frequency", "description": "Probability represents relative frequency in long run / repeated selections" }, { "id": "generalizability", "description": "Explains that the proportion communicates an approximate long-run rate and can support predictions over many future selections, unlike a raw count tied to the 2,105 observed sales" }, { "id": "prediction-utility", "description": "Can make predictions about future selections or different sample sizes" }, { "id": "with-replacement", "description": "With replacement maintains the probability for each selection" } ], "commonMistakes": [ "Saying exactly 20% will be jazz (rather than approximately or on average)", "Not connecting probability to long-run behavior", "Focusing only on the specific calculation without interpreting what it means" ], "worksheet": "u4_lesson1-2-3_live.html" }, "WS-U4L1-3-exitTicket": { "elements": [ { "id": "sample-space", "description": "Identifies sample space as {red, blue, green} or all 20 individual marbles" }, { "id": "total-outcomes", "description": "Correctly identifies total as 20 marbles (8+5+7)" }, { "id": "p-blue-calculation", "description": "P(blue) = 5/20 = 0.25 or 25%" }, { "id": "p-blue-interpretation", "description": "Interprets P(blue) as long-run relative frequency (about 25% of many selections would be blue)" }, { "id": "complement-rule", "description": "Uses complement rule: P(not red) = 1 - P(red) = 1 - 8/20 = 12/20 = 0.60" }, { "id": "expected-green", "description": "Expected green = 100 × (7/20) = 35 marbles" } ], "commonMistakes": [ "Giving P(blue) without the interpretation as a long-run relative frequency", "Not interpreting probability as relative frequency in long run", "Calculating P(not red) by adding other probabilities instead of using 1 - P(red)", "Not showing complement rule formula even if answer is correct", "Confusing expected value with guaranteed exact count" ], "worksheet": "u4_lesson1-2-3_live.html" }, "WS-U4L1-2-reflect1": { "elements": [ { "id": "humans-avoid-streaks", "description": "Humans tend to avoid long streaks/runs of the same outcome" }, { "id": "real-random-has-streaks", "description": "Truly random sequences often contain longer streaks than expected" }, { "id": "humans-alternate", "description": "Humans tend to alternate too frequently (HTHTHTHT pattern)" }, { "id": "streaks-are-normal", "description": "Acknowledges that streaks of 5, 6, 7, or 8+ are actually common in 100 flips" } ], "commonMistakes": [ "Saying random means 'equal' or 'balanced' (e.g., exactly 50-50)", "Not recognizing that streaks are normal in random data", "Focusing on total counts rather than patterns/streaks" ], "worksheet": "u4_lesson1-2_live.html" }, "WS-U4L1-2-reflect2": { "elements": [ { "id": "independence", "description": "Each flip is independent - past flips don't affect future flips" }, { "id": "no-due-effect", "description": "The coin is not 'due' for heads; probability remains 50%" }, { "id": "lln-long-run", "description": "Law of Large Numbers applies to LONG RUN, not next single flip" }, { "id": "gamblers-fallacy", "description": "This is called the gambler's fallacy (optional term)" } ], "commonMistakes": [ "Saying the Law of Large Numbers means things 'balance out' in the short run", "Not clearly stating that each flip remains 50-50", "Confusing long-run patterns with predictions for the next trial" ], "worksheet": "u4_lesson1-2_live.html" }, "WS-U4L1-2-exitTicket": { "elements": [ { "id": "digit-assignment", "description": "Assigns digits to represent correct/incorrect (e.g., 1=correct, 2-4=incorrect; or 1-25=correct, 26-100=incorrect)" }, { "id": "trial-definition", "description": "One trial = generating 5 random numbers (one per question)" }, { "id": "recording", "description": "Count how many of the 5 are 'correct' in each trial" }, { "id": "success-criterion", "description": "Success = 3 or more correct out of 5" }, { "id": "many-trials", "description": "Repeat the five-question trial many times and use the proportion with at least 3 correct as the probability estimate" } ], "commonMistakes": [ "Using digits 1-4 where 1=correct gives 25% but not explaining this clearly", "Not specifying that one trial consists of exactly 5 random selections", "Forgetting to mention the success threshold (at least 3 correct)", "Not mentioning the need for many trials to estimate probability" ], "worksheet": "u4_lesson1-2_live.html" }, "WS-U4L10-12-reflect1": { "elements": [ { "id": "binomial-identification", "description": "Correctly identifies binomial for the first scenario (fixed n, counting successes)" }, { "id": "binomial-parameters", "description": "States binomial parameters: n = 10, p = 0.30" }, { "id": "binomial-setup", "description": "Sets up calculation: P(X=4) = C(10,4) · (0.30)^4 · (0.70)^6" }, { "id": "geometric-identification", "description": "Correctly identifies geometric for the second scenario (counting trials to first success)" }, { "id": "geometric-parameters", "description": "States geometric parameter: p = 0.30" }, { "id": "geometric-setup", "description": "Sets up calculation: P(X=3) = (0.70)^2 · (0.30)" }, { "id": "distinction-explanation", "description": "Explains that both scenarios have binary outcomes, independent trials, and constant p = 0.30; binomial fixes n = 10 and counts successes, while geometric counts trials until the first success" } ], "commonMistakes": [ "Using geometric when n is fixed (counting successes out of n trials)", "Using binomial when counting trials until first success", "Forgetting to state parameters (n, p for binomial; p for geometric)", "Setting up geometric with p^(x-1) instead of (1-p)^(x-1)", "Not explaining WHY each distribution applies", "Writing calculator commands instead of formula setup" ], "worksheet": "u4_lesson10-12_live.html" }, "WS-U4L10-12-exitTicket": { "elements": [ { "id": "part-a-binomial", "description": "Part (a): Identifies binomial with n=8, p=0.40" }, { "id": "part-a-calculation", "description": "Part (a): P(X=3) = C(8,3) · (0.40)^3 · (0.60)^5 ≈ 0.279" }, { "id": "part-b-mean", "description": "Part (b): μ = np = 8(0.40) = 3.2 hurricanes" }, { "id": "part-b-interpretation", "description": "Part (b): Interprets as 'on average' or 'in the long run' over many seasons" }, { "id": "part-c-geometric", "description": "Part (c): Identifies geometric with p=0.40" }, { "id": "part-c-calculation", "description": "Part (c): P(X=4) = (0.60)^3 · (0.40) ≈ 0.0864" }, { "id": "part-d-mean", "description": "Part (d): μ = 1/p = 1/0.40 = 2.5 hurricanes" }, { "id": "part-d-interpretation", "description": "Part (d): Over many such sequences, the first landfall occurs on about the 2.5th hurricane on average, counting the hurricane that makes landfall" } ], "commonMistakes": [ "Using geometric for part (a) or binomial for part (c)", "Forgetting the (1-p)^5 term in binomial calculation", "Using (0.60)^4 instead of (0.60)^3 for geometric (should be x-1 failures)", "Not interpreting the means in context", "Saying 'the probability is 3.2' instead of 'expected value'", "Confusing the two means (np vs 1/p)" ], "worksheet": "u4_lesson10-12_live.html" }, "WS-U4L3-5-reflect1": { "elements": [ { "id": "long-run-interpretation", "description": "If we randomly select many albums, about 20% would be jazz" }, { "id": "relative-frequency", "description": "Probability represents relative frequency in long run / repeated selections" }, { "id": "generalizability", "description": "Explains that the proportion communicates an approximate long-run rate and can support predictions over many future selections, unlike a raw count tied to the 2,105 observed sales" }, { "id": "prediction-utility", "description": "Can make predictions about future selections or different sample sizes" }, { "id": "with-replacement", "description": "With replacement maintains the probability for each selection" } ], "commonMistakes": [ "Saying exactly 20% will be jazz (rather than approximately or on average)", "Not connecting probability to long-run behavior", "Focusing only on the specific calculation without interpreting what it means" ], "worksheet": "u4_lesson3-4-5_live.html" }, "WS-U4L3-5-reflect2": { "elements": [ { "id": "mutually-exclusive-definition", "description": "Mutually exclusive means events CANNOT occur together (P(A ∩ B) = 0)" }, { "id": "low-probability-definition", "description": "Low probability means events CAN occur together but rarely do (P(A ∩ B) > 0 but small)" }, { "id": "mutually-exclusive-example", "description": "Gives a correct example of mutually exclusive from the table (e.g., Famous and Telepathy where intersection is 0)" }, { "id": "low-probability-example", "description": "Gives a correct example of non-mutually exclusive events with small intersection (e.g., Famous and Fly with 5 students)" }, { "id": "impossibility-vs-rarity", "description": "Distinguishes between impossible (zero probability) and just unlikely (small probability)" } ], "commonMistakes": [ "Thinking mutually exclusive means 'unlikely' rather than 'impossible'", "Not checking the actual intersection values in the table", "Confusing 'independent' with 'mutually exclusive'", "Saying events with small intersection are mutually exclusive" ], "worksheet": "u4_lesson3-4-5_live.html" }, "WS-U4L3-5-reflect3": { "elements": [ { "id": "without-replacement-changes-composition", "description": "Without replacement: the first selection changes the composition of remaining marbles" }, { "id": "fewer-remaining", "description": "If first marble is red, there are fewer red marbles AND fewer total marbles for second draw" }, { "id": "conditional-probability", "description": "The probability of the second event depends on (is conditional on) the first outcome" }, { "id": "with-replacement-independence", "description": "With replacement: the composition stays the same, so second probability is unchanged" }, { "id": "specific-values", "description": "Mentions specific values: 3/9 vs 4/10 without replacement, or 4/10 each time with replacement" } ], "commonMistakes": [ "Not mentioning that both the numerator AND denominator change", "Saying probabilities are always the same regardless of replacement", "Claiming the draws are independent without explaining that replacement restores the original 4-red-out-of-10 composition and keeps the second-draw probability unchanged.", "Not connecting 'conditional probability' to the concept of changing composition" ], "worksheet": "u4_lesson3-4-5_live.html" }, "WS-U4L3-5-exitTicket": { "elements": [ { "id": "p-math-calculation", "description": "P(Math) = 105/200 = 0.525 or 52.5%" }, { "id": "p-math-interpretation", "description": "Interprets P(Math) as long-run relative frequency (about 52.5% of randomly selected students prefer Math)" }, { "id": "joint-probability", "description": "P(Freshman ∩ Math) = 45/200 = 0.225 or 22.5%" }, { "id": "not-mutually-exclusive", "description": "Freshman and Math are NOT mutually exclusive because their intersection is not zero (45 students are both)" }, { "id": "conditional-freshman", "description": "P(Math | Freshman) = 45/100 = 0.45 or 45%" }, { "id": "conditional-sophomore", "description": "P(Math | Sophomore) = 60/100 = 0.60 or 60%" }, { "id": "comparison", "description": "Interprets the comparison: sophomores in this survey were more likely to prefer Math than freshmen (60% versus 45%)" } ], "commonMistakes": [ "Forgetting to divide by 200 for overall probability vs row/column total for conditional", "Not interpreting probability as relative frequency in long run", "Saying Freshman and Math ARE mutually exclusive because some students chose English", "Confusing P(Freshman ∩ Math) with P(Math | Freshman)", "Using wrong denominator for conditional probabilities" ], "worksheet": "u4_lesson3-4-5_live.html" }, "WS-U4L6-reflect1": { "elements": [ { "id": "independent-def", "description": "Independent: knowing one occurred doesn't change probability of other (P(A|B) = P(A))" }, { "id": "mutually-exclusive-def", "description": "Mutually exclusive: events cannot occur together (P(A and B) = 0)" }, { "id": "cannot-be-both", "description": "Events with non-zero probabilities cannot be both independent AND mutually exclusive" }, { "id": "reasoning", "description": "Explains that for positive-probability events, mutual exclusivity gives P(A and B)=0 while independence would require P(A and B)=P(A)P(B)>0" } ], "commonMistakes": [ "Claiming two positive-probability events can be both independent and mutually exclusive.", "Thinking independent means the events don't affect each other so they can't happen together", "Not recognizing that P(A and B) = 0 for mutually exclusive but P(A)·P(B) > 0 creates a contradiction" ], "worksheet": "u4_lesson6_live.html" }, "WS-U4L6-reflect2": { "elements": [ { "id": "one-check-sufficient", "description": "Yes, checking P(A|B) = P(A) IS sufficient to establish independence" }, { "id": "equivalent-checks", "description": "Could also check P(B|A) = P(B) or P(A and B) = P(A)·P(B) — all are equivalent" }, { "id": "any-one-works", "description": "If one condition holds, all others automatically hold for independence" }, { "id": "calculation-accuracy", "description": "Main concern is ensuring the calculation is done correctly" } ], "commonMistakes": [ "Thinking you must check ALL conditions (P(A|B)=P(A) AND P(B|A)=P(B) AND product rule)", "Not recognizing that the three conditions are mathematically equivalent", "Confusing what needs to be checked vs. what could be checked" ], "worksheet": "u4_lesson6_live.html" }, "WS-U4L6-exitTicket": { "elements": [ { "id": "independence-check", "description": "Checks if P(A)·P(B) = P(A and B): 0.3 × 0.5 = 0.15" }, { "id": "independence-conclusion", "description": "Concludes events ARE independent since 0.15 = 0.15" }, { "id": "addition-rule", "description": "Uses addition rule: P(A or B) = P(A) + P(B) - P(A and B)" }, { "id": "correct-calculation", "description": "Calculates P(A or B) = 0.3 + 0.5 - 0.15 = 0.65" }, { "id": "shows-work", "description": "Shows the arithmetic steps clearly" } ], "commonMistakes": [ "Forgetting to subtract the intersection in the addition rule", "Using the simplified P(A) + P(B) without checking if events are mutually exclusive", "Calculating P(A)·P(B) but not comparing it to the given P(A and B)", "Not showing work/formula before plugging in numbers" ], "worksheet": "u4_lesson6_live.html" }, "WS-U4L7-8-reflect1": { "elements": [ { "id": "discrete-definition", "description": "Discrete = countable values with gaps/spaces between them" }, { "id": "continuous-definition", "description": "Continuous = infinite values in an interval (no gaps)" }, { "id": "discrete-example", "description": "Provides a valid original example of a discrete random variable" }, { "id": "continuous-example", "description": "Provides a valid original example of a continuous random variable" }, { "id": "example-justification", "description": "Explains why the discrete example has countable separated values and why the continuous example can take any value in an interval" } ], "commonMistakes": [ "Giving an example of only one type", "Confusing 'countable' with 'finite' (discrete can be infinite but countable)", "Saying continuous means 'any number' without mentioning 'in an interval'", "Giving examples without explaining why they fit the category" ], "worksheet": "u4_lesson7-8_live.html" }, "WS-U4L7-8-reflect2": { "elements": [ { "id": "probability-distribution", "description": "Sets up correct probabilities (1/3 for each outcome)" }, { "id": "expected-value-calculation", "description": "Correctly calculates E(X) = (1/3)(10000) + (1/3)(1000) + (1/3)(0) = $3,666.67" }, { "id": "mean-interpretation", "description": "Interprets expected value as long-run average over many games" }, { "id": "decision-with-reasoning", "description": "Makes recommendation based on comparing expected value ($3,666.67) to cost ($3,000)" }, { "id": "expected-profit", "description": "Correctly identifies expected profit is $666.67 (winnings minus fee)" } ], "commonMistakes": [ "Using wrong probabilities (not recognizing equal 1/3 chance for each)", "Forgetting to interpret what the expected value means", "Saying 'yes' or 'no' without comparing to the $3,000 fee", "Confusing expected value with guaranteed outcome" ], "worksheet": "u4_lesson7-8_live.html" }, "WS-U4L7-8-exitTicket": { "elements": [ { "id": "net-profit-values", "description": "Correctly identifies net profits: roll 6 → $8 net, roll 1 → $1 net, other → -$2 net" }, { "id": "probabilities", "description": "Correct probabilities: P(6)=1/6, P(1)=1/6, P(other)=4/6" }, { "id": "distribution-format", "description": "Creates clear probability distribution table or list" }, { "id": "expected-value", "description": "Calculates E(X) = (1/6)(8) + (1/6)(1) + (4/6)(-2) = $0.17 (approximately)" }, { "id": "fair-game-interpretation", "description": "Interprets: positive expected value means game favors player; not 'fair' from carnival's perspective" }, { "id": "long-run-context", "description": "Explains that over many plays the player's net profit would average about +$0.17 per play" } ], "commonMistakes": [ "Using gross winnings ($10, $3, $0) instead of net profit ($8, $1, -$2)", "Forgetting that 'other' outcomes (2,3,4,5) have probability 4/6, not 1/6", "Saying expected value of $0.17 means you'll win 17 cents every game", "Confusing 'fair to the player' with 'fair to the carnival'" ], "worksheet": "u4_lesson7-8_live.html" }, "WS-U4L9-reflect1": { "elements": [ { "id": "identify-error", "description": "Identifies that the student incorrectly added standard deviations directly" }, { "id": "variance-rule", "description": "States that variances (not SDs) add for independent random variables" }, { "id": "correct-calculation", "description": "Shows correct calculation: σ = √(3² + 4²) = √(9 + 16) = √25 = 5" }, { "id": "conceptual-explanation", "description": "Explains WHY variances add (uncertainty/variability increases when combining)" } ], "commonMistakes": [ "Agreeing with the student that 3 + 4 = 7 is correct", "Knowing to use variances but making arithmetic errors", "Subtracting variances instead of adding", "Not taking the square root at the end" ], "worksheet": "u4_lesson9_live.html" }, "WS-U4L9-reflect2": { "elements": [ { "id": "mean-calculation", "description": "Correctly calculates mean: μ(A+B) = 5 + 12 = 17 lbs" }, { "id": "variance-step", "description": "Shows variance calculation: σ² = 0.8² + 1.5² = 0.64 + 2.25 = 2.89" }, { "id": "sd-calculation", "description": "Correctly calculates SD: σ = √2.89 ≈ 1.7 lbs" }, { "id": "explains-process", "description": "Explains that means add directly but SDs require variance addition then square root" }, { "id": "independence-note", "description": "Mentions that independence is required for the variance addition rule" } ], "commonMistakes": [ "Adding standard deviations directly: 0.8 + 1.5 = 2.3 lbs (THE TRAP!)", "Forgetting to take the square root after adding variances", "Not showing the variance step", "Subtracting instead of adding for the mean" ], "worksheet": "u4_lesson9_live.html" }, "WS-U4L9-exitTicket": { "elements": [ { "id": "total-mean", "description": "Calculates total mean: μ = 4.50 + 3.25 = $7.75" }, { "id": "total-variance", "description": "Calculates variance: σ² = 0.60² + 0.45² = 0.36 + 0.2025 = 0.5625" }, { "id": "total-sd", "description": "Calculates SD: σ = √0.5625 = $0.75" }, { "id": "discount-mean", "description": "Applies discount to mean: 0.9 × $7.75 = $6.975 (or ≈$6.98)" }, { "id": "discount-sd", "description": "Applies discount to SD: |0.9| × $0.75 = $0.675 (or ≈$0.68)" }, { "id": "shows-work", "description": "Shows clear step-by-step work for each calculation" } ], "commonMistakes": [ "Adding SDs directly: 0.60 + 0.45 = 1.05 (THE TRAP!)", "Forgetting to apply discount to standard deviation", "Subtracting 10% instead of multiplying by 0.9", "Not showing variance calculation step", "Thinking the constant in a transformation (like discount) affects SD differently than multiplication" ], "worksheet": "u4_lesson9_live.html" }, "WS-U5L1-2-reflect1": { "elements": [ { "id": "define-variability", "description": "Defines sampling variability as the natural variation in statistics from different samples of the same population" }, { "id": "example-connection", "description": "Connects explanation to a specific example (koala weights clustering around 15 lbs, or election polls varying around 51%)" }, { "id": "single-sample-limitation", "description": "Explains why a single sample is insufficient — it's unpredictable like a dice roll, could be far from the truth" }, { "id": "many-samples-pattern", "description": "Notes that many samples form a predictable distribution/pattern that reveals the population truth" } ], "commonMistakes": [ "Confusing sampling variability with measurement error", "Saying larger samples are always better without explaining why", "Saying only that larger samples are better without saying what varies", "Using 'it' without specifying what varies" ], "worksheet": "u5_lesson1-2_live.html" }, "WS-U5L1-2-reflect2": { "elements": [ { "id": "define-variable", "description": "Defines the random variable (X = length of a randomly selected bolt in cm)" }, { "id": "z-score-work", "description": "Correctly calculates Z-scores: z = (4.8-5.0)/0.1 = -2 and z = (5.2-5.0)/0.1 = 2" }, { "id": "probability-calc", "description": "Correctly determines P(rejected) = P(X < 4.8) + P(X > 5.2) ≈ 0.0228 + 0.0228 = 0.0456 (or uses complement: 1 - 0.9544 = 0.0456)" }, { "id": "interpretation", "description": "Interprets the result in context: approximately 4.56% of bolts will be rejected" } ], "commonMistakes": [ "Forgetting to include both tails (only calculating one side)", "Calculating P(4.8 < X < 5.2) instead of P(rejected)", "Reporting a probability with no in-context interpretation", "Using calculator syntax without showing Z-scores or identifying normal distribution" ], "worksheet": "u5_lesson1-2_live.html" }, "WS-U5L1-2-exitTicket": { "elements": [ { "id": "part-a-zscore", "description": "Calculates z = (9.5 - 10.0)/0.3 = -1.67 and finds P(X < 9.5) ≈ 0.0475" }, { "id": "part-b-mean", "description": "Correctly finds μ(X-Y) = 10.0 - 9.5 = 0.5 oz" }, { "id": "part-b-sd", "description": "Correctly finds σ(X-Y) = √(0.3² + 0.2²) = √(0.09 + 0.04) = √0.13 ≈ 0.361 oz (does NOT subtract SDs)" }, { "id": "part-c-setup", "description": "Recognizes that Y > X means X - Y < 0, finds z = (0 - 0.5)/0.361 ≈ -1.39" }, { "id": "part-c-probability", "description": "Correctly finds P(X - Y < 0) ≈ 0.0823" }, { "id": "normal-justification", "description": "Notes that X-Y is approximately normal because both X and Y are independently normal" } ], "commonMistakes": [ "Subtracting standard deviations instead of adding variances", "Forgetting to take the square root after adding variances", "Not recognizing that Y > X means X - Y < 0", "Not justifying that the difference distribution is normal" ], "worksheet": "u5_lesson1-2_live.html" }, "WS-U5L3-reflect1": { "elements": [ { "id": "clt-definition", "description": "States that sampling distribution of means becomes approximately normal with large enough sample size" }, { "id": "regardless-population", "description": "Mentions that CLT works regardless of population shape/distribution" }, { "id": "sample-size-matters", "description": "Explains that larger sample sizes make the sampling distribution more normal" }, { "id": "simulation-example", "description": "References specific examples from the simulations (skewed population, U-shaped population, etc.)" }, { "id": "why-amazing", "description": "Explains why it's 'amazing' - no matter what the population looks like, we get normal sampling distribution with large n" } ], "commonMistakes": [ "Confusing the population distribution with the sampling distribution", "Thinking CLT only works for normal populations", "Describing the distribution of individual values instead of the distribution of sample means", "Thinking larger samples have MORE variability (opposite is true)" ], "worksheet": "u5_lesson3_live.html" }, "WS-U5L3-reflect2": { "elements": [ { "id": "probability-meaning", "description": "Explains that 0.012 means only 1.2% of random reallocations produced a difference as extreme as 1.69" }, { "id": "chance-alone-interpretation", "description": "Connects to 'by chance alone' - if treatment had no effect, we'd rarely see this result" }, { "id": "evidence-for-effect", "description": "Explains that small probability means the result is unlikely due to chance, suggesting treatment is effective" }, { "id": "larger-probability-contrast", "description": "Explains that 0.30 would mean a result at least this extreme is fairly common under chance alone and would not provide convincing evidence that melatonin is effective" }, { "id": "context", "description": "Uses context of melatonin/sleep hours in explanation" } ], "commonMistakes": [ "Thinking 0.012 is the probability melatonin works (it's not - it's probability of observing result if it doesn't work)", "Confusing statistical significance with practical significance", "Not understanding what 'by chance alone' means in randomization context", "Thinking larger probability = stronger evidence (opposite is true)" ], "worksheet": "u5_lesson3_live.html" }, "WS-U5L3-exitTicket": { "elements": [ { "id": "difference-calculation", "description": "Calculates observed difference as 8.5 - 6.2 = 2.3 cm" }, { "id": "probability-calculation", "description": "Calculates probability as 25/1000 = 0.025 (or 2.5%)" }, { "id": "probability-interpretation", "description": "Explains that only 2.5% of random reallocations produced difference this large or larger" }, { "id": "conclusion", "description": "Concludes that fertilizer likely does increase growth because result is unlikely by chance (small p-value)" }, { "id": "reasoning", "description": "Shows clear reasoning connecting small probability to evidence for effectiveness" } ], "commonMistakes": [ "Subtracting in wrong order (6.2 - 8.5)", "Not converting 25/1000 to decimal or percentage", "Concluding that 0.025 means fertilizer definitely works (should say 'provides evidence')", "Misinterpreting what the randomization distribution represents" ], "worksheet": "u5_lesson3_live.html" }, "WS-U5L4-reflect1": { "elements": [ { "id": "unbiased-means-on-average", "description": "Explains that 'unbiased' means the average across ALL possible samples equals the parameter, not that any single sample is correct" }, { "id": "single-sample-can-miss", "description": "Acknowledges that individual samples can be far from the parameter — unbiased does NOT guarantee accuracy for one sample" }, { "id": "centered-at-parameter", "description": "Describes the sampling distribution as centered at μ, so the estimator has no systematic tendency to overestimate or underestimate" }, { "id": "variability-is-natural", "description": "Mentions that variability from sample to sample is expected and natural" }, { "id": "uses-example", "description": "References the Weimaraner example (2.3 vs 5) to illustrate the concept" } ], "commonMistakes": [ "Thinking 'unbiased' means every sample gives the right answer", "Confusing bias (systematic error) with variability (random error)", "Thinking a single bad estimate proves the estimator is biased", "Not understanding what 'on average' refers to (all possible samples, not repeated measurements)" ], "worksheet": "u5_lesson4_live.html" }, "WS-U5L4-reflect2": { "elements": [ { "id": "mean-not-equal-parameter", "description": "States that the mean of all sample ranges (7.8) does not equal the population range (10), so it's biased" }, { "id": "direction-of-bias", "description": "Explains that the sample range systematically underestimates — it tends to be LESS than the population range" }, { "id": "why-one-direction", "description": "Explains WHY: a sample can never have a range larger than the population range (can't get values outside population boundaries), so sample ranges can only be ≤ population range" }, { "id": "overestimation-possible", "description": "Explains that a different estimator can be biased upward if its sampling distribution is centered above the parameter, so it systematically overestimates" }, { "id": "dot-plot-reference", "description": "References the dot plot clustering to the left of 10" } ], "commonMistakes": [ "Saying the range is biased 'because 7.8 is close to 10' (closeness doesn't matter — not equal means biased)", "Confusing the direction — thinking range overestimates", "Not understanding WHY the bias is one-directional", "Thinking all statistics are biased, or that bias can be fixed by taking a larger sample" ], "worksheet": "u5_lesson4_live.html" }, "WS-U5L4-exitTicket": { "elements": [ { "id": "mean-is-unbiased", "description": "States yes, the mean of all sample means should equal 75 because x-bar is an unbiased estimator of mu" }, { "id": "range-is-biased", "description": "States no, the mean of all sample ranges will NOT equal 30 because the sample range is a biased estimator that underestimates" }, { "id": "range-reasoning", "description": "Explains why range underestimates: samples of 2 from 4 values are unlikely to capture both the min (60) and max (90) every time" }, { "id": "mean-more-trustworthy", "description": "Concludes the sample mean is more trustworthy because it's unbiased — it targets the parameter on average" }, { "id": "clear-reasoning", "description": "Shows clear connection between the Weimaraner lesson and this new context" } ], "commonMistakes": [ "Thinking both statistics are unbiased because 'samples are random'", "Thinking the range would equal 30 because it's a 'simple' statistic", "Not connecting back to the Weimaraner lesson (this is the same pattern)", "Confusing 'trustworthy' with 'always correct' instead of 'unbiased'" ], "worksheet": "u5_lesson4_live.html" }, "WS-U5L5-reflect1": { "elements": [ { "id": "sd-in-context", "description": "Interprets sigma_p-hat = 0.065 in context: sample proportions of students with driver's licenses typically vary by about 0.065 from the population proportion of 0.30" }, { "id": "references-all-samples", "description": "References 'all random samples of size 50 from this population' — not just one sample" }, { "id": "uses-typically", "description": "Uses language like 'typically' or 'on average' when describing the variability" }, { "id": "importance-connection", "description": "Explains that knowing typical sample-to-sample variability helps judge how close a statistic may be to the truth and supports later confidence intervals or significance tests." }, { "id": "not-single-sample", "description": "Distinguishes that the standard deviation describes sample-to-sample variation, not variation within a single sample" } ], "commonMistakes": [ "Interpreting 0.065 as the standard deviation of the population rather than the sampling distribution", "Forgetting to say 'typically' or 'on average' — implying every sample is exactly 0.065 away", "Not referencing all possible samples of this size", "Giving a generic definition of standard deviation without context (driver's licenses, this population)", "Confusing variation within one sample with variation between different sample proportions" ], "worksheet": "u5_lesson5_live.html" }, "WS-U5L5-reflect2": { "elements": [ { "id": "correct-mean", "description": "Correctly calculates mu_p-hat = 0.40" }, { "id": "correct-sd", "description": "Correctly calculates sigma_p-hat = sqrt(0.40 × 0.60 / 100) = 0.049 (or approximately 0.049)" }, { "id": "large-counts-check", "description": "Checks Large Counts: np = 100(0.40) = 40 >= 10 and n(1-p) = 100(0.60) = 60 >= 10" }, { "id": "z-score-calculation", "description": "Calculates z = (0.52 - 0.40) / 0.049 = 2.45 (approximately)" }, { "id": "interprets-unusual", "description": "Interprets: 0.52 is about 2.45 standard deviations above the mean, which would be unusual/surprising" }, { "id": "probability-value", "description": "Calculates or estimates P(p-hat >= 0.52) ≈ 0.007 (very small probability)" } ], "commonMistakes": [ "Using the wrong formula (e.g., sigma = p(1-p)/n without the square root)", "Forgetting to check the Large Counts condition", "Computing the z-score with the wrong values in the numerator or denominator", "Saying a z-score of 2.45 is 'not unusual' — it is quite unusual (beyond 2 SD)", "Not interpreting in context (just giving numbers without explanation)" ], "worksheet": "u5_lesson5_live.html" }, "WS-U5L5-exitTicket": { "elements": [ { "id": "center", "description": "States mu_p-hat = 0.05" }, { "id": "spread", "description": "Calculates sigma_p-hat = sqrt(0.05 × 0.95 / 200) ≈ 0.0154" }, { "id": "shape-conditions", "description": "Checks Large Counts: np = 200(0.05) = 10 >= 10 and n(1-p) = 200(0.95) = 190 >= 10, so approximately normal" }, { "id": "z-score", "description": "Calculates z = (0.08 - 0.05) / 0.0154 ≈ 1.95" }, { "id": "probability-interpretation", "description": "Finds P(p-hat >= 0.08) ≈ 0.026 (about 2.6%) and interprets: this would be somewhat unusual if p truly equals 0.05, suggesting the inspector might have reason for concern" }, { "id": "ten-percent-condition", "description": "Notes that the 10% condition cannot be verified from the information given; it holds if the relevant population contains at least 2,000 bulbs (or it is reasonable to assume production is that large)" } ], "commonMistakes": [ "Forgetting to check that np = 10 is EXACTLY at the boundary — it just barely meets the condition", "Using 16 instead of 0.08 in the z-score formula (must use proportions, not counts)", "Forgetting that sigma_p-hat uses the POPULATION proportion p = 0.05, not the sample proportion 0.08", "Not providing a clear conclusion — just calculating without interpreting", "Saying 'it's fine' without considering that z ≈ 1.95 is close to the typical threshold of 2" ], "worksheet": "u5_lesson5_live.html" }, "WS-U5L6-reflect1": { "elements": [ { "id": "sd-meaning", "description": "Explains that the SD tells us how much the difference in sample proportions typically varies from the true difference" }, { "id": "practical-value", "description": "Explains the practical value: without knowing typical variation, we can't judge whether an observed difference is close to or far from the truth" }, { "id": "mean-insufficient", "description": "Explains why the mean alone is insufficient — it tells us the center but not how spread out the values are around it" }, { "id": "connection-to-inference", "description": "Connects to later inference: SD helps us build confidence intervals or determine if results are unusual (significance tests)" }, { "id": "typically-language", "description": "Uses language like 'typically' or 'on average' when describing the SD" } ], "commonMistakes": [ "Confusing the SD of the sampling distribution with the SD of individual observations", "Thinking the mean is sufficient because it tells us the 'right answer'", "Not explaining WHY knowing variation matters (just stating that it does)", "Confusing standard deviation with standard error" ], "worksheet": "u5_lesson6_live.html" }, "WS-U5L6-reflect2": { "elements": [ { "id": "probability-meaning", "description": "Explains that 0.18 means about 18% of all possible sample pairs would produce School B with a higher proportion than School A" }, { "id": "not-unusual", "description": "Concludes this is not unusual — 18% is a reasonable probability, so this could easily happen by chance" }, { "id": "small-prob-contrast", "description": "Explains that 0.001 would mean only 0.1% of samples produce this result — very unusual and surprising" }, { "id": "unusual-definition", "description": "Connects to the concept of 'unusual': small probabilities suggest the result is unlikely by chance, large probabilities suggest it's not surprising" }, { "id": "context", "description": "Uses context of driver's licenses and schools in the explanation" } ], "commonMistakes": [ "Interpreting 0.18 as the probability that School B actually has more licensed students", "Not referencing 'all possible samples' in the interpretation", "Thinking any probability below 0.50 is 'unusual'", "Not explaining the contrast — just stating 0.001 is 'smaller'" ], "worksheet": "u5_lesson6_live.html" }, "WS-U5L6-exitTicket": { "elements": [ { "id": "mean-calculation", "description": "Calculates mean = 0.58 - 0.45 = 0.13" }, { "id": "mean-interpretation", "description": "Interprets mean in context: for all random samples of these sizes, the differences in sample proportions of support will have a mean of about 0.13" }, { "id": "sd-calculation", "description": "Calculates SD = sqrt[0.58(0.42)/100 + 0.45(0.55)/80] ≈ 0.074" }, { "id": "sd-interpretation", "description": "Interprets SD: the differences typically vary by about 0.074 from the true difference of 0.13" }, { "id": "large-counts-check", "description": "Checks all four: 100(0.58)=58, 100(0.42)=42, 80(0.45)=36, 80(0.55)=44 — all >= 10, so approximately normal" }, { "id": "z-score", "description": "Calculates z = (0 - 0.13) / 0.074 ≈ -1.76" }, { "id": "probability", "description": "Finds P(p-hat_1 - p-hat_2 < 0) ≈ 0.039 using Table A or technology" }, { "id": "unusual-conclusion", "description": "Concludes this IS somewhat unusual — only about 3.9% of sample pairs would show City 2 with equal or more support" } ], "commonMistakes": [ "Subtracting in wrong order (0.45 - 0.58)", "Forgetting to take the square root in the SD formula", "Only checking 2 of the 4 Large Counts products (treating it as single-proportion check)", "Not interpreting calculations in the context of cities and ballot measure support", "Forgetting to calculate the z-score before finding the probability", "Confusing which direction to shade on the normal curve" ], "worksheet": "u5_lesson6_live.html" }, "WS-U5L7-reflect1": { "elements": [ { "id": "mean-similarity", "description": "Explains that both sampling distributions have means equal to the population parameter (mu_x-bar = mu, mu_p-hat = p)" }, { "id": "sd-similarity", "description": "Explains that both SDs decrease as sample size increases — both have n in the denominator (sigma/sqrt(n) for means, sqrt(p(1-p)/n) for proportions)" }, { "id": "normality-difference", "description": "Correctly states the different normality conditions: for x-bar, population normal OR n >= 30; for p-hat, Large Counts condition (np >= 10 and n(1-p) >= 10)" }, { "id": "why-different", "description": "Offers a reasonable explanation for the different conditions, such as proportions arising from binary outcomes while sample means can originate from populations of many shapes" }, { "id": "both-unbiased", "description": "Notes that both statistics are unbiased estimators of their respective parameters" } ], "commonMistakes": [ "Applying Large Counts condition to sample means instead of CLT/normal population", "Thinking both distributions use the same normality conditions", "Not recognizing that both SDs share the pattern of decreasing with larger n", "Confusing the 10% condition (for SD formula validity) with the normality condition" ], "worksheet": "u5_lesson7_live.html" }, "WS-U5L7-reflect2": { "elements": [ { "id": "averaging-effect", "description": "Explains that when averaging multiple values, extreme highs and lows tend to cancel out, pulling the average closer to the population mean" }, { "id": "sd-formula-connection", "description": "Connects to the formula: sigma_x-bar = sigma/sqrt(n), so as n increases, the SD decreases, making the distribution narrower" }, { "id": "n50-prediction", "description": "Correctly predicts that with n = 50, P(x-bar >= 4.5) would be even smaller (much more unusual) because the SD decreases" }, { "id": "n50-calculation-or-reasoning", "description": "Provides reasoning or approximate calculation: sigma_x-bar for n=50 would be 0.5/sqrt(50) ≈ 0.071, making z = (4.5-4)/0.071 ≈ 7.07, which gives essentially 0 probability" }, { "id": "inference-connection", "description": "Connects to inference: the fact that averages are less variable is what makes them useful for making conclusions about populations" } ], "commonMistakes": [ "Thinking larger samples are MORE variable (confusing variability within a sample with variability of sample means)", "Not connecting the conceptual explanation to the formula sigma/sqrt(n)", "Thinking P(x-bar >= 4.5) would increase with larger n", "Confusing individual values with sample means" ], "worksheet": "u5_lesson7_live.html" }, "WS-U5L7-exitTicket": { "elements": [ { "id": "mean-calculation", "description": "Calculates mean = mu = 64 inches" }, { "id": "mean-interpretation", "description": "Interprets: for all random samples of 36 women, the sample mean heights will have a mean of 64 inches" }, { "id": "sd-calculation", "description": "Calculates SD = 3/sqrt(36) = 3/6 = 0.5 inches" }, { "id": "sd-interpretation", "description": "Interprets: the sample mean heights will typically vary by about 0.5 inches from the population mean of 64 inches" }, { "id": "normality-justification", "description": "States the population is normally distributed, so the sampling distribution is also normal (could also note n >= 30 applies as a second reason)" }, { "id": "z-score", "description": "Calculates z = (65 - 64) / 0.5 = 2.00" }, { "id": "probability", "description": "Finds P(x-bar >= 65) = P(z >= 2.00) ≈ 0.0228" }, { "id": "unusual-conclusion", "description": "Concludes this is unusual — only about 2.28% of samples would produce a mean of 65+ inches" } ], "commonMistakes": [ "Using sigma = 3 instead of sigma_x-bar = 0.5 when calculating the z-score", "Forgetting to take the square root of n in the SD formula", "Not interpreting in context with units (inches)", "Only citing CLT (n >= 30) without noting the population is already normal", "Forgetting to reference 'all possible samples' in interpretations", "Shading the wrong direction on the normal curve" ], "worksheet": "u5_lesson7_live.html" }, "WS-U5L8-reflect1": { "elements": [ { "id": "mean-similarity", "description": "Explains that both sampling distributions have means equal to the difference in population parameters (mu_1 - mu_2 for means, p_1 - p_2 for proportions)" }, { "id": "sd-similarity", "description": "Explains that both SD formulas add variability from the two groups under a square root — both involve adding terms (one per group) under a radical" }, { "id": "normality-difference", "description": "Correctly states that normality conditions differ: for x-bar_1 - x-bar_2, both populations normal OR both n >= 30; for p-hat_1 - p-hat_2, Large Counts condition (four checks of np >= 10)" }, { "id": "variance-addition", "description": "Notes that both SD formulas ADD components even though the statistic involves subtraction — variability accumulates for independent random variables" }, { "id": "sd-decrease-with-n", "description": "Notes that both SDs decrease as sample sizes increase" } ], "commonMistakes": [ "Applying Large Counts condition to difference in sample means instead of CLT/normal populations", "Thinking the SD formulas subtract terms because the statistic involves subtraction", "Not recognizing that both distributions share the additive variance structure", "Confusing the 10% condition (for SD formula validity) with the normality condition" ], "worksheet": "u5_lesson8_live.html" }, "WS-U5L8-reflect2": { "elements": [ { "id": "sd-decreases", "description": "Explains that increasing both sample sizes from 6 to 60 would make the SD of the sampling distribution smaller (denominators in the SD formula get larger)" }, { "id": "z-increases", "description": "Explains that a smaller SD means the same difference of 0.2 (numerator of z-score stays 1.2 - 1 = 0.2) is divided by a smaller number, making the z-score larger" }, { "id": "probability-decreases", "description": "Correctly predicts that a larger z-score means the probability P(x-bar_1 - x-bar_2 > 1.2) would decrease (smaller tail area)" }, { "id": "unusual-prediction", "description": "Concludes that with both sample sizes increased to 60, a difference of 1.2 ounces would become unusual because its tail probability becomes small" }, { "id": "formula-reasoning", "description": "References the SD formula: sqrt(sigma_1^2/n_1 + sigma_2^2/n_2) — dividing by larger n values reduces each term" } ], "commonMistakes": [ "Thinking larger samples increase variability (confusing within-sample spread with sampling distribution spread)", "Not connecting the SD change to the z-score numerator staying the same", "Thinking the probability would increase with larger samples", "Not referencing the formula to explain why SD decreases" ], "worksheet": "u5_lesson8_live.html" }, "WS-U5L8-exitTicket": { "elements": [ { "id": "mean-calculation", "description": "Calculates mean = 50 - 45 = 5 mm" }, { "id": "mean-interpretation", "description": "Interprets: for all random samples of 16 acorns from each species, the differences in sample mean lengths will have a mean of 5 mm" }, { "id": "sd-calculation", "description": "Calculates SD = sqrt(16/16 + 9/16) = sqrt(1 + 0.5625) = sqrt(1.5625) = 1.25 mm" }, { "id": "sd-interpretation", "description": "Interprets: the differences in sample mean lengths will typically vary by about 1.25 mm from the true mean difference of 5 mm" }, { "id": "normality-justification", "description": "States both population distributions are approximately normal, so the sampling distribution of x-bar_1 - x-bar_2 is also approximately normal" }, { "id": "z-score", "description": "Calculates z = (7.5 - 5) / 1.25 = 2.00" }, { "id": "probability", "description": "Finds P(x-bar_1 - x-bar_2 > 7.5) = P(z > 2.00) ≈ 0.0228" }, { "id": "unusual-conclusion", "description": "Concludes this is unusual — only about 2.28% of samples would produce a difference of 7.5+ mm" } ], "commonMistakes": [ "Using sigma values instead of sigma^2 in the SD formula (forgetting to square)", "Forgetting to take the square root at the end of the SD calculation", "Not interpreting in context with units (mm)", "Using the individual population SD instead of the SD of the sampling distribution for the z-score", "Forgetting to reference 'all possible samples' in interpretations", "Not specifying which normality condition applies" ], "worksheet": "u5_lesson8_live.html" }, "WS-U6L1-2-reflect1": { "elements": [ { "id": "identify-evidence", "description": "Identifies the evidence: p-hat = 0.60 (18/30 chose green), which is greater than the expected 0.50 if no effect" }, { "id": "two-explanations", "description": "States both explanations: (1) no real effect, result by chance alone; (2) students truly associate green with more natural" }, { "id": "simulation-purpose", "description": "Explains the simulation: coin flips (50/50) repeated 100 times with n=30 to estimate how often p-hat >= 0.60 occurs by chance" }, { "id": "conclusion-reasoning", "description": "Explains why not convincing: 16/100 is somewhat likely, so cannot rule out random chance as an explanation" }, { "id": "general-logic", "description": "Articulates the general principle: evidence is convincing only if you can eliminate random chance as a plausible explanation" } ], "commonMistakes": [ "Saying the evidence IS convincing (it is not — 16/100 is too likely)", "Confusing p-hat (0.60) with the probability from the simulation (0.16)", "Not explaining WHY 16/100 means we can't rule out chance", "Forgetting to mention that the simulation assumes no effect (p = 0.50)" ], "worksheet": "u6_lesson1-2_live.html" }, "WS-U6L1-2-reflect2": { "elements": [ { "id": "random-condition", "description": "Checks random: students were randomly selected from the school" }, { "id": "ten-percent-condition", "description": "Checks 10%: 150 <= 10% of 2,400 (i.e., 150 <= 240)" }, { "id": "large-counts-condition", "description": "Checks Large Counts: successes = 96 >= 10 and failures = 54 >= 10" }, { "id": "ci-calculation", "description": "Calculates CI: p-hat = 0.64, SE = sqrt(0.64 x 0.36/150) ≈ 0.0392, ME = 1.96 x 0.0392 ≈ 0.0769, CI ≈ (0.563, 0.717)" }, { "id": "interpretation", "description": "Interprets: We are 95% confident that the true proportion of all students at this school who have received the flu vaccine is between approximately 0.563 and 0.717" } ], "commonMistakes": [ "Checking conditions superficially without calculations (e.g., just writing 'SRS' without referencing the problem)", "Forgetting the 10% condition or not showing the calculation", "Using p instead of p-hat in the standard error formula", "Interpreting as 'there is a 95% probability that p is in the interval' instead of using confidence language", "Not mentioning the specific population in the interpretation" ], "worksheet": "u6_lesson1-2_live.html" }, "WS-U6L1-2-exitTicket": { "elements": [ { "id": "conditions-verified", "description": "Verifies all three conditions: random sample, 400 <= 10% of 50,000 (400 <= 5,000), successes = 224 >= 10 and failures = 176 >= 10" }, { "id": "ci-calculation", "description": "Calculates CI: SE = sqrt(0.56 x 0.44/400) ≈ 0.0248, ME = 1.96 x 0.0248 ≈ 0.0487, CI ≈ (0.511, 0.609)" }, { "id": "evidence-evaluation", "description": "Evaluates the claim: since the entire interval is above 0.50, there IS convincing evidence that more than half support the project" }, { "id": "proper-interpretation", "description": "Provides a proper CI interpretation: We are 95% confident that the proportion of all city residents who support the park project is between approximately 0.511 and 0.609" } ], "commonMistakes": [ "Confusing this with a significance test (using p0 instead of p-hat in SE)", "Saying there is no convincing evidence because 0.56 is close to 0.50", "Not connecting the interval bounds to the claimed value of 0.50", "Forgetting to verify conditions before calculating" ], "worksheet": "u6_lesson1-2_live.html" }, "WS-U6L10-reflect1": { "elements": [ { "id": "parameter-definitions", "description": "Defines p1 and p2 as true population cure proportions for similar patients (azithromycin and placebo groups)" }, { "id": "null-equality", "description": "States the null hypothesis as p1 = p2 or p1 - p2 = 0" }, { "id": "alternative-direction", "description": "States the alternative as p1 > p2 or p1 - p2 > 0" }, { "id": "one-sided-justification", "description": "Explains that one-sided is used because the question asks whether azithromycin is more effective" }, { "id": "parameter-language", "description": "Uses population-parameter language rather than just sample outcomes" }, { "id": "no-sample-statistics", "description": "Avoids writing hypotheses with sample statistics such as p-hat" } ], "commonMistakes": [ "Using p-hat symbols instead of population parameters", "Writing a two-sided alternative when the claim is more effective", "Reversing the inequality direction", "Leaving parameters undefined", "Stating hypotheses about sample counts instead of population proportions" ], "worksheet": "u6_lesson10_live.html" }, "WS-U6L10-reflect2": { "elements": [ { "id": "parameter-definitions", "description": "Defines pS and pB as true population proportions of residents in Soltown and Brightville who purchased sunglasses" }, { "id": "null-equality", "description": "States the null hypothesis as pS = pB or pS - pB = 0" }, { "id": "two-sided-alternative", "description": "States the alternative as pS != pB or pS - pB != 0" }, { "id": "difference-rationale", "description": "Explains that two-sided is used because the question asks whether there is a difference without direction" }, { "id": "parameter-not-statistic", "description": "Explains that hypotheses should be written using population parameters, not sample statistics" }, { "id": "equivalent-forms", "description": "May show both equivalent forms (comparison form and difference-equals-zero form)" } ], "commonMistakes": [ "Using a one-sided alternative even though the question asks for a difference", "Failing to define pS and pB", "Using sample proportions in hypotheses", "Confusing null and alternative roles", "Not connecting hypotheses to population context" ], "worksheet": "u6_lesson10_live.html" }, "WS-U6L10-exitTicket": { "elements": [ { "id": "parameter-definitions", "description": "Defines pA and pB as the true population benchmark-success proportions for students using App A and App B" }, { "id": "null-hypothesis", "description": "States H0 as pA = pB or pA - pB = 0" }, { "id": "alternative-hypothesis", "description": "States Ha as pA > pB or pA - pB > 0" }, { "id": "procedure-name", "description": "Identifies the method as a two-sample z-test for a difference in population proportions" }, { "id": "pooled-proportion", "description": "Computes pooled proportion as p-hat-c = (78 + 66)/(120 + 120) = 144/240 = 0.60" }, { "id": "expected-counts", "description": "Checks expected counts using pooled proportion: 120(0.60)=72, 120(0.40)=48 for each group, all at least 10" }, { "id": "conditions-conclusion", "description": "Concludes conditions are met by noting that the two random samples are from separate app groups and are treated as independent, citing the given 10% condition, and verifying the pooled expected counts" }, { "id": "parameter-focus", "description": "Keeps hypotheses in terms of parameters rather than sample statistics" } ], "commonMistakes": [ "Using a two-sided alternative when the claim is that App A is higher", "Forgetting to pool successes for expected-count checks", "Using incorrect pooled-proportion arithmetic", "Checking large counts with separate sample proportions instead of pooled proportion", "Not explicitly concluding whether conditions are met" ], "worksheet": "u6_lesson10_live.html" }, "WS-U6L11-reflect1": { "elements": [ { "id": "pooled-proportion", "description": "Computes the pooled proportion as (82 + 74) / (130 + 149) = 156/279 = 0.559" }, { "id": "test-statistic", "description": "States or computes the standardized test statistic as z = 2.25 using the difference in sample proportions and pooled standard error" }, { "id": "p-value", "description": "States the p-value as 0.0122" }, { "id": "right-tail-justification", "description": "Explains that the p-value is a right-tail probability because the alternative is p1 > p2" }, { "id": "p-value-interpretation", "description": "Interprets the p-value by assuming the null hypothesis is true and describing the probability of getting a difference of 0.134 or greater by chance alone in the random assignment" }, { "id": "context-language", "description": "Uses the context of azithromycin and placebo cure proportions for patients like those in the study" } ], "commonMistakes": [ "Using the separate sample proportions instead of the pooled proportion in the standard error", "Giving the wrong z statistic or p-value", "Using a two-sided or left-tail p-value when the alternative is greater than", "Interpreting the p-value as the probability that the null hypothesis is true", "Leaving the interpretation out of context" ], "worksheet": "u6_lesson11_live.html" }, "WS-U6L11-reflect2": { "elements": [ { "id": "null-assumption", "description": "Begins the interpretation by assuming the null hypothesis is true, meaning the population proportions are equal or their difference is zero" }, { "id": "two-sided-interpretation", "description": "Interprets 0.1556 as the probability of getting a sample difference of -0.037 or one more different in either direction by chance alone in the random samples" }, { "id": "compare-to-alpha", "description": "Compares 0.1556 to alpha = 0.10 and notes that the p-value is larger" }, { "id": "decision", "description": "States the decision to fail to reject H0" }, { "id": "context-conclusion", "description": "Concludes there is not convincing statistical evidence that the Soltown and Brightville population proportions differ" }, { "id": "not-proving-equality", "description": "Explains that failing to reject H0 does not prove the population proportions are equal" } ], "commonMistakes": [ "Forgetting to assume the null hypothesis is true when interpreting the p-value", "Not mentioning that the result is two-sided or in either direction", "Rejecting H0 even though 0.1556 is greater than 0.10", "Saying the data prove the two city proportions are equal", "Leaving the conclusion out of context" ], "worksheet": "u6_lesson11_live.html" }, "WS-U6L11-exitTicket": { "elements": [ { "id": "parameter-definitions", "description": "Defines pA and pB as the true population benchmark-success proportions for students using App A and App B" }, { "id": "hypotheses", "description": "States H0 as pA = pB or pA - pB = 0 and Ha as pA > pB or pA - pB > 0" }, { "id": "procedure", "description": "Identifies the method as a two-sample z-test for a difference in population proportions" }, { "id": "conditions", "description": "Verifies conditions by citing random samples, the given 10% condition, and large counts using the pooled proportion" }, { "id": "pooled-proportion", "description": "Computes the pooled proportion as (84 + 66) / (120 + 120) = 150/240 = 0.625" }, { "id": "expected-counts", "description": "Checks expected counts with pooled proportion: 120(0.625)=75 and 120(0.375)=45 in each group, all at least 10" }, { "id": "test-statistic", "description": "Computes the standardized test statistic as z = 2.40" }, { "id": "p-value", "description": "Computes or states the right-tail p-value as about 0.0082" }, { "id": "decision-and-conclusion", "description": "Compares 0.0082 to alpha = 0.05, rejects H0, and concludes there is convincing statistical evidence that App A has a higher true success proportion" }, { "id": "context-language", "description": "Keeps the conclusion in context rather than only reporting a symbolic decision" } ], "commonMistakes": [ "Using a two-sided alternative instead of a greater-than alternative", "Forgetting to use the pooled proportion in the standard error and expected counts", "Giving incorrect pooled proportion, z, or p-value calculations", "Not checking the large-count condition with expected successes and failures", "Failing to compare the p-value to alpha before stating the conclusion" ], "worksheet": "u6_lesson11_live.html" }, "WS-U6L3-reflect1": { "elements": [ { "id": "ci-interpretation", "description": "Correctly interprets CI: 'We are 95% confident that the interval from 0.558 to 0.682 captures the proportion of all adults in the city who support expanding public transit'" }, { "id": "at-least-50-claim", "description": "Correctly evaluates the 50% claim: Since ALL values in the interval (0.558 to 0.682) are above 0.50, there IS convincing evidence that at least 50% support expansion" }, { "id": "more-than-70-claim", "description": "Correctly evaluates the 70% claim: Since 0.70 is NOT in the interval (all values are below 0.682), there is NOT convincing evidence that more than 70% support expansion" }, { "id": "justification-logic", "description": "Explains the general logic: if all values are consistent with the claim, there is convincing evidence; if some are inconsistent, there is not" } ], "commonMistakes": [ "Using probability language: 'There is a 95% probability that...' instead of confidence language", "Not referencing the sample or population in the interpretation", "Saying 0.70 IS in the interval (it is not — the upper bound is 0.682)", "Confusing 'at least 50%' with 'more than 50%' (for this interval, both yield the same conclusion)", "Failing to explain WHY the interval supports or doesn't support each claim" ], "worksheet": "u6_lesson3_live.html" }, "WS-U6L3-reflect2": { "elements": [ { "id": "repeated-sampling", "description": "Explains that in repeated random sampling with the same sample size, approximately 95% of the resulting 95% CIs will capture the true population proportion" }, { "id": "not-probability", "description": "Correctly states NO — a specific CI either captures the true proportion (probability 1) or doesn't (probability 0); the 95% refers to the long-run capture rate across many intervals, not any single one" }, { "id": "increase-sample-size", "description": "Identifies increasing sample size as a way to decrease ME, explaining that n is in the denominator of the SE formula (width ∝ 1/√n)" }, { "id": "decrease-confidence-level", "description": "Identifies decreasing the confidence level as a way to decrease ME, explaining that a lower confidence level means a smaller critical value z*" }, { "id": "quadruple-rule", "description": "Mentions that you must quadruple the sample size to cut the margin of error in half" } ], "commonMistakes": [ "Saying '95% probability that the interval contains p' — this is the key misconception", "Not distinguishing between what happens across many samples vs. one specific interval", "Saying 'increase confidence level' decreases ME (it's the opposite)", "Forgetting to explain WHY larger n or lower confidence level decreases ME", "Confusing standard error with standard deviation of the population" ], "worksheet": "u6_lesson3_live.html" }, "WS-U6L3-exitTicket": { "elements": [ { "id": "define-parameter", "description": "Step 1 — Defines the parameter: p = the proportion of all songs on the playlist that are jazz" }, { "id": "identify-procedure", "description": "Step 2 — Identifies the procedure: one-sample z-interval for a population proportion" }, { "id": "check-conditions", "description": "Step 3 — Checks all three conditions: Random (randomly selected ✓), 10% (80 ≤ 10% of 1200 = 120 ✓), Large Counts (24 successes ≥ 10 and 56 failures ≥ 10 ✓)" }, { "id": "calculate-ci", "description": "Step 4 — Calculates: p-hat = 24/80 = 0.30, z* = 1.645, SE = √(0.30×0.70/80) ≈ 0.0512, ME ≈ 0.0843, CI ≈ (0.216, 0.384)" }, { "id": "interpret-ci", "description": "Step 5 — Interprets: 'We are 90% confident that the interval from approximately 0.216 to 0.384 captures the proportion of all songs on the playlist that are jazz'" }, { "id": "justify-claim", "description": "Justifies the claim: Since all values in the CI (0.216 to 0.384) are greater than 0.20, there IS convincing evidence that more than 20% of songs are jazz" } ], "commonMistakes": [ "Forgetting to define the parameter as the first step", "Using z* = 1.96 instead of 1.645 for a 90% CI", "Checking conditions superficially without showing calculations", "Using p instead of p-hat in the standard error formula", "Interpreting with probability language instead of confidence language", "Saying there is NOT convincing evidence for 20% when the entire interval is above 0.20" ], "worksheet": "u6_lesson3_live.html" }, "WS-U6L4-reflect1": { "elements": [ { "id": "define-parameter", "description": "Defines the parameter in context as p = the proportion of all teens who would prefer the new granola bar to the old version" }, { "id": "null-hypothesis", "description": "States the null hypothesis correctly as H0: p = 0.50" }, { "id": "alternative-hypothesis", "description": "States the alternative hypothesis correctly as Ha: p > 0.50" }, { "id": "one-sided-reasoning", "description": "Explains that the alternative is one-sided because the research question asks whether the proportion is higher than 50%, so evidence is only being considered in one direction" }, { "id": "equality-rule", "description": "Notes that the null contains the equality and the alternative uses a strict inequality" } ], "commonMistakes": [ "Using p-hat instead of p in the hypotheses", "Writing H0: p > 0.50 and Ha: p = 0.50", "Defining the parameter in terms of the sample rather than all teens", "Using a two-sided alternative when the claim is specifically 'higher than 50%'", "Forgetting that the null hypothesis contains the equality" ], "worksheet": "u6_lesson4_live.html" }, "WS-U6L4-reflect2": { "elements": [ { "id": "identify-procedure", "description": "Identifies the correct procedure as a one-sample z test for a population proportion" }, { "id": "random-condition", "description": "States that the random condition is met because the sample of 120 adults is selected at random" }, { "id": "ten-percent-condition", "description": "States that the 10% condition cannot be verified without the town's adult population size and that it would be met if the town has at least 1,200 adults" }, { "id": "large-counts", "description": "Checks large counts using the null value: 120(0.35) = 42 and 120(0.65) = 78, both at least 10" }, { "id": "use-p-zero", "description": "Explains that the large counts check uses 0.35 because significance tests assume the null hypothesis is true, so the check uses p0 rather than p-hat" } ], "commonMistakes": [ "Calling the procedure a confidence interval instead of a significance test", "Checking large counts with p-hat instead of p0", "Ignoring the 10% condition entirely", "Stating the sample is random without linking it to the condition", "Forgetting to calculate both n*p0 and n*(1-p0)" ], "worksheet": "u6_lesson4_live.html" }, "WS-U6L4-exitTicket": { "elements": [ { "id": "define-parameter", "description": "Defines the parameter as p = the proportion of all students at the school who would say they usually get at least 8 hours of sleep on school nights" }, { "id": "state-hypotheses", "description": "States the hypotheses correctly as H0: p = 0.65 and Ha: p < 0.65" }, { "id": "identify-procedure", "description": "Identifies the procedure as a one-sample z test for a population proportion" }, { "id": "check-random", "description": "Checks the random condition using the fact that the principal randomly selected 90 students" }, { "id": "check-ten-percent", "description": "States that the 10% condition cannot be verified from the information given and would hold if the school has at least 900 students" }, { "id": "check-large-counts", "description": "Checks large counts using the null value: 90(0.65) = 58.5 and 90(0.35) = 31.5, both at least 10" }, { "id": "proceed-and-parameter", "description": "Concludes that the test may proceed if the school has at least 900 students, since the other conditions are met, and explains that hypotheses use the population proportion p rather than the sample statistic p̂" } ], "commonMistakes": [ "Writing a two-sided or greater-than alternative instead of Ha: p < 0.65", "Using p-hat in the hypotheses", "Checking large counts with 50/90 instead of 0.65", "Forgetting to assume the school is large enough for the 10% condition", "Defining the parameter as the proportion of sampled students rather than all students at the school" ], "worksheet": "u6_lesson4_live.html" }, "WS-U6L5-reflect1": { "elements": [ { "id": "identifies-error", "description": "Identifies that the p-value is NOT the probability that H0 is true — this is a fundamental misinterpretation" }, { "id": "assumes-h0-true", "description": "Explains that the p-value assumes H0 IS true from the start (it is a conditional probability given H0 is true)" }, { "id": "correct-interpretation", "description": "Provides a correct interpretation: assuming 50% of all students would choose the green cup, there is a 0.1357 probability of getting a sample proportion of 0.60 or greater by chance alone in a random sample of 30 students" }, { "id": "as-extreme-or-more", "description": "Uses 'as extreme as or more extreme' or 'or greater' language (not just 'exactly 0.60')" }, { "id": "by-chance-alone", "description": "States that the probability describes what could occur by chance alone under the null model" } ], "commonMistakes": [ "Saying the p-value is the probability of the alternative hypothesis being true", "Giving a correct interpretation but forgetting to explain WHY the student's version is wrong", "Writing 'probability of getting exactly 0.60' instead of '0.60 or greater'", "Omitting the assumption that H0 is true from the interpretation", "Omitting 'by chance alone' from the interpretation" ], "worksheet": "u6_lesson5_live.html" }, "WS-U6L5-reflect2": { "elements": [ { "id": "identifies-football-stronger", "description": "Identifies the football study (p-value = 0.0244) as providing stronger evidence against H0" }, { "id": "smaller-pvalue-stronger", "description": "Explains that a smaller p-value means stronger evidence against H0 because the observed result would be less likely to occur by chance if H0 were true" }, { "id": "lemonade-not-unusual", "description": "Explains that the lemonade p-value of 0.1357 means the result (p-hat = 0.60) is not that unusual if H0 is true — about a 13.6% chance" }, { "id": "football-unusual", "description": "Explains that the football p-value of 0.0244 means the result is quite unusual if H0 is true — only about a 2.4% chance" }, { "id": "connects-to-z-scores", "description": "Optionally connects the comparison to the z-scores (1.10 vs -2.25) showing the football result is farther from the null value in standard deviation units" } ], "commonMistakes": [ "Thinking that a larger p-value means stronger evidence", "Simply stating 'football is stronger' without explaining why smaller p-values indicate stronger evidence", "Confusing the p-value with the probability that H0 is true", "Not connecting the p-value comparison to what would happen by chance under H0" ], "worksheet": "u6_lesson5_live.html" }, "WS-U6L5-exitTicket": { "elements": [ { "id": "correct-z-calculation", "description": "Correctly calculates z = (0.06 - 0.10) / sqrt(0.10 * 0.90 / 150) = -0.04 / 0.02449 = -1.63 (approximately)" }, { "id": "uses-p0-not-phat", "description": "Uses p0 = 0.10 (not p-hat = 0.06) in the denominator of the test statistic" }, { "id": "correct-pvalue", "description": "Calculates the p-value as P(z <= -1.63) which is approximately 0.0516 (area in the LEFT tail, since Ha is p < 0.10)" }, { "id": "one-sided-left", "description": "Recognizes this is a one-sided test (Ha: p < 0.10) and finds area in the left tail only, not both tails" }, { "id": "correct-interpretation", "description": "Interprets the p-value: Assuming 10% of all students at the university are left-handed, there is approximately a 0.0516 probability of getting a sample proportion of 0.06 or less by chance alone in a random sample of 150 students" }, { "id": "assumes-h0", "description": "Interpretation includes the assumption that H0 is true" }, { "id": "or-more-extreme", "description": "Interpretation includes 'or less' / 'or more extreme' language" }, { "id": "by-chance-alone", "description": "Interpretation includes 'by chance alone'" } ], "commonMistakes": [ "Using p-hat = 0.06 instead of p0 = 0.10 in the denominator", "Finding the right-tail area instead of the left-tail area (Ha is <, not >)", "Doubling the one-tail area (treating it as two-sided when Ha is one-sided)", "Interpreting the p-value as the probability that H0 is true", "Forgetting 'or less' / 'or more extreme' in the interpretation", "Forgetting to assume H0 is true in the interpretation" ], "worksheet": "u6_lesson5_live.html" }, "WS-U6L6-reflect1": { "elements": [ { "id": "error-accept-h0", "description": "Identifies that saying 'accept H0' is wrong — we should say 'fail to reject H0' because failing to find convincing evidence against H0 is not the same as proving it true" }, { "id": "error-h0-true", "description": "Identifies that concluding 'H0 is true' is wrong — a large p-value means the data are consistent with H0, but it does not prove H0 is true" }, { "id": "explicit-comparison", "description": "Corrected conclusion explicitly compares p-value to alpha (e.g., 'Because the p-value of 0.23 is greater than alpha = 0.05')" }, { "id": "correct-decision", "description": "Corrected conclusion says 'we fail to reject H0' (not 'we accept H0')" }, { "id": "ha-in-context", "description": "Corrected conclusion states there is NOT convincing statistical evidence for Ha (though context is generic here, the structure should reference Ha)" } ], "commonMistakes": [ "Only identifying one of the two errors (accepting H0 vs. concluding H0 is true) — these are related but distinct mistakes", "Writing a corrected conclusion that still says 'we accept H0' instead of 'we fail to reject H0'", "Forgetting to explicitly compare the p-value to alpha in the corrected conclusion", "Saying the conclusion should be about rejecting Ha instead of failing to reject H0", "Confusing 'fail to reject H0' with 'reject Ha' — the correct phrasing is always about H0" ], "worksheet": "u6_lesson6_live.html" }, "WS-U6L6-reflect2": { "elements": [ { "id": "fail-to-reject-meaning", "description": "Explains that failing to reject H0 means the data do not provide convincing evidence AGAINST H0 — the observed result could plausibly have happened by chance if H0 were true" }, { "id": "not-proof", "description": "Explains that this does not prove H0 is true — there could still be a real difference, but the sample evidence was not strong enough to detect it" }, { "id": "cereal-example", "description": "Uses the cereal example to illustrate: p-hat = 0.169 IS less than 0.20, so there is some evidence the proportion is less, but the difference could be due to random chance (p-value = 0.2676 is too large)" }, { "id": "evidence-vs-proof", "description": "Distinguishes between 'lack of convincing evidence' and 'proof' — statistics deals in probabilities, not absolute proof" }, { "id": "plausible-explanation", "description": "Notes that a 26.76% chance of getting this result under H0 means it's quite plausible the data happened by random chance alone" } ], "commonMistakes": [ "Saying that failing to reject H0 means there is no evidence at all against H0 — there was some evidence (p-hat = 0.169 < 0.20), it just wasn't convincing enough", "Saying p-hat = 0.169 proves the proportion is less than 0.20", "Not connecting the p-value of 0.2676 to the idea that the result could plausibly have happened by chance", "Saying 'we can never prove anything in statistics' without explaining what we CAN conclude (we can have convincing evidence)", "Confusing the p-value with the probability that H0 is true" ], "worksheet": "u6_lesson6_live.html" }, "WS-U6L6-exitTicket": { "elements": [ { "id": "correct-hypotheses", "description": "States H0: p = 0.60 and Ha: p < 0.60" }, { "id": "defines-parameter", "description": "Defines p as the proportion of all residents who support the new park (or equivalent)" }, { "id": "explicit-comparison-05", "description": "Part (b): Explicitly compares p-value of 0.1357 to alpha = 0.05 (e.g., 'Because the p-value of 0.1357 is greater than alpha = 0.05')" }, { "id": "fail-to-reject-05", "description": "Part (b): Concludes we fail to reject H0" }, { "id": "conclusion-in-context", "description": "Part (b): States there is not convincing statistical evidence that the proportion of all residents who support the new park is less than 0.60" }, { "id": "alpha-20-changes", "description": "Part (c): Recognizes that with alpha = 0.20, the conclusion changes because 0.1357 < 0.20, so we would reject H0" }, { "id": "alpha-20-conclusion", "description": "Part (c): With alpha = 0.20, states there IS convincing statistical evidence that the proportion is less than 0.60" } ], "commonMistakes": [ "Writing Ha: p != 0.60 instead of Ha: p < 0.60 (the council member suspects it is LESS than 0.60)", "Forgetting to define the parameter p", "Not explicitly comparing the p-value to alpha — just jumping to 'fail to reject'", "Saying 'accept H0' instead of 'fail to reject H0'", "Thinking the conclusion stays the same for alpha = 0.20 — it changes because 0.1357 < 0.20", "Forgetting to state the conclusion about Ha in context (must mention the park and residents)" ], "worksheet": "u6_lesson6_live.html" }, "WS-U6L7-reflect1": { "elements": [ { "id": "type1-context", "description": "Describes a Type I error in context: concluding that more than 50% of students prefer the new lunch option when the true proportion is actually 0.50" }, { "id": "type2-context", "description": "Describes a Type II error in context: failing to find convincing evidence that more than 50% prefer the new lunch option when the true proportion really is greater than 0.50" }, { "id": "uses-context", "description": "States the errors in the actual lunch-option context rather than only giving abstract definitions" }, { "id": "consequential-choice", "description": "Identifies which error is more consequential in this context and gives a reasonable justification tied to real consequences" }, { "id": "ha-language", "description": "Frames the conclusion around the alternative hypothesis being true or not supported, not around 'proving' hypotheses" } ], "commonMistakes": [ "Swapping the two errors so Type I is described as failing to reject H0 or Type II is described as rejecting a true H0", "Forgetting that the null value here is p = 0.50, so a Type I error means the true proportion is exactly 0.50", "Writing generic definitions without describing the lunch preference scenario", "Saying an error is more consequential without explaining the practical impact", "Claiming one error is always more consequential in every context" ], "worksheet": "u6_lesson7_live.html" }, "WS-U6L7-reflect2": { "elements": [ { "id": "power-meaning", "description": "Explains in context that if the true proportion who would choose the green cup is 0.64, there is an 0.80 probability of finding convincing evidence that more than 50% of students would choose the green cup" }, { "id": "beta-value", "description": "Finds the probability of a Type II error as 1 - 0.80 = 0.20" }, { "id": "increase-power", "description": "Names at least one valid way to increase power, such as increasing sample size, increasing alpha, or decreasing standard error" }, { "id": "not-h0-probability", "description": "Does not confuse power with the probability that H0 is false or the probability that the study is correct" }, { "id": "factor-explanation", "description": "Briefly explains why the named change increases the chance of rejecting a false H0" } ], "commonMistakes": [ "Saying power is the probability that H0 is false", "Saying the Type II error probability is the same as power instead of 1 minus power", "Interpreting power without mentioning the specific true value p = 0.64", "Listing a factor that does not increase power", "Forgetting to connect the interpretation to finding convincing evidence for more than 50%" ], "worksheet": "u6_lesson7_live.html" }, "WS-U6L7-exitTicket": { "elements": [ { "id": "correct-hypotheses", "description": "States H0: p = 0.40 and Ha: p > 0.40" }, { "id": "type1-context", "description": "Describes a Type I error in context: concluding that more than 40% of high school students exercise at least 5 days per week when the true proportion is actually 0.40" }, { "id": "type2-context", "description": "Describes a Type II error in context: failing to find convincing evidence that more than 40% exercise at least 5 days per week when the true proportion really is greater than 0.40" }, { "id": "power-interpretation", "description": "Interprets the power in context: if the true proportion is 0.55, there is a 0.70 probability of finding convincing evidence that more than 40% exercise at least 5 days per week" }, { "id": "type1-probability", "description": "Finds the probability of a Type I error as alpha = 0.01" }, { "id": "type2-probability", "description": "Finds the probability of a Type II error as 1 - 0.70 = 0.30" }, { "id": "two-power-factors", "description": "Gives two valid changes that would increase power, such as increasing sample size, increasing alpha, decreasing standard error, or having the true parameter farther from the null, with brief explanation" } ], "commonMistakes": [ "Writing Ha: p != 0.40 instead of Ha: p > 0.40", "Describing Type I and Type II errors without using the exercise context", "Saying the probability of a Type II error is 0.70 instead of 0.30", "Interpreting power without mentioning the specific true proportion p = 0.55", "Giving only one factor that increases power when the question asks for two", "Naming a change without explaining how it makes rejecting a false H0 more likely" ], "worksheet": "u6_lesson7_live.html" }, "WS-U6L8-reflect1": { "elements": [ { "id": "procedure-name", "description": "Identifies the correct procedure as a two-sample z-interval for a difference in proportions" }, { "id": "independence", "description": "Explains that the data come from two independent random samples, one from each park" }, { "id": "ten-percent", "description": "States that the 10% condition requires Park A to contain at least 2,100 trees and Park B at least 1,900 trees; the word 'large' makes this plausible but does not provide exact totals" }, { "id": "large-counts", "description": "Checks the large counts condition using 42, 168, 30, and 160, and notes that all are at least 10" }, { "id": "conclusion", "description": "Concludes that the two-sample z-interval is appropriate provided the stated 10% population-size requirements hold" }, { "id": "context-language", "description": "Uses the park and leaf-damage context rather than only giving abstract definitions" } ], "commonMistakes": [ "Naming a one-sample interval instead of a two-sample z-interval for a difference in proportions", "Forgetting to check the 10% condition when the setting uses random samples", "Checking only the successes and not the failures", "Saying the conditions are met without supporting the claim with the actual counts", "Giving only generic definitions and not referring to the two parks" ], "worksheet": "u6_lesson8_live.html" }, "WS-U6L8-reflect2": { "elements": [ { "id": "interval-interpretation", "description": "Interprets the interval in context: we are 95% confident that the true proportion of similar dogs that would get ticks with the new formula is between 0.0343 and 0.2907 lower than with the old formula" }, { "id": "parameter-context", "description": "Refers to the true proportions of similar dogs that would get ticks with the two formulas, not just the sample results" }, { "id": "negative-values-meaning", "description": "Explains that the negative values mean p_new - p_old is likely less than 0, so the new formula likely leads to a lower tick rate than the old formula" }, { "id": "direction-language", "description": "States the direction correctly by describing the new formula as lower or better at preventing ticks" }, { "id": "not-probability", "description": "Does not say there is a 95% probability that the parameter is in the interval" }, { "id": "percentage-points", "description": "May express the interpretation in percentage points lower" } ], "commonMistakes": [ "Saying the interval is about sample proportions instead of population proportions", "Reversing the subtraction and claiming the old formula is lower", "Saying the interval means there is a 95% chance the parameter is in this one computed interval", "Ignoring the context of dogs getting ticks", "Treating the negative values as if they are impossible instead of meaningful for a difference" ], "worksheet": "u6_lesson8_live.html" }, "WS-U6L8-exitTicket": { "elements": [ { "id": "procedure-name", "description": "Identifies the correct procedure as a two-sample z-interval for a difference in proportions" }, { "id": "independence", "description": "Explains that the data come from two independent random samples, one from each high school" }, { "id": "ten-percent", "description": "Checks the 10% condition by noting that each sample of 150 is no more than 10% of each school's population because each school has far more than 1,500 students" }, { "id": "large-counts", "description": "Checks the large counts condition using 60, 90, 39, and 111, and notes that all are at least 10" }, { "id": "point-estimate", "description": "Finds the point estimate p-hat_A - p-hat_B = 0.40 - 0.26 = 0.14" }, { "id": "confidence-interval", "description": "Calculates a 95% confidence interval approximately equal to (0.035, 0.245), allowing for reasonable rounding" }, { "id": "interpretation", "description": "Interprets the interval in context: we are 95% confident that the proportion who buy lunch at High School A is about 3.5 to 24.5 percentage points higher than at High School B" }, { "id": "difference-meaning", "description": "Notes that the interval is entirely positive, suggesting High School A likely has the higher true lunch-buying proportion" }, { "id": "calculation-details", "description": "May show intermediate work such as the standard error or margin of error" } ], "commonMistakes": [ "Using a one-sample procedure instead of a two-sample z-interval for a difference in proportions", "Forgetting the 10% condition even though the data come from random samples", "Checking only the numbers who bought lunch and not the numbers who did not", "Computing the point estimate incorrectly or reversing A - B", "Giving an interval but not interpreting it in context", "Ignoring that the interval is entirely positive when describing the conclusion" ], "worksheet": "u6_lesson8_live.html" }, "WS-U6L9-reflect1": { "elements": [ { "id": "interval-interpretation", "description": "Interprets the interval in context: we are 90% confident that the true difference high minus low in the proportions of trees that died from the disease is between -0.029 and 0.079" }, { "id": "parameter-context", "description": "Refers to the true population proportions of trees in the two forests or elevations, not just the sample proportions" }, { "id": "zero-plausible", "description": "Explains that 0 is in the interval, so 0 is a plausible value for the true difference" }, { "id": "claim-conclusion", "description": "Concludes that there is not convincing evidence that the disease is more lethal at one elevation" }, { "id": "direction-language", "description": "Uses the stated direction high minus low correctly" }, { "id": "positive-negative-mention", "description": "May note that the interval includes both negative and positive values" } ], "commonMistakes": [ "Talking only about the sample proportions instead of the population proportions", "Saying there is a 90% probability that the true difference is in this particular interval", "Ignoring the fact that 0 is inside the interval", "Claiming the interval proves one elevation is more lethal", "Reversing the direction of high minus low" ], "worksheet": "u6_lesson9_live.html" }, "WS-U6L9-reflect2": { "elements": [ { "id": "interval-interpretation", "description": "Interprets the interval in context: we are 95% confident that the true proportion of similar dogs that would get ticks with the new formula is between 0.0343 and 0.2907 lower than with the old formula" }, { "id": "parameter-context", "description": "Refers to the true proportions of similar dogs that would get ticks with the two formulas" }, { "id": "all-negative", "description": "Explains that all values in the interval are negative, so p_new - p_old is less than 0" }, { "id": "claim-supported", "description": "Concludes that there is convincing evidence that the new formula is better at preventing ticks than the old formula" }, { "id": "direction-language", "description": "States the direction correctly by describing the new formula as lower or better, not higher" }, { "id": "percentage-points", "description": "May describe the interval as a range of percentage points lower" } ], "commonMistakes": [ "Reversing new minus old and saying the old formula is better", "Describing only the sample results instead of the population parameter", "Ignoring that every value in the interval is negative", "Saying negative values are impossible rather than meaningful for a difference", "Treating the confidence interval as a probability statement about this one interval" ], "worksheet": "u6_lesson9_live.html" }, "WS-U6L9-exitTicket": { "elements": [ { "id": "interval-interpretation", "description": "Interprets the confidence interval in context: we are 96% confident that the true difference day minus night in the proportions of all parts produced within specifications is between -0.016 and 0.096" }, { "id": "parameter-context", "description": "Refers to the population proportions for all parts from the day and night shifts, not just the sampled parts" }, { "id": "zero-plausible", "description": "Explains that 0 is in the interval, so 0 is a plausible value for the true difference" }, { "id": "not-significant", "description": "Concludes that there is not convincing evidence that the true difference is significantly different from 0" }, { "id": "confidence-level", "description": "Correctly interprets the 96% confidence level using repeated random sampling and many intervals" }, { "id": "repeated-samples-details", "description": "Specifies that many pairs of random samples of 200 parts from each shift would be taken and about 96% of the resulting intervals would capture the true difference" }, { "id": "not-probability", "description": "Does not say there is a 96% probability that the true difference is in this one interval" } ], "commonMistakes": [ "Saying there is a 96% probability that the true difference is in this interval", "Ignoring that 0 is included in the interval", "Claiming the shifts are significantly different because the endpoints are not equal", "Describing only the sample proportions instead of the population proportions", "Leaving out the repeated-sampling language when interpreting the confidence level" ], "worksheet": "u6_lesson9_live.html" }, "WS-U7L1-reflect1": { "elements": [ { "id": "evidence-statistic", "description": "Identifies the evidence as the observed positive difference in sample means" }, { "id": "observed-value", "description": "States that the observed difference was $12.49, or notes that 22.04 is greater than 9.55" }, { "id": "expected-zero", "description": "May explain that if wording made no difference, the expected difference in means would be 0" }, { "id": "chance-explanation", "description": "Describes explanation 1 as no real wording effect with the observed difference caused by chance variation in random assignment" }, { "id": "treatment-explanation", "description": "Describes explanation 2 as the bonus wording causing students to spend more on average" }, { "id": "context", "description": "Keeps the explanation in the context of the bonus vs. rebate college student study" } ], "commonMistakes": [ "Treating the evidence as a conclusion instead of the observed difference in means", "Leaving out that the expected difference is 0 if wording makes no difference", "Describing only one explanation instead of both", "Ignoring chance variation in random assignment", "Not connecting the answer to bonus vs. rebate spending" ], "worksheet": "u7_lesson1_live.html" }, "WS-U7L1-reflect2": { "elements": [ { "id": "assumption", "description": "States that the simulation assumes the wording makes no difference" }, { "id": "rerandomize", "description": "Explains that the subjects are randomly reassigned or rerandomized into groups many times" }, { "id": "calculate-statistic", "description": "Explains that each trial computes the difference in means" }, { "id": "many-trials", "description": "Notes that the process is repeated many times, with 1000 trials shown in the lesson" }, { "id": "tail-count", "description": "States that 13 out of 1000 trials were 12.49 or greater, or gives the probability 0.013" }, { "id": "unlikely-chance", "description": "Explains that 0.013 is a very small probability under chance alone" }, { "id": "conclusion", "description": "Concludes that the chance-alone explanation is unlikely and the data provide convincing evidence for the bonus wording effect" } ], "commonMistakes": [ "Forgetting that the simulation assumes no wording effect", "Describing sampling new students instead of rerandomizing the existing subjects", "Not mentioning the difference in means as the statistic", "Treating 0.013 as a large probability", "Failing to use the small probability to justify the conclusion" ], "worksheet": "u7_lesson1_live.html" }, "WS-U7L1-exitTicket": { "elements": [ { "id": "evidence", "description": "States that the evidence is the observed difference in means of $12.49, with the bonus group spending more on average" }, { "id": "expected-zero", "description": "May note that the expected difference would be 0 if wording made no difference" }, { "id": "chance-explanation", "description": "Describes one explanation as no wording effect with chance variation in random assignment producing the observed difference" }, { "id": "effect-explanation", "description": "Describes the other explanation as the bonus wording causing higher average spending" }, { "id": "simulation-probability", "description": "Uses the simulation result 13 out of 1000 or 0.013 as the estimated probability by chance alone" }, { "id": "small-probability", "description": "Explains that this probability is very small or unlikely" }, { "id": "convincing-evidence", "description": "Concludes that the data provide convincing evidence that the bonus wording causes college students like those in the study to spend more on average" } ], "commonMistakes": [ "Calling the evidence just 'the claim' instead of the observed difference in means", "Only giving one explanation for the observed difference", "Ignoring the 13 out of 1000 simulation result", "Not explaining why a small probability matters", "Giving a conclusion without context about students like those in the study" ], "worksheet": "u7_lesson1_live.html" }, "WS-U7L2-reflect1": { "elements": [ { "id": "procedure", "description": "Identifies a one-sample t-interval for a population mean" }, { "id": "single-mean", "description": "Explains that the goal is to estimate the mean of one population" }, { "id": "random-condition", "description": "States that the 10 bags were randomly selected" }, { "id": "ten-percent", "description": "States that it is reasonable that 10 bags is less than or equal to 10% of all bags from the wholesaler" }, { "id": "small-sample-shape", "description": "Notes that n = 10 is less than 30, so a graph must be checked for shape" }, { "id": "shape-evidence", "description": "Uses the boxplot evidence that there is no strong skewness or outliers" }, { "id": "conditions-met", "description": "Concludes that the conditions are met" } ], "commonMistakes": [ "Calling the procedure a z interval instead of a t interval", "Failing to mention that the sample is random", "Skipping the 10% condition", "Saying n = 10 is automatically large enough without checking shape", "Not using graph-based evidence for no strong skewness or outliers" ], "worksheet": "u7_lesson2_live.html" }, "WS-U7L2-reflect2": { "elements": [ { "id": "sample-stats", "description": "States the sample statistics x-bar = 906.8, s = 8.22, and n = 10" }, { "id": "degrees-freedom", "description": "Computes or states df = 9" }, { "id": "critical-value", "description": "States the 95% critical value t* = 2.262" }, { "id": "formula", "description": "Uses the confidence interval form x-bar plus or minus t* times s over square root of n" }, { "id": "margin-error", "description": "States or supports that the margin of error is 5.88" }, { "id": "interval", "description": "Gives the interval 900.92 to 912.68" }, { "id": "context-interpretation", "description": "Interprets the interval in context for the mean weight of all bags from the wholesaler" }, { "id": "confidence-language", "description": "May explicitly say we are 95% confident the true mean lies between the bounds" } ], "commonMistakes": [ "Using a z critical value instead of a t critical value", "Using the wrong degrees of freedom", "Leaving out the margin of error step", "Giving the bounds without interpreting the interval in context", "Interpreting the interval as applying to individual bags rather than the population mean" ], "worksheet": "u7_lesson2_live.html" }, "WS-U7L2-exitTicket": { "elements": [ { "id": "procedure", "description": "Identifies the procedure as a one-sample t-interval for a population mean" }, { "id": "random-sample", "description": "States that the crabs came from a random sample" }, { "id": "ten-percent", "description": "States that it is reasonable that 40 crabs is no more than 10% of all fiddler crabs of this species" }, { "id": "large-sample", "description": "Uses n = 40 to justify the large-sample shape condition" }, { "id": "degrees-freedom", "description": "Computes or states df = 39" }, { "id": "critical-value", "description": "States the 90% technology critical value t* = 1.685" }, { "id": "formula", "description": "Uses x-bar plus or minus t* times s over square root of n" }, { "id": "margin-error", "description": "Finds a margin of error of about 1.76" }, { "id": "interval", "description": "Gives an interval of about 65.89 to 69.41 scoops per 30 seconds" }, { "id": "context-interpretation", "description": "Interprets the interval as a confidence interval for the mean feeding rate of all fiddler crabs of this species" }, { "id": "table-note", "description": "May note that Table B would use df = 30 and t* = 1.697 if technology were unavailable" } ], "commonMistakes": [ "Using a z interval instead of a t interval", "Not using n = 40 to justify the shape condition", "Using the wrong degrees of freedom", "Forgetting to calculate or report the interval bounds", "Not interpreting the interval as the population mean feeding rate" ], "worksheet": "u7_lesson2_live.html" }, "WS-U7L3-reflect1": { "elements": [ { "id": "inside-plausible", "description": "Explains that a claimed mean inside the interval is a plausible value for the population mean" }, { "id": "no-evidence-against", "description": "States that if the claimed value is inside the interval, there is not convincing evidence against that claim" }, { "id": "sugar-interval", "description": "Uses the powdered sugar example to explain that 907 is inside its confidence interval (exact endpoints are not required)" }, { "id": "sugar-conclusion", "description": "Concludes there is not convincing evidence that the powdered sugar bags are underfilled" }, { "id": "all-above-logic", "description": "Explains that if the entire interval is above the comparison value, the data support a greater-than claim" }, { "id": "unit-conversion", "description": "Notes that 2 scoops per second must be converted to 60 scoops per 30 seconds" }, { "id": "crab-example", "description": "Uses the fiddler crab example to explain that its interval is entirely above 60 scoops per 30 seconds (exact endpoints are not required)" }, { "id": "crab-conclusion", "description": "Concludes there is convincing evidence the mean feeding rate is greater than 2 scoops per second" } ], "commonMistakes": [ "Saying a value inside the interval proves the claim is true instead of saying it is plausible", "Claiming there is convincing evidence against the powdered sugar claim even though 907 is inside the interval", "Forgetting to convert 2 scoops per second to 60 scoops per 30 seconds", "Using only one example instead of explaining both interval situations", "Failing to connect the conclusion to the population mean in context" ], "worksheet": "u7_lesson3_live.html" }, "WS-U7L3-reflect2": { "elements": [ { "id": "repeated-sampling", "description": "Interprets confidence level using repeated random sampling with the same sample size" }, { "id": "context", "description": "States that many random samples of 10 people would be used to build 95% confidence intervals for the mean human body temperature of all people" }, { "id": "capture-rate", "description": "States that about 95% of those intervals would capture the population mean" }, { "id": "not-probability", "description": "Explains that 95% is not the probability that one completed interval contains mu" }, { "id": "sample-size-effect", "description": "States that increasing sample size decreases the margin of error" }, { "id": "confidence-effect", "description": "States that decreasing the confidence level decreases the critical value and the margin of error" }, { "id": "precision", "description": "May note that a smaller margin of error makes the interval narrower or more precise" }, { "id": "quadruple-rule", "description": "May note that quadrupling the sample size cuts the margin of error in half" } ], "commonMistakes": [ "Saying there is a 95% chance that the one computed interval contains mu", "Leaving out repeated random sampling", "Forgetting to mention the sample size of 10 in the body temperature context", "Saying increasing confidence level makes the interval narrower", "Saying larger sample sizes increase margin of error" ], "worksheet": "u7_lesson3_live.html" }, "WS-U7L3-exitTicket": { "elements": [ { "id": "procedure", "description": "Identifies the procedure as a one-sample t-interval for a population mean" }, { "id": "random-condition", "description": "States that the data came from a random sample of 23 crows" }, { "id": "ten-percent", "description": "States that 23 is reasonably less than 10% of all crows in the region" }, { "id": "shape-condition", "description": "Uses the stemplot evidence of no strong skewness or outliers" }, { "id": "sample-stats", "description": "Uses the sample statistics x-bar = 4.90, s = 1.12, and n = 23" }, { "id": "degrees-freedom", "description": "Computes or states df = 22" }, { "id": "critical-value", "description": "States the 95% critical value t* = 2.074" }, { "id": "formula", "description": "Uses the one-sample t-interval form x-bar plus or minus t* times s over square root of n" }, { "id": "margin-error", "description": "States or supports that the margin of error is 0.484" }, { "id": "interval", "description": "Gives the interval 4.416 to 5.384 ppm" }, { "id": "interpretation", "description": "Interprets the interval in context for the mean lead level of all crows in the region" }, { "id": "claim-logic", "description": "Explains that because the entire interval is below 6.0 ppm, there is convincing evidence the population mean lead level is less than 6.0 ppm" } ], "commonMistakes": [ "Using a z interval instead of a t interval", "Leaving out one of the required conditions", "Using the wrong degrees of freedom or critical value", "Interpreting the interval for individual crows instead of the population mean", "Failing to explain that 6.0 is outside the interval and therefore not plausible for the mean" ], "worksheet": "u7_lesson3_live.html" }, "WS-U7L4-reflect1": { "elements": [ { "id": "got-hops-parameter", "description": "Defines mu for Got Hops as the mean vertical jump for all students at the high school" }, { "id": "got-hops-null", "description": "States the Got Hops null hypothesis as H0: mu = 15 inches" }, { "id": "got-hops-alt", "description": "States the Got Hops alternative hypothesis as Ha: mu != 15 inches because the question asks whether the mean differs from 15" }, { "id": "null-equality", "description": "Explains that the null hypothesis uses equality and represents no difference or no change" }, { "id": "alt-inequality", "description": "Explains that the alternative hypothesis uses a strict inequality and represents the claim the data are meant to support" }, { "id": "two-sided", "description": "Identifies the Got Hops test as two-sided because the question asks whether the mean differs" }, { "id": "tread40-parameter", "description": "Defines mu for Tread40 as the mean mileage for all Tread40 tires" }, { "id": "tread40-hypotheses", "description": "States H0: mu = 40000 and Ha: mu > 40000 for the tire example" }, { "id": "one-sided", "description": "Explains that the tire test is one-sided because the question asks whether the mean is more than 40000 miles" }, { "id": "no-xbar", "description": "May note that hypotheses should use mu rather than x-bar" } ], "commonMistakes": [ "Writing x-bar instead of mu in the hypotheses", "Using an equals sign in the alternative hypothesis", "Calling the Got Hops test one-sided even though the question asks whether the mean differs", "Using Ha: mu != 40000 for the tire example instead of Ha: mu > 40000", "Failing to define the population mean in context" ], "worksheet": "u7_lesson4_live.html" }, "WS-U7L4-reflect2": { "elements": [ { "id": "procedure", "description": "Identifies the correct procedure as a one-sample t-test for a population mean" }, { "id": "why-procedure", "description": "Explains that this procedure is used when testing a claim about one population mean from one sample of quantitative data" }, { "id": "random-condition", "description": "States that the data should come from a random sample or randomized experiment" }, { "id": "ten-percent", "description": "States that if sampling without replacement, the sample size should be less than 10% of the population" }, { "id": "shape-condition", "description": "States that either n is at least 30 or, if n is smaller, the sample data should show no strong skewness or outliers" }, { "id": "graph-needed", "description": "Explains that when n is less than 30, a graph of the sample data should be shown as evidence" }, { "id": "jump-example", "description": "Uses the vertical jump example to say the conditions were met because the sample was random, 20 is less than 10% of the school, and the graph showed no strong skewness or outliers" }, { "id": "tablet-example", "description": "Uses the tablet example to say the conditions were not met because there was no random sample and the dotplot showed strong skewness with a potential outlier" } ], "commonMistakes": [ "Naming a z-test instead of a one-sample t-test", "Forgetting the 10% condition when sampling without replacement", "Saying a graph is optional for small samples", "Claiming the tablet example meets the conditions", "Ignoring the role of strong skewness or outliers" ], "worksheet": "u7_lesson4_live.html" }, "WS-U7L4-exitTicket": { "elements": [ { "id": "parameter", "description": "Defines mu as the mean mileage for all Tread40 tires" }, { "id": "null", "description": "States the null hypothesis as H0: mu = 40000 miles" }, { "id": "alternative", "description": "States the alternative hypothesis as Ha: mu > 40000 miles" }, { "id": "procedure", "description": "Identifies the procedure as a one-sample t-test for a population mean" }, { "id": "random", "description": "Uses the fact that the data came from a random sample of 35 tires" }, { "id": "ten-percent", "description": "States that 35 is reasonably less than 10% of all Tread40 tires" }, { "id": "large-sample", "description": "Explains that n = 35 is at least 30, so the large-sample condition for shape is satisfied" }, { "id": "one-sided", "description": "Explains that the test is one-sided because the question asks whether the mean is more than 40000 miles" }, { "id": "no-xbar", "description": "Explains that x-bar should not appear because hypotheses are about the population parameter mu, not the sample statistic" } ], "commonMistakes": [ "Using Ha: mu != 40000 instead of Ha: mu > 40000", "Writing x-bar = 40000 in the null hypothesis", "Omitting the large-sample condition from n = 35", "Forgetting to justify the 10% condition", "Calling the test two-sided even though the question asks whether the mean is greater than 40000" ], "worksheet": "u7_lesson4_live.html" }, "WS-U7L5-reflect1": { "elements": [ { "id": "general-formula", "description": "Explains that the standardized test statistic is statistic minus parameter divided by standard error" }, { "id": "t-formula", "description": "States the one-sample t-test formula t = (x-bar - mu0) / (s / sqrt(n))" }, { "id": "degrees-of-freedom", "description": "States that the degrees of freedom are n - 1" }, { "id": "got-hops-t", "description": "May report t = 1.535 with 19 degrees of freedom for Got Hops" }, { "id": "got-hops-p", "description": "Uses Got Hops as a two-sided P-value example; the exact decimal is optional" }, { "id": "tread40-t", "description": "May report t = 6.491 with 34 degrees of freedom for Tread40" }, { "id": "tread40-p", "description": "Uses Tread40 as a right-tail P-value example; the exact decimal is optional" }, { "id": "tail-direction", "description": "Connects the direction of the alternative hypothesis to the tail area used for the p-value" }, { "id": "technology-tableb", "description": "May mention that technology or Table B can be used to find or approximate the p-value" } ], "commonMistakes": [ "Using a z-test instead of a one-sample t-test", "Forgetting that the degrees of freedom are n - 1", "Using one tail for Got Hops even though the alternative is two-sided", "Doubling the Tread40 tail probability even though the alternative is one-sided", "Not connecting the alternative hypothesis to the p-value direction" ], "worksheet": "u7_lesson5_live.html" }, "WS-U7L5-reflect2": { "elements": [ { "id": "assume-null", "description": "States that a p-value is interpreted assuming the null hypothesis is true" }, { "id": "chance-alone", "description": "Explains that the p-value is the probability of getting a sample mean or test statistic as extreme or more extreme by chance alone" }, { "id": "got-hops-interpretation", "description": "Interprets the Got Hops p-value of 0.1412 in context as getting a sample mean jump as extreme as 15.8 inches in either direction if mu = 15" }, { "id": "got-hops-decision", "description": "Compares 0.1412 to alpha = 0.05 and concludes fail to reject H0 for Got Hops" }, { "id": "got-hops-conclusion", "description": "States there is not convincing statistical evidence that the mean vertical jump for all students at the school differs from 15 inches" }, { "id": "small-large-pvalues", "description": "Explains that small p-values lead toward rejecting H0 while large p-values lead toward failing to reject H0" }, { "id": "tread40-decision", "description": "Compares 9.99 x 10^-8 to alpha = 0.01 and concludes reject H0 for Tread40" }, { "id": "tread40-conclusion", "description": "States there is convincing statistical evidence that the mean mileage for Tread40 tires is greater than 40000 miles" }, { "id": "no-accept-null", "description": "May note that we reject or fail to reject H0 rather than accept it" } ], "commonMistakes": [ "Saying the p-value is the probability that H0 is true", "Writing accept H0 instead of fail to reject H0", "Rejecting H0 for Got Hops even though 0.1412 is greater than 0.05", "Failing to reject H0 for Tread40 even though the p-value is far below 0.01", "Leaving the conclusion out of context" ], "worksheet": "u7_lesson5_live.html" }, "WS-U7L5-exitTicket": { "elements": [ { "id": "paired-data", "description": "Recognizes that this is a matched-pairs setting analyzed with one sample of differences" }, { "id": "parameter", "description": "Defines mu_D as the true mean difference in cooked weight (with seasoning minus without seasoning) for packages like those in the study" }, { "id": "null", "description": "States the null hypothesis as H0: mu_D = 0" }, { "id": "alternative", "description": "States the alternative hypothesis as Ha: mu_D > 0 because the claim is that seasoning helps bacon retain more weight" }, { "id": "procedure", "description": "Identifies the procedure as a one-sample t-test for mu_D or for the mean difference" }, { "id": "random-assignment", "description": "Explains that the random assignment condition is met" }, { "id": "no-ten-percent", "description": "Explains that there is no 10% condition because this was not a random sample without replacement" }, { "id": "shape", "description": "Uses the graph of differences to say there is no strong skewness or outliers" }, { "id": "decision", "description": "Uses p = 0.0199 and alpha = 0.05 to reject H0" }, { "id": "conclusion", "description": "States there is convincing statistical evidence that the seasoning causes bacon to retain more weight on average for packages like those in the study" }, { "id": "alpha-default", "description": "May mention that alpha = 0.05 is used because no significance level was given" } ], "commonMistakes": [ "Treating the data as two independent samples instead of matched pairs", "Defining the parameter as mu instead of mu_D or forgetting the direction with minus without", "Claiming a 10% condition is required for this randomized experiment", "Ignoring the graph of differences for the small sample", "Failing to reject H0 even though 0.0199 is less than 0.05" ], "worksheet": "u7_lesson5_live.html" }, "WS-U7L6-reflect1": { "elements": [ { "id": "procedure-match", "description": "Explains that two groups with quantitative data and a confidence interval for a difference in population means leads to a 2-sample t-interval" }, { "id": "independence", "description": "States that independence requires two independent random samples or a randomized experiment" }, { "id": "ten-percent", "description": "States that when sampling without replacement, each sample should be no more than 10% of its respective population" }, { "id": "normal-condition", "description": "Explains that both sample sizes must be at least 30, or otherwise both sample distributions should be free from extreme skewness and outliers" }, { "id": "spider-example", "description": "Uses the spider example to note random samples of 14 females and 14 males and that the histograms were roughly unimodal and symmetric" }, { "id": "slow-internet-assignment", "description": "Uses the slow-internet example to explain that random assignment satisfies the first condition" }, { "id": "slow-internet-shape", "description": "Explains that the slow-internet example fails because of clear skewness and an outlier, and that you cannot just combine the groups into 36" }, { "id": "large-sample-variation", "description": "May mention that if both spider samples were 35, the normal condition would be easier because both sample sizes exceed 30" } ], "commonMistakes": [ "Using a 1-sample t-interval instead of a 2-sample t-interval", "Forgetting that each sample must satisfy the 10% condition separately", "Combining 18 and 18 into 36 to justify normality", "Thinking random selection is required even when there is random assignment", "Ignoring the skewness and outlier in the slow-internet example" ], "worksheet": "u7_lesson6_live.html" }, "WS-U7L6-reflect2": { "elements": [ { "id": "general-formula", "description": "States the general confidence interval form statistic plus or minus critical value times standard error" }, { "id": "two-sample-formula", "description": "States or describes the 2-sample t-interval formula using the difference in sample means and the standard error sqrt((s1^2 / n1) + (s2^2 / n2))" }, { "id": "point-estimate", "description": "Identifies the point estimate as the difference in sample means and gives 10.043 mm for female minus male" }, { "id": "technology-df", "description": "Explains that the degrees of freedom are found with technology and that the calculator uses pooled set to no" }, { "id": "interval-values", "description": "Gives the interval (7.956, 12.129) or equivalent values from the example" }, { "id": "critical-value-or-moe", "description": "Includes t* about 2.13 or the margin of error 2.086 mm" }, { "id": "interpretation", "description": "Interprets the interval in context as plausible values for how much longer female Argiope spiders are on average than male Argiope spiders" }, { "id": "positive-interval", "description": "Recognizes that because the interval is entirely positive, females are estimated to be larger on average than males" }, { "id": "equivalent-form", "description": "May mention that the interval can also be written as 10.043 +/- 2.086" } ], "commonMistakes": [ "Using a z interval instead of a t interval", "Forgetting the square root standard error formula for two means", "Using n - 1 for the degrees of freedom instead of technology", "Turning pooled on even though the lesson says to keep pooled set to no", "Interpreting the interval as a probability that the parameter is in the interval" ], "worksheet": "u7_lesson6_live.html" }, "WS-U7L6-exitTicket": { "elements": [ { "id": "parameter", "description": "Defines the parameter as mu_F - mu_M, the true difference in mean body lengths of female and male Argiope spiders" }, { "id": "procedure", "description": "Identifies the correct procedure as a 2-sample t-interval for the difference in population means" }, { "id": "independence", "description": "Explains that the independence condition is met because the data came from two random samples" }, { "id": "ten-percent", "description": "Explains that each sample of 14 spiders is less than or equal to 10% of its respective population" }, { "id": "shape-check-needed", "description": "Explains that the shape check is needed because both sample sizes are less than 30" }, { "id": "shape-result", "description": "States that the histograms were roughly unimodal and symmetric with no extreme skewness or outliers, so the normal condition is reasonable" }, { "id": "interpretation", "description": "Interprets the 95% confidence interval as plausible values for the true difference female minus male in mean body length" }, { "id": "contextual-values", "description": "States that female Argiope spiders are estimated to be about 7.956 to 12.129 millimeters longer on average than male Argiope spiders" }, { "id": "positive-endpoints", "description": "Explains that the positive endpoints suggest females tend to be larger on average than males" }, { "id": "pooled-no", "description": "May mention that technology should be used with pooled set to no" } ], "commonMistakes": [ "Defining the parameter as a single mean instead of mu_F - mu_M", "Using a 1-sample procedure instead of a 2-sample t-interval", "Forgetting that both sample sizes are under 30, so a shape check is needed", "Saying the confidence interval proves the result rather than giving plausible values", "Failing to connect the positive interval to females being larger on average" ], "worksheet": "u7_lesson6_live.html" }, "WS-U7L7-reflect1": { "elements": [ { "id": "general-template", "description": "States that we are C% confident that the interval captures the true population difference being estimated" }, { "id": "spider-context", "description": "Uses the spider context and identifies the parameter as the true difference in mean body lengths of female and male Argiope spiders" }, { "id": "direction", "description": "Includes the subtraction direction female minus male or explains that female spiders are 7.956 to 12.129 millimeters larger on average" }, { "id": "order-matters", "description": "Explains that reversing the subtraction order negates the interval and changes the interpretation" }, { "id": "sample-error", "description": "Explains that an interpretation should not refer to the sample or use wording like these spiders" }, { "id": "probability-error", "description": "Explains that a 95% confidence level is not a 95% chance that the fixed parameter is in this interval" }, { "id": "alternate-interpretation", "description": "May give the alternate valid wording that females are between 7.956 and 12.129 millimeters larger than males" }, { "id": "reverse-interval", "description": "May mention the reversed interval (-12.129, -7.956) for male minus female" } ], "commonMistakes": [ "Referring to the sample instead of the population parameter", "Saying there is a 95% chance the parameter is in the interval", "Leaving out the subtraction order", "Ignoring that reversing the subtraction negates the interval", "Describing the interval as the difference in sample means rather than true means" ], "worksheet": "u7_lesson7_live.html" }, "WS-U7L7-reflect2": { "elements": [ { "id": "zero-means-no-difference", "description": "Explains that 0 represents no difference between the two population means" }, { "id": "spider-support", "description": "Explains that 0 is not in the spider interval, so 0 is not plausible and the interval supports the claim that female spiders are larger on average" }, { "id": "sign-matches-claim", "description": "Connects the positive spider interval to a directional claim that female minus male is greater than 0" }, { "id": "firestation-no-support", "description": "Explains that 0 is in the fire-station interval, so no difference is plausible and the interval does not support the council member's belief of different mean response times" }, { "id": "contextual-conclusion", "description": "Makes both conclusions in context rather than as abstract rules only" }, { "id": "do-not-say-wrong", "description": "Notes that when 0 is in the interval you should not say the council member is wrong, only that the interval does not support the claim" }, { "id": "restaurant-connection", "description": "Explains that 0 is in the restaurant interval (-9.3, 3.2), so no difference is plausible and there is not convincing evidence of a difference between foam and plastic container mean temperatures" }, { "id": "plausible-language", "description": "Uses the language of plausible or not plausible values for the parameter" } ], "commonMistakes": [ "Forgetting that 0 means no difference", "Claiming that 0 in the interval proves the means are equal", "Saying the council member is wrong instead of saying the interval does not support the claim", "Ignoring the sign of the spider interval when making the directional conclusion", "Giving conclusions without context" ], "worksheet": "u7_lesson7_live.html" }, "WS-U7L7-exitTicket": { "elements": [ { "id": "parameter", "description": "Defines the parameter as mu_N - mu_S, the true difference in mean response times for calls from the northern and southern fire stations" }, { "id": "procedure", "description": "Identifies the correct procedure as a 2-sample t-interval for a difference in population means" }, { "id": "random-samples", "description": "Explains that the independence condition is met because there are random samples from both fire stations" }, { "id": "ten-percent", "description": "States that each sample of 50 calls is reasonably no more than 10% of all calls for its station" }, { "id": "large-sample", "description": "Explains that both sample sizes are greater than 30, so the sampling distribution is approximately normal by the Central Limit Theorem" }, { "id": "interval-interpretation", "description": "Interprets the interval in context as plausible values for the true difference northern minus southern in mean response times, between about -2.37 and 0.37 minutes" }, { "id": "zero-plausible", "description": "Explains that because 0 is in the interval, no difference is plausible" }, { "id": "claim-conclusion", "description": "Concludes that the interval does not support the council member's belief that the stations have different mean response times" }, { "id": "confidence-level", "description": "Interprets the 95% confidence level in repeated sampling language, saying about 95% of such intervals would capture the true parameter" }, { "id": "do-not-say-wrong", "description": "May note that you should not say the council member is wrong or that the stations are definitely the same" } ], "commonMistakes": [ "Defining the parameter as a single mean instead of mu_N - mu_S", "Using a 1-sample interval instead of a 2-sample t-interval", "Forgetting to justify the 10% condition for each station", "Saying the interval proves the stations are equal because 0 is in the interval", "Interpreting the confidence level as the probability that this interval contains the parameter" ], "worksheet": "u7_lesson7_live.html" }, "WS-U7L8-reflect1": { "elements": [ { "id": "define-parameters", "description": "Defines mu_Y and mu_N as the true population mean number of yawns for the exposed and non-exposed groups" }, { "id": "null-no-effect", "description": "Explains that the null hypothesis represents no treatment effect, no difference, or no change" }, { "id": "null-equality", "description": "States a correct null hypothesis such as mu_Y = mu_N or mu_Y - mu_N = 0" }, { "id": "alternative-directional", "description": "States a correct directional alternative such as mu_Y > mu_N or mu_Y - mu_N > 0" }, { "id": "yawn-more-context", "description": "Connects the alternative to the contextual claim that people yawn more on average when watching someone yawn" }, { "id": "parameters-not-statistics", "description": "Explains that hypotheses must use population parameters rather than sample statistics like x-bar values" }, { "id": "words-or-symbols", "description": "May give both a verbal and symbolic version of the hypotheses" }, { "id": "skeptic-language", "description": "May explicitly describe the skeptic as shrugging off the claim or not expecting a treatment effect" } ], "commonMistakes": [ "Writing the null hypothesis with an inequality instead of equality", "Using sample means or other statistics in the hypotheses", "Failing to define mu_Y and mu_N in context", "Using a non-directional alternative when the claim is specifically that yawning increases", "Leaving the hypotheses disconnected from the population means" ], "worksheet": "u7_lesson8_live.html" }, "WS-U7L8-reflect2": { "elements": [ { "id": "one-sided-definition", "description": "Explains that a one-sided alternative is used when the claim is directional and asks whether one mean is larger or smaller" }, { "id": "yawning-direction", "description": "Explains that the yawning study uses a one-sided alternative because the claim is that exposure increases the mean number of yawns" }, { "id": "not-two-sided-yawning", "description": "States that it would not make sense to use the opposite inequality or a two-sided alternative for the yawning study" }, { "id": "two-sided-definition", "description": "Explains that a two-sided alternative is used when the question is simply whether the means are different" }, { "id": "textbook-example", "description": "Uses the chemistry-vs.-physics textbook example as a two-sided test because the question asks whether the mean word lengths differ" }, { "id": "reverse-order", "description": "Explains that reversing the subtraction order reverses the direction of a one-sided inequality" }, { "id": "not-equal-symbol", "description": "May mention that a two-sided alternative uses the not-equal-to symbol" }, { "id": "context-controls-direction", "description": "May explicitly say that context determines the direction of the alternative hypothesis" } ], "commonMistakes": [ "Using a two-sided alternative when the claim has a specific direction", "Using a one-sided alternative when the question only asks whether the means differ", "Forgetting that reversing subtraction order also reverses the inequality", "Ignoring the role of context in choosing the alternative", "Claiming that the same inequality works after reversing the group order" ], "worksheet": "u7_lesson8_live.html" }, "WS-U7L8-exitTicket": { "elements": [ { "id": "define-mu-c", "description": "Defines mu_C as the true mean word length for the chemistry textbook population of words" }, { "id": "define-mu-p", "description": "Defines mu_P as the true mean word length for the physics textbook population of words" }, { "id": "null-correct", "description": "States a correct null hypothesis such as mu_C = mu_P or mu_C - mu_P = 0" }, { "id": "alternative-correct", "description": "States a correct two-sided alternative such as mu_C not equal mu_P or mu_C - mu_P not equal 0" }, { "id": "two-sided-reason", "description": "Explains that the alternative should be two-sided because the question asks whether the mean word lengths differ without specifying a direction" }, { "id": "parameters-not-samples", "description": "Explains that hypotheses should use population parameters rather than the sample means 5.71 and 6.03" }, { "id": "consistent-order", "description": "Uses a consistent subtraction order between the null and alternative hypotheses" }, { "id": "statistics-descriptive", "description": "May note that the sample means are descriptive statistics from the samples, not the parameters being tested" } ], "commonMistakes": [ "Using the sample means 5.71 and 6.03 directly in the hypotheses", "Writing a one-sided alternative when the question asks whether the means differ", "Failing to define mu_C and mu_P in context", "Writing the null hypothesis without equality", "Mixing subtraction order between the null and alternative hypotheses" ], "worksheet": "u7_lesson8_live.html" }, "WS-U7L9-reflect1": { "elements": [ { "id": "correct-hypotheses", "description": "States correct hypotheses such as mu_C - mu_P = 0 and mu_C - mu_P not equal 0" }, { "id": "random-condition", "description": "Checks or explains the random condition for each sample" }, { "id": "independence-condition", "description": "Checks or explains the independence or 10% condition" }, { "id": "large-sample-condition", "description": "Checks or explains that the sample sizes are large enough or the distributions are approximately normal" }, { "id": "test-statistic-setup", "description": "Explains that the test statistic compares the observed sample difference to 0 using the standard error of x-bar_C minus x-bar_P" }, { "id": "observed-difference", "description": "Uses or identifies the observed sample difference as -0.32" }, { "id": "test-statistic-value", "description": "Gives an approximate test statistic of about -0.97 and/or interprets the negative sign correctly" }, { "id": "p-value", "description": "Gives an approximate two-sided P-value of about 0.33" }, { "id": "decision-conclusion", "description": "States fail to reject H0 at alpha = 0.05 and concludes there is not convincing evidence that the mean word lengths differ" }, { "id": "standard-error-language", "description": "May explicitly name the denominator as the estimated standard error" }, { "id": "statistical-significance", "description": "May note that the result is not statistically significant at the 5% level" } ], "commonMistakes": [ "Using a one-sided alternative even though the question asks whether the means differ", "Forgetting to check the 10% condition or the large-sample condition", "Using the sample means alone without describing the test statistic setup", "Giving the wrong decision by rejecting H0 even though the P-value is about 0.33", "Writing a conclusion that claims the means are proven equal" ], "worksheet": "u7_lesson9_live.html" }, "WS-U7L9-reflect2": { "elements": [ { "id": "p-value-meaning", "description": "Explains that the P-value is the probability, assuming the null hypothesis is true, of getting a result at least as extreme as the one observed" }, { "id": "two-sided-reason", "description": "Explains that the test is two-sided because the question asks whether the mean word lengths differ without naming a direction" }, { "id": "negative-sign", "description": "Explains that the negative test statistic means the chemistry sample mean was lower than the physics sample mean because the subtraction was chemistry minus physics" }, { "id": "compare-to-alpha", "description": "Explains that the P-value is greater than 0.05, so the result is not statistically significant and we fail to reject H0" }, { "id": "not-proof-equal", "description": "Explains that failing to reject H0 does not prove the population means are equal; it only means there is not convincing evidence of a difference" }, { "id": "contextual-conclusion", "description": "States the conclusion in context about the population mean word lengths of chemistry and physics textbooks" }, { "id": "reverse-order", "description": "May mention that reversing the subtraction order would reverse the sign of the test statistic but not the two-sided P-value" }, { "id": "sampling-language", "description": "May emphasize that the sample evidence was not strong enough to rule out random variation" } ], "commonMistakes": [ "Describing the P-value as the probability that H0 is true", "Saying the test is one-sided when the question only asks whether the means differ", "Ignoring what the negative sign means for the chosen subtraction order", "Claiming that fail to reject H0 proves the two population means are equal", "Leaving the conclusion disconnected from chemistry and physics textbook word lengths" ], "worksheet": "u7_lesson9_live.html" }, "WS-U7L9-exitTicket": { "elements": [ { "id": "null-hypothesis", "description": "States a correct null hypothesis such as mu_C - mu_P = 0" }, { "id": "alternative-hypothesis", "description": "States a correct two-sided alternative such as mu_C - mu_P not equal 0" }, { "id": "conditions", "description": "Checks the needed conditions, including random sampling, the 10% condition or independence, and a large sample or roughly normal distributions" }, { "id": "test-statistic", "description": "Gives or describes a test statistic of about -0.97 using the difference in sample means and the standard error" }, { "id": "p-value", "description": "Gives an approximate two-sided P-value of about 0.33" }, { "id": "decision", "description": "States the correct decision to fail to reject H0 at alpha = 0.05" }, { "id": "context-conclusion", "description": "Writes a conclusion in context that there is not convincing evidence that the textbook mean word lengths differ" }, { "id": "not-proof-equal", "description": "Explains that failing to reject H0 does not prove the population means are equal" }, { "id": "negative-sign-meaning", "description": "May explain that the negative sign comes from chemistry minus physics being negative in the sample" }, { "id": "not-significant", "description": "May note that the result is not statistically significant at the 5% level" } ], "commonMistakes": [ "Using a one-sided alternative instead of a two-sided one", "Skipping the conditions entirely or only naming one of them", "Reporting the wrong P-value or rejecting H0 at alpha = 0.05", "Writing a conclusion that says the textbooks definitely have the same mean word length", "Forgetting to state the conclusion in context" ], "worksheet": "u7_lesson9_live.html" }, "WS-U8L1-reflect1": { "elements": [ { "id": "expected-counts", "description": "States that a fair 10-sided die gives expected counts of 10 in each category" }, { "id": "cancellation-problem", "description": "Explains that Obs - Exp values sum to 0 or cancel, so adding them is not a useful overall measure" }, { "id": "squaring-reason", "description": "Explains that the differences are squared so the contributions are positive and larger discrepancies get more weight" }, { "id": "divide-by-expected", "description": "Explains that dividing by expected count makes the discrepancy relative to what was expected" }, { "id": "chi-square-formula", "description": "Describes chi-square as the sum of (Obs - Exp)^2 / Exp across categories" }, { "id": "chi-square-value", "description": "Identifies the observed chi-square statistic as 4.2" }, { "id": "larger-chi-square", "description": "Explains that a larger chi-square value means a bigger mismatch or poorer fit between observed and expected counts" }, { "id": "absolute-values-issue", "description": "May mention that absolute values still depend heavily on sample size" }, { "id": "squared-sum", "description": "May mention that the sum of squared differences was 42 before dividing by expected counts" } ], "commonMistakes": [ "Saying the observed minus expected differences can just be added directly", "Leaving out why the differences are squared", "Leaving out why each squared difference is divided by the expected count", "Reporting the wrong chi-square value instead of 4.2", "Treating a larger chi-square value as a better fit rather than a worse fit" ], "worksheet": "u8_lesson1_live.html" }, "WS-U8L1-reflect2": { "elements": [ { "id": "fair-die-model", "description": "States that the simulation assumes the die is fair and rolls 100 outcomes under that model" }, { "id": "sampling-distribution", "description": "Explains that the dotplot shows the sampling distribution of chi-square values from many simulated samples" }, { "id": "p-value-estimate", "description": "Explains that the P-value is found from the proportion of simulated chi-square values at least as large as 4.2" }, { "id": "approximate-p-value", "description": "Gives the approximate P-value as about 0.9110 or 932/1023" }, { "id": "not-surprising", "description": "Explains that such a result is not surprising under the fair-die model because it happens over 90% of the time" }, { "id": "contextual-conclusion", "description": "Concludes that the results are consistent with random variation and there is not convincing evidence that the die is weighted" }, { "id": "good-fit-language", "description": "May describe the observed counts as a good fit to the expected counts" }, { "id": "large-p-value-meaning", "description": "May explicitly say that a large P-value means the observed statistic is typical rather than unusual under the model" } ], "commonMistakes": [ "Describing the P-value as the probability that the die is fair", "Forgetting that the simulation assumes the fair-die model is true", "Leaving out that the P-value counts chi-square values at least as large as 4.2", "Calling 0.911 a small P-value or saying the result is statistically significant", "Concluding that the die is weighted even though the simulated result is very common" ], "worksheet": "u8_lesson1_live.html" }, "WS-U8L1-exitTicket": { "elements": [ { "id": "expected-count", "description": "States that the expected count is 10 for each of the 10 outcomes under a fair die" }, { "id": "obs-exp-problem", "description": "Explains that adding Obs - Exp is not useful because the positive and negative differences cancel" }, { "id": "chi-square-computation", "description": "Describes chi-square as adding (Obs - Exp)^2 / Exp across categories" }, { "id": "observed-statistic", "description": "Identifies the observed chi-square statistic as 4.2" }, { "id": "simulation-distribution", "description": "Explains that the simulation distribution is made from many chi-square values from fair-die samples of 100 rolls" }, { "id": "p-value", "description": "Estimates the P-value as about 0.9110 or 932/1023 by counting simulated values at least as large as 4.2" }, { "id": "conclusion", "description": "Concludes that the result is not surprising and there is not convincing evidence that the die is weighted" }, { "id": "random-variation", "description": "Explains that the large P-value means random variation is a very reasonable explanation for the observed counts" }, { "id": "good-fit", "description": "May describe the observed counts as a good fit to the fair-die model" }, { "id": "extreme-weighting", "description": "May note that larger chi-square values would indicate more evidence against the fair-die model" } ], "commonMistakes": [ "Using the observed counts as the expected counts", "Forgetting that Obs - Exp sums to 0 because of cancellation", "Leaving out the division by expected count in the chi-square statistic", "Interpreting 0.911 as evidence against the fair die", "Claiming the die is weighted even though the result is very common under the fair-die model" ], "worksheet": "u8_lesson1_live.html" }, "WS-U8L2-reflect1": { "elements": [ { "id": "expected-counts", "description": "Explains that expected counts come from multiplying the hypothesized proportions (0.452, 0.292, 0.256) by the sample size of 40" }, { "id": "contributions", "description": "Describes contributions as (Obs − Exp)²/Exp for each category, measuring relative discrepancy" }, { "id": "chi-square-value", "description": "Identifies the observed chi-square statistic as 7.746" }, { "id": "chi-square-meaning", "description": "Explains that chi-square measures the distance between observed and expected counts relative to expected counts" }, { "id": "df-calculation", "description": "States that degrees of freedom = number of categories minus 1, so df = 3 − 1 = 2" }, { "id": "df-not-sample-size", "description": "Notes that chi-square df are based on number of categories, not sample size (unlike t-distributions)" }, { "id": "df-shape-effect", "description": "Explains that as degrees of freedom increase the chi-square distribution becomes less right-skewed" }, { "id": "non-negative", "description": "May mention that chi-square values are always non-negative" }, { "id": "simulation-result", "description": "May mention that only about 2.2% of simulated samples had chi-square ≥ 7.746" } ], "commonMistakes": [ "Saying degrees of freedom depend on sample size like t-distributions", "Forgetting to explain that contributions divide by expected count to make discrepancies relative", "Reporting the wrong chi-square value", "Saying the chi-square distribution becomes more right-skewed as df increase (opposite is true)", "Using equal proportions (1/3 each) instead of the census-based proportions" ], "worksheet": "u8_lesson2_live.html" }, "WS-U8L2-reflect2": { "elements": [ { "id": "null-hypothesis", "description": "States that the null hypothesis says the proportions for each category equal specified values" }, { "id": "alternative-hypothesis", "description": "States that the alternative says at least one proportion is not as specified in the null" }, { "id": "alternative-in-words", "description": "Explains that the alternative must be stated in words because directional inequalities do not apply with multiple categories" }, { "id": "procedure-identification", "description": "Identifies chi-square GOF test as appropriate for one categorical variable with one sample" }, { "id": "random-condition", "description": "States data must come from a random sample or randomized experiment" }, { "id": "ten-percent-condition", "description": "States the sample must be ≤ 10% of the population when sampling without replacement" }, { "id": "large-counts-condition", "description": "States that all expected counts must be at least 5" }, { "id": "expected-count-formula", "description": "May explain that expected counts are found by multiplying null proportions by sample size" }, { "id": "parameters-not-statistics", "description": "May note that hypotheses are about population parameters, never sample statistics like p-hat" } ], "commonMistakes": [ "Writing the alternative with directional inequalities (< or >) instead of words", "Stating the alternative as if all proportions are different rather than at least one", "Using sample proportions (p-hat) in the hypotheses instead of population parameters", "Forgetting the expected-count condition or using an incorrect threshold", "Confusing chi-square GOF (one variable, one sample) with chi-square independence (two variables)" ], "worksheet": "u8_lesson2_live.html" }, "WS-U8L2-exitTicket": { "elements": [ { "id": "null-hypothesis", "description": "States H₀: p₁ = p₂ = p₃ = p₄ = 0.25 or equivalent wording that all quadrants have equal proportions" }, { "id": "alternative-hypothesis", "description": "States Hₐ in words: at least one proportion is not 0.25 or the distribution is not equal across quadrants" }, { "id": "procedure", "description": "Identifies chi-square goodness-of-fit test because there is one categorical variable (quadrant) and one sample" }, { "id": "random-condition", "description": "Checks that a random sample of 100 Battleship players was selected" }, { "id": "ten-percent", "description": "States 100 ≤ 10% of all Battleship players (at least 1000 exist)" }, { "id": "expected-counts", "description": "Calculates expected counts as 0.25 × 100 = 25 for each quadrant and notes all are > 5" }, { "id": "degrees-of-freedom", "description": "States df = 4 − 1 = 3 for four categories" }, { "id": "distribution-shape", "description": "Describes the chi-square distribution with 3 df as right-skewed with only non-negative values" }, { "id": "larger-chi-square", "description": "Explains that a larger chi-square value means observed counts deviate more from expected, indicating a poorer fit to the equal-proportions model" }, { "id": "define-parameters", "description": "May define p₁ through p₄ as the proportions of players placing most ships in each quadrant" } ], "commonMistakes": [ "Stating the alternative with inequalities instead of words", "Using 100 as the expected count instead of 25", "Forgetting to check the 10% condition", "Calculating df from sample size instead of number of categories", "Saying a larger chi-square means a better fit (opposite is true)" ], "worksheet": "u8_lesson2_live.html" }, "WS-U8L3-reflect1": { "elements": [ { "id": "null-model", "description": "States the hypothesized Dallas household proportions 0.452, 0.292, and 0.256 for the three income brackets" }, { "id": "expected-counts", "description": "Gives expected counts 18.08, 11.68, and 10.24 for the sample of 40" }, { "id": "chi-square", "description": "Explains that the category contributions (Observed - Expected)^2 / Expected are summed and reports chi-square = 7.746" }, { "id": "degrees-freedom", "description": "States df = 3 - 1 = 2" }, { "id": "p-value", "description": "Reports the upper-tail p-value of about 0.0208" }, { "id": "decision-conclusion", "description": "Because 0.0208 < 0.05, rejects H0 and concludes there is convincing evidence that the distribution of predatory lending businesses is not the same as the specified Dallas household proportions" }, { "id": "largest-contribution", "description": "Identifies the $100,000-and-above bracket as having the largest contribution" }, { "id": "practical-meaning", "description": "Explains that predatory lending businesses were underrepresented in the highest-income areas relative to the null model" } ], "commonMistakes": [ "Using equal proportions instead of 0.452, 0.292, and 0.256", "Omitting division by expected count in the contributions", "Using df = 3 instead of 2", "Failing to reject H0 even though 0.0208 < 0.05", "Choosing the category with the largest observed count instead of the largest contribution" ], "worksheet": "u8_lesson3_live.html" }, "WS-U8L3-reflect2": { "elements": [ { "id": "p-value-template", "description": "Explains that a P-value is the probability, assuming H0 is true, of a chi-square result as large as or larger than the observed result" }, { "id": "predatory-interpretation", "description": "Interprets 0.0208 as the chance of chi-square 7.746 or greater under the specified Dallas household-proportion model" }, { "id": "predatory-decision", "description": "Compares 0.0208 to 0.05 and rejects H0" }, { "id": "predatory-conclusion", "description": "Concludes there is convincing evidence that the predatory-lending-business distribution differs from the specified household proportions" }, { "id": "battleship-interpretation", "description": "Interprets 0.079 as the chance of a result this extreme or more extreme if quadrant preferences are equally distributed" }, { "id": "battleship-decision", "description": "Compares 0.079 to 0.05 and fails to reject H0" }, { "id": "battleship-conclusion", "description": "Concludes there is not convincing evidence that Battleship players' quadrant preferences differ from equal proportions" } ], "commonMistakes": [ "Saying the p-value is the probability that the null hypothesis is true", "Omitting as-extreme-or-more-extreme language", "Failing to reject H0 for predatory lending", "Rejecting H0 for Battleship", "Claiming that failing to reject proves equal quadrant preferences" ], "worksheet": "u8_lesson3_live.html" }, "WS-U8L3-exitTicket": { "elements": [ { "id": "hypotheses", "description": "States H0 with the six specified district proportions and Ha that at least one district proportion differs" }, { "id": "parameter-definition", "description": "May define the parameters as the population participation proportions for districts A through F" }, { "id": "procedure", "description": "Identifies the procedure as a chi-square goodness-of-fit test for one sample and one categorical variable with six categories" }, { "id": "conditions", "description": "Checks the simple random sample condition, the 10% condition for n = 240, and that all expected counts are greater than 5" }, { "id": "expected-counts", "description": "Uses or reports the expected counts 76.8, 28.8, 24, 64.8, 12, and 33.6" }, { "id": "mechanics", "description": "Reports chi-square = 47.48, df = 5, and p-value = 4.53 x 10^-9 or approximately 0" }, { "id": "decision-and-conclusion", "description": "Rejects H0 at alpha = 0.05 and concludes that the district participation proportions are not the same as those used by the city" }, { "id": "district-d", "description": "Identifies District D as having the greatest change because it has the largest contribution, about 28.27" }, { "id": "sample-size", "description": "May mention that the total sample size is 240 households" }, { "id": "full-test-structure", "description": "May organize the answer into hypotheses, procedure/conditions, mechanics, and conclusion" } ], "commonMistakes": [ "Using df = 6 instead of 5", "Omitting the specific null proportions or failing to define the parameters", "Skipping the expected counts when checking conditions", "Failing to reject even though the p-value is essentially 0", "Choosing the district with the largest observed count instead of the district with the largest contribution" ], "worksheet": "u8_lesson3_live.html" }, "WS-U8L4-reflect1": { "elements": [ { "id": "no-relationship", "description": "States that expected counts are calculated assuming there is no relationship between year and school type" }, { "id": "totals", "description": "Identifies the row totals 429, 37, and 68, the column totals 320 and 214, and the table total 534" }, { "id": "formula", "description": "Uses or states the formula expected count = (row total * column total) / table total" }, { "id": "public-2019", "description": "Gives the expected count for Public in 2019 as 257.1, or shows the calculation (429 * 320) / 534" }, { "id": "private-2019", "description": "Gives the expected count for Private/Parochial/Charter in 2019 as about 22.2" }, { "id": "subtraction-shortcut", "description": "Explains that subtraction with row totals or column totals can be used once some expected counts are known" }, { "id": "remaining-counts", "description": "Identifies the remaining expected counts as about 40.7, 171.9, 14.8, and 27.3" }, { "id": "no-total-cells", "description": "May note that totals do not get expected counts" }, { "id": "key-parentheses", "description": "May note that a key should identify expected counts if they are shown in parentheses" } ], "commonMistakes": [ "Using the observed count in the cell instead of the row total or column total in the formula", "Forgetting to divide by the table total", "Assuming the expected counts should all be equal instead of using the no-relationship model", "Leaving out the subtraction shortcut for the remaining cells", "Treating totals cells as if they also need expected counts" ], "worksheet": "u8_lesson4_live.html" }, "WS-U8L4-reflect2": { "elements": [ { "id": "no-relationship", "description": "States that expected counts are based on the assumption of no relationship between education level and employment status" }, { "id": "totals", "description": "Identifies the row totals 1940 and 60, the column totals 220, 570, and 1210, and the table total 2000" }, { "id": "formula", "description": "Uses or states the formula expected count = (row total * column total) / table total" }, { "id": "first-expected-count", "description": "Gives the expected count for Employed and No High School Diploma as 213.4" }, { "id": "second-expected-count", "description": "Gives the expected count for Employed and High School Diploma No College as 552.9" }, { "id": "remaining-counts", "description": "Identifies the remaining expected counts as 1173.7, 6.6, 17.1, and 36.3" }, { "id": "subtraction-shortcut", "description": "Explains that subtraction from row totals or column totals can be used to find the remaining cells after calculating a few expected counts" }, { "id": "no-total-cells", "description": "Explains that totals do not get expected counts because they are already fixed sums of the interior cells" }, { "id": "variable-placement", "description": "May mention that education level is in the columns and employment status is in the rows" } ], "commonMistakes": [ "Using 206 or 548 directly in the expected-count formula instead of the row and column totals", "Mixing observed counts with expected counts", "Forgetting that the table total is 2000", "Not using subtraction to finish the table after the first calculations", "Claiming that totals should also have expected counts" ], "worksheet": "u8_lesson4_live.html" }, "WS-U8L4-exitTicket": { "elements": [ { "id": "row-and-table-totals", "description": "Finds the row totals as 1940 and 60 and the table total as 2000" }, { "id": "assumption", "description": "States that expected counts are calculated assuming no relationship between education level and employment status" }, { "id": "formula", "description": "Uses the formula expected count = (row total * column total) / table total" }, { "id": "first-two-expected-counts", "description": "Calculates the first two expected counts as 213.4 and 552.9" }, { "id": "remaining-four-counts", "description": "Finds the remaining expected counts as 1173.7, 6.6, 17.1, and 36.3" }, { "id": "totals-explanation", "description": "Explains that totals do not get expected counts because they are already the row and column sums of the interior cells" }, { "id": "key-explanation", "description": "Explains that a key helps the reader know that numbers in parentheses are expected counts" }, { "id": "column-totals", "description": "May explicitly restate the column totals 220, 570, and 1210" }, { "id": "subtraction-language", "description": "May explicitly say that subtraction preserves the row totals and column totals" } ], "commonMistakes": [ "Getting the row totals wrong by not adding across correctly", "Using the wrong denominator instead of the table total 2000", "Stopping after the first two expected counts and not finishing the table", "Saying totals need expected counts", "Not explaining the purpose of the key for expected counts in parentheses" ], "worksheet": "u8_lesson4_live.html" }, "WS-U8L5-reflect1": { "elements": [ { "id": "goal-matters", "description": "Explains that the choice of test depends on the goal of the study and how the data were collected" }, { "id": "homogeneity-purpose", "description": "States that homogeneity is used to compare distributions of one categorical variable across multiple populations or treatment groups" }, { "id": "homogeneity-collection", "description": "States that homogeneity uses independent random samples from each population or multiple groups in a randomized experiment" }, { "id": "independence-purpose", "description": "States that independence is used to decide whether two categorical variables are associated" }, { "id": "independence-collection", "description": "States that independence uses one random sample and measures both variables on each individual" }, { "id": "school-example", "description": "Identifies the school-type-by-year example as a chi-square test for homogeneity because there were separate random samples from 2019 and 2020" }, { "id": "employment-example", "description": "Identifies the education-level-by-employment example as a chi-square test for independence because one random sample of 2000 adults was used" }, { "id": "two-way-table", "description": "May mention that both situations are summarized in a two-way table of categorical data" } ], "commonMistakes": [ "Choosing homogeneity just because there are two variables instead of because distributions are compared across populations or groups", "Choosing independence even when separate random samples were taken from different populations", "Leaving out the role of data collection when deciding the procedure", "Mixing up the school example and the employment example", "Failing to mention that independence uses one random sample with both variables measured on each person" ], "worksheet": "u8_lesson5_live.html" }, "WS-U8L5-reflect2": { "elements": [ { "id": "homogeneity-hypotheses", "description": "States that for homogeneity the null says no difference in distributions and the alternative says there is a difference in distributions" }, { "id": "independence-hypotheses", "description": "States that for independence the null says no association or that the variables are independent, and the alternative says there is an association or that the variables are not independent" }, { "id": "no-parameter", "description": "Explains that these chi-square hypotheses do not define a single numerical parameter" }, { "id": "many-sided", "description": "Explains that the homogeneity alternative is many-sided because several category proportions could be higher or lower" }, { "id": "design-condition", "description": "States that homogeneity uses independent random samples from the populations or groups, or multiple groups from a randomized experiment, while independence uses one random sample" }, { "id": "ten-percent", "description": "States the 10% condition when sampling without replacement" }, { "id": "expected-counts", "description": "States that all expected counts must be at least 5" }, { "id": "expected-not-observed", "description": "States that expected counts, not observed counts, are checked" }, { "id": "school-minimum", "description": "May mention that the smallest expected count in the school example is 14.8" }, { "id": "conditions-met", "description": "May explicitly conclude that the conditions are met in both lesson examples" } ], "commonMistakes": [ "Writing hypotheses about a single proportion or mean instead of about distributions or association", "Forgetting that no parameter is defined in these chi-square hypotheses", "Leaving out the 10% condition for sampling without replacement", "Checking observed counts instead of expected counts", "Saying only some expected counts need to be above 5 rather than all of them" ], "worksheet": "u8_lesson5_live.html" }, "WS-U8L5-exitTicket": { "elements": [ { "id": "school-test", "description": "Identifies the school study as a chi-square test for homogeneity because it compares school-type distributions across two years using separate random samples" }, { "id": "school-hypotheses", "description": "States H0 as no difference in school-type distributions between years and Ha as a difference in distributions" }, { "id": "employment-test", "description": "Identifies the education-by-employment study as a chi-square test for independence because two categorical variables were measured on one random sample" }, { "id": "employment-hypotheses", "description": "States H0 as no association/independence between education and employment status and Ha as an association/not independent" }, { "id": "design-condition", "description": "States the appropriate random-sampling or randomized-experiment condition: independent random samples or treatment groups for homogeneity, and one random sample for independence" }, { "id": "ten-percent", "description": "States the 10% condition when sampling without replacement" }, { "id": "large-counts", "description": "States that all expected cell counts must be at least 5" }, { "id": "expected-not-observed", "description": "Explains that expected counts are checked because the chi-square approximation is evaluated under the null model, which determines the expected counts" } ], "commonMistakes": [ "Calling the school study independence instead of homogeneity", "Calling the employment study homogeneity instead of independence", "Reversing the null and alternative language", "Checking observed rather than expected counts", "Using greater than 5 instead of at least 5" ], "worksheet": "u8_lesson5_live.html" }, "WS-U8L6-reflect1": { "elements": [ { "id": "chi-square-formula", "description": "States that the chi-square statistic is computed by summing (observed minus expected) squared divided by expected across all cells" }, { "id": "contributions", "description": "Explains that each cell makes a contribution and the contributions add to the overall chi-square statistic" }, { "id": "observed-expected", "description": "Makes clear that both observed counts and expected counts are used in the calculation" }, { "id": "df-formula", "description": "States that degrees of freedom are calculated with (rows minus 1) times (columns minus 1)" }, { "id": "exclude-totals", "description": "States that the total row and total column are not counted when finding rows and columns for degrees of freedom" }, { "id": "p-value-methods", "description": "Explains that the p-value is found from the right tail of a chi-square distribution and can be estimated with Table C or found exactly with chi-squared CDF or other technology" }, { "id": "school-example", "description": "Uses the school example with chi-square statistic 5.55, degrees of freedom 2, and p-value about 0.062 or between 0.05 and 0.10" }, { "id": "employment-example", "description": "May mention the employment example with chi-square statistic 14.30 and a very small p-value of about 0.0008" } ], "commonMistakes": [ "Forgetting to divide by the expected count in each term", "Using only observed counts and leaving out expected counts", "Counting the total row or total column when finding degrees of freedom", "Describing the p-value as a left-tail or two-tail probability instead of a right-tail probability", "Leaving out the school example values from the explanation" ], "worksheet": "u8_lesson6_live.html" }, "WS-U8L6-reflect2": { "elements": [ { "id": "interpretation-template", "description": "Explains that a p-value interpretation starts by assuming the null hypothesis is true" }, { "id": "chance-alone", "description": "States that the p-value is the probability of getting the observed chi-square statistic or greater by chance alone in the random sample or random assignment" }, { "id": "compare-to-alpha", "description": "Explains that the statistical decision is made by comparing the p-value to alpha" }, { "id": "large-pvalue", "description": "States that if the p-value is greater than alpha, fail to reject the null hypothesis and say there is not convincing statistical evidence for the alternative in context" }, { "id": "small-pvalue", "description": "States that if the p-value is less than or equal to alpha, reject the null hypothesis and say there is convincing statistical evidence for the alternative in context" }, { "id": "school-conclusion", "description": "Uses the school example: p-value 0.062 is greater than 0.05, so fail to reject and conclude there is not convincing statistical evidence of a difference in school-type distributions from 2019 to 2020" }, { "id": "employment-conclusion", "description": "Uses the employment example: p-value 0.0008 is less than or equal to 0.01, so reject and conclude there is convincing statistical evidence of an association between education level and employment status" }, { "id": "follow-up-analysis", "description": "Explains that follow-up analysis examines the largest cell contribution to identify which cells most strongly explain the chi-square result; may use the employment example as illustration" } ], "commonMistakes": [ "Saying the p-value is the probability that the null hypothesis is true", "Rejecting the null hypothesis when the p-value is larger than alpha", "Failing to state the conclusion in context of the problem", "Mixing up the school and employment conclusions", "Ignoring the largest contribution in the follow-up analysis" ], "worksheet": "u8_lesson6_live.html" }, "WS-U8L6-exitTicket": { "elements": [ { "id": "procedure", "description": "Identifies the procedure as a chi-square test for independence because the problem asks about association between two categorical variables in one random sample" }, { "id": "null-hypothesis", "description": "States the null hypothesis that age group at diagnosis and gender are independent or not associated for the population of people currently being treated for schizophrenia" }, { "id": "alternative-hypothesis", "description": "States the alternative hypothesis that age group at diagnosis and gender are not independent or are associated for that population" }, { "id": "alpha", "description": "States that alpha = 0.05 can be used when no significance level is given" }, { "id": "conditions", "description": "Verifies the conditions using the random sample of 207 people, the 10 percent condition, and the fact that all expected counts are at least 5" }, { "id": "statistic-and-df", "description": "Reports the chi-square statistic 10.884 and degrees of freedom 3" }, { "id": "p-value", "description": "Reports the p-value of 0.012" }, { "id": "decision-and-conclusion", "description": "States that because 0.012 is less than 0.05 we reject the null hypothesis and conclude there is convincing statistical evidence of an association between age group at diagnosis and gender for the population currently being treated for schizophrenia" }, { "id": "calculator", "description": "Explains one way technology helps, such as using a chi-square test with observed counts in matrix A and reading expected counts from matrix B" }, { "id": "interpretation-note", "description": "May mention that a p-value interpretation is not required unless the question specifically asks for it" } ], "commonMistakes": [ "Calling the procedure homogeneity instead of independence", "Writing the null hypothesis as there is an association", "Leaving out alpha when it must be assumed from context", "Forgetting one of the three conditions", "Reporting the statistic or p-value incorrectly or giving the wrong final decision" ], "worksheet": "u8_lesson6_live.html" }, "WS-U9L1-reflect1": { "elements": [ { "id": "population", "description": "Identifies the July 1995 Old Faithful data as the population or starting model for the simulation" }, { "id": "sample-size", "description": "States that repeated random samples of 25 eruptions are taken from the 1995 population" }, { "id": "sample-regression-line", "description": "Explains that a sample regression line is calculated for each simulated sample" }, { "id": "record-slope", "description": "Explains that the slope from each sample regression line is recorded" }, { "id": "random-variation", "description": "States that slopes vary from sample to sample because of random variation" }, { "id": "trial-values", "description": "Includes sample slope examples such as 14.46 and 11.97 from the first two trials" }, { "id": "sampling-distribution", "description": "Explains that the repeated slopes form a simulated sampling distribution of the slope" }, { "id": "old-model", "description": "May mention that the simulation is based on the older July 1995 regression model" } ], "commonMistakes": [ "Treating the 2019 sample as the population used for simulation", "Leaving out that many random samples of size 25 are taken", "Forgetting that each simulated sample produces its own regression line and slope", "Ignoring random variation as the reason slopes differ", "Leaving out the sample slope examples from the first two trials" ], "worksheet": "u9_lesson1_live.html" }, "WS-U9L1-reflect2": { "elements": [ { "id": "observed-slope", "description": "Identifies 7.79 as the observed slope from the July 2019 sample regression line" }, { "id": "compare-to-distribution", "description": "Explains that 7.79 is compared to the simulated sampling distribution built from the July 1995 model" }, { "id": "unusual-language", "description": "Uses the idea of a slope at least as unusual or surprising as 7.79" }, { "id": "either-direction", "description": "States that unusualness is judged in either direction from the center of the distribution" }, { "id": "estimated-probability", "description": "States that the estimated probability is approximately 0" }, { "id": "interpretation", "description": "Explains that a probability near 0 means the observed slope would be very unlikely if the 1995 model were still valid" }, { "id": "contextual-conclusion", "description": "Concludes in context that it is not believable that the July 1995 model still applies for predicting wait time from duration in July 2019" }, { "id": "left-tail", "description": "May mention that 7.79 is far to the left of the simulated distribution" } ], "commonMistakes": [ "Saying the slope 7.79 is compared directly to the intercept instead of to simulated slopes", "Leaving out the idea of at least as unusual in either direction", "Interpreting the estimated probability as exactly impossible instead of approximately 0 from simulation", "Concluding that the 1995 model is believable even though the observed slope is far outside the simulated results", "Failing to state the conclusion in the context of predicting Old Faithful wait times" ], "worksheet": "u9_lesson1_live.html" }, "WS-U9L1-exitTicket": { "elements": [ { "id": "population-and-sample", "description": "Describes the July 1995 Old Faithful population and the July 2019 random sample of 25 eruptions" }, { "id": "regression-lines", "description": "Reports the July 1995 population regression model and the July 2019 sample regression line, including the key slopes 13.29 and 7.79" }, { "id": "simulation-process", "description": "Explains that many random samples of size 25 are simulated from the 1995 population and a sample regression line is computed for each one" }, { "id": "sampling-distribution", "description": "Explains that the repeated slopes create a simulated sampling distribution of the slope" }, { "id": "unusual-slope", "description": "Explains that the observed slope 7.79 is unusually far from the center of the simulated distribution" }, { "id": "estimated-probability", "description": "States that the estimated probability of getting a slope that unusual is about 0" }, { "id": "meaning-of-probability", "description": "Explains that such a result would be very unlikely if the July 1995 population regression model were still valid" }, { "id": "contextual-conclusion", "description": "Concludes in context that it is not believable that the July 1995 model still applies for predicting wait time from duration in July 2019" }, { "id": "trial-slopes", "description": "May mention the example simulated slopes 14.46 and 11.97 from the first two trials" }, { "id": "either-direction", "description": "May mention that unusualness is judged in either direction from the center of the distribution" } ], "commonMistakes": [ "Leaving out either the 1995 population or the 2019 sample", "Not reporting the regression lines or confusing the two slopes", "Forgetting that the simulation repeatedly samples from the 1995 population", "Saying a probability of about 0 proves the model is impossible rather than not believable", "Failing to state the conclusion in context about predicting wait time from duration in 2019" ], "worksheet": "u9_lesson1_live.html" }, "WS-U9L2-reflect1": { "elements": [ { "id": "linear-condition", "description": "States that the true relationship between x and y must be linear" }, { "id": "equal-variance-condition", "description": "States that the standard deviation of the y-values must stay the same for all x-values" }, { "id": "normal-condition", "description": "States that for a particular x-value the responses are approximately normally distributed" }, { "id": "normal-shape", "description": "Explains that when the conditions are met, the sampling distribution of the sample slope is approximately normal" }, { "id": "center-beta", "description": "Explains that the center of the sampling distribution is the true population slope beta" }, { "id": "variability-formula", "description": "Gives or correctly describes the variability formula sigma_b = sigma / (sigma_x sqrt(n))" }, { "id": "old-faithful-values", "description": "Includes the Old Faithful values mean 13.29 and standard deviation 1.10 for the simulated sampling distribution" }, { "id": "ten-percent", "description": "May mention the 10% condition when sampling without replacement" } ], "commonMistakes": [ "Listing only one or two of the three required population regression model conditions", "Saying the sampling distribution is always normal without tying that to the conditions being met", "Failing to say that the center of the sampling distribution is beta", "Leaving out the variability formula or confusing it with a confidence interval formula", "Omitting the Old Faithful values 13.29 and 1.10" ], "worksheet": "u9_lesson2_live.html" }, "WS-U9L2-reflect2": { "elements": [ { "id": "scatterplot-linearity", "description": "Uses the scatterplot and/or residual plot to say the relationship looks linear and there is no leftover curved pattern" }, { "id": "equal-spread", "description": "Explains that the spread appears roughly similar across x-values" }, { "id": "residual-normality", "description": "States that the residuals show no obvious skewness or outliers, or explains the normality check" }, { "id": "random-sample", "description": "States that the data came from a random sample of 25 eruptions" }, { "id": "estimate-sigma-x", "description": "Explains that s_x = 0.213 is used to estimate sigma_x" }, { "id": "estimate-sigma", "description": "Explains that s = 4.2097 is used to estimate sigma" }, { "id": "t-interval-reason", "description": "States that the correct procedure is a t-interval for slope because sigma is estimated using sample information" }, { "id": "independence", "description": "May mention independence and the 10% condition" }, { "id": "n-greater-than-30", "description": "May mention that if there are clear departures from normality, the sample size should be greater than 30" } ], "commonMistakes": [ "Ignoring the residual plot when discussing linearity", "Leaving out the equal-spread or normality check", "Confusing s_x with s or swapping what each one estimates", "Saying z-interval instead of t-interval", "Forgetting to mention that the data came from a random sample" ], "worksheet": "u9_lesson2_live.html" }, "WS-U9L2-exitTicket": { "elements": [ { "id": "procedure", "description": "Identifies the procedure as a t-interval for the slope of a population regression line" }, { "id": "why-appropriate", "description": "Explains that the procedure is appropriate because the conditions are satisfied and sigma is estimated from sample data" }, { "id": "point-estimate", "description": "States that the point estimate is the sample slope b = 7.79" }, { "id": "df-and-critical", "description": "Gives df = 23 and t* = 2.069 for the 95% interval" }, { "id": "se-and-me", "description": "Reports the standard error 4.03 and the margin of error 8.34" }, { "id": "final-interval", "description": "Gives the final interval -0.55 to 16.13" }, { "id": "narrower-interval", "description": "Explains that a larger sample or a lower confidence level would make the interval narrower" }, { "id": "sample-based-estimates", "description": "May mention that s_x = 0.213 estimates sigma_x and s = 4.2097 estimates sigma" }, { "id": "conditions", "description": "May summarize the graphical and data-collection conditions that were checked" } ], "commonMistakes": [ "Using the intercept instead of the slope as the point estimate", "Giving the wrong degrees of freedom or critical value", "Confusing standard error with margin of error", "Leaving out the final interval or reporting the wrong endpoints", "Saying the interval gets narrower by increasing the confidence level" ], "worksheet": "u9_lesson2_live.html" }, "WS-U9L3-reflect1": { "elements": [ { "id": "confidence-statement", "description": "States that we are 95% confident the interval captures the parameter" }, { "id": "interval-values", "description": "Includes the interval endpoints -0.55 and 16.13" }, { "id": "parameter-context", "description": "Identifies the parameter as the slope of the population regression line for predicting wait time from eruption duration for all Old Faithful eruptions in July 2019" }, { "id": "confidence-level-meaning", "description": "Explains that the confidence level is about repeated random sampling, not about this one interval alone" }, { "id": "long-run-capture", "description": "States that about 95% of 95% confidence intervals from many random samples of size 25 would capture the true slope" }, { "id": "sample-size-25", "description": "May mention the repeated samples are size 25" }, { "id": "parameter-not-statistic", "description": "May clearly distinguish the population slope from the sample slope" } ], "commonMistakes": [ "Saying there is a 95% chance the true slope is in this one computed interval", "Leaving out the interval endpoints -0.55 and 16.13", "Failing to identify the parameter as the slope of the population regression line in context", "Not mentioning repeated random sampling when interpreting the confidence level", "Explaining the sample slope instead of the population slope" ], "worksheet": "u9_lesson3_live.html" }, "WS-U9L3-reflect2": { "elements": [ { "id": "beta-zero-meaning", "description": "Explains that beta = 0 means a horizontal line or no linear relationship" }, { "id": "old-faithful-contains-zero", "description": "States that the Old Faithful interval contains 0" }, { "id": "old-faithful-conclusion", "description": "Concludes that there is not convincing evidence of a linear relationship for Old Faithful" }, { "id": "raoul-interval-values", "description": "Includes Raoul's interval 0.013 to 0.080 or notes that it contains only positive values" }, { "id": "raoul-conclusion", "description": "Concludes that there is convincing evidence of a positive linear relationship for Raoul's experiment" }, { "id": "justify-claim-rule", "description": "Explains that claims are justified by checking whether the interval contains values consistent with the claim" }, { "id": "plausible-language", "description": "May use language like plausible value or consistent with the data" }, { "id": "negative-interval-case", "description": "May note that an interval containing only negative values would support a negative linear relationship" } ], "commonMistakes": [ "Saying Old Faithful shows convincing evidence even though the interval contains 0", "Ignoring the meaning of beta = 0", "Failing to mention that Raoul's interval contains only positive values", "Using p-value reasoning instead of interval reasoning", "Not connecting the conclusion to values consistent with the claim" ], "worksheet": "u9_lesson3_live.html" }, "WS-U9L3-exitTicket": { "elements": [ { "id": "parameter", "description": "Defines the parameter as the slope of the population regression line for predicting selling price from distance from the city center for all three-bedroom houses near this city" }, { "id": "procedure", "description": "Identifies the procedure as a t-interval for the slope of a population regression line and notes that conditions are assumed met" }, { "id": "b-df-tstar", "description": "Gives b = -2.158, df = 18, and t* = 2.101" }, { "id": "se-and-me", "description": "Reports the standard error 0.149 and the margin of error 0.313" }, { "id": "final-interval", "description": "Gives the final interval -2.471 to -1.845" }, { "id": "interpretation", "description": "Interprets the interval in context as a decline of between $1,845 and $2,471 in selling price for each additional mile from the city center" }, { "id": "claim-justification", "description": "Explains that the interval contains -2, so the data do not contradict the agent's belief about a $2,000 decrease per mile" }, { "id": "units", "description": "May mention the slope is in thousands of dollars per mile" }, { "id": "sample-size", "description": "May mention the random sample size is 20 houses" } ], "commonMistakes": [ "Using the intercept instead of the slope as the point estimate", "Giving the wrong degrees of freedom or critical value", "Confusing the standard error with the margin of error", "Interpreting the slope without context or units", "Saying the interval contradicts the agent even though -2 is inside the interval" ], "worksheet": "u9_lesson3_live.html" }, "WS-U9L4-reflect1": { "elements": [ { "id": "prevent-bias", "description": "States that choosing Ha after seeing data would bias the test or allow researchers to pick a convenient direction" }, { "id": "pre-registration", "description": "Explains that the research question should drive the hypothesis, not the observed pattern" }, { "id": "integrity", "description": "May mention that choosing after data collection undermines the integrity or validity of the significance test" }, { "id": "one-vs-two", "description": "May connect this to the choice between one-sided and two-sided alternatives" } ], "commonMistakes": [ "Saying it does not matter when you choose the alternative", "Confusing the null hypothesis with the alternative hypothesis", "Not explaining how post-hoc hypothesis selection introduces bias" ], "worksheet": "u9_lesson4_live.html" }, "WS-U9L4-reflect2": { "elements": [ { "id": "list-conditions", "description": "Names or references the four conditions: linearity, constant spread, normality, and independence" }, { "id": "validity-reason", "description": "Explains that the conditions ensure the t-test produces valid and reliable results" }, { "id": "violation-consequence", "description": "Gives at least one example of what could go wrong if a condition is violated (e.g., curved relationship makes the linear model meaningless, non-constant spread distorts standard errors, non-normality affects the t-distribution, non-independence invalidates the sampling distribution)" }, { "id": "each-serves-purpose", "description": "May explain that each condition addresses a different aspect of the statistical model" } ], "commonMistakes": [ "Listing the conditions without explaining why they matter", "Giving only a vague consequence like \"the test would be wrong\" without specifics", "Confusing the conditions for slope inference with conditions for a different test" ], "worksheet": "u9_lesson4_live.html" }, "WS-U9L4-exitTicket": { "elements": [ { "id": "null-hypothesis", "description": "States H0: beta = 0" }, { "id": "alternative-hypothesis", "description": "States Ha: beta < 0 because higher temperature is expected to be associated with slower growth (negative relationship)" }, { "id": "define-beta", "description": "Defines beta as the slope of the population regression line for predicting coral growth rate from water temperature for all coral reefs" }, { "id": "test-name", "description": "Names the procedure as a t-test for slope" }, { "id": "condition-linear", "description": "Lists linearity: the true relationship between temperature and growth rate is linear" }, { "id": "condition-constant-sd", "description": "Lists constant standard deviation: the spread of growth rates does not vary with temperature" }, { "id": "condition-normality", "description": "Lists normality: for a particular temperature, growth rates are approximately normally distributed" }, { "id": "condition-independence", "description": "Lists independence: random sample and n <= 10% of all coral reefs" } ], "commonMistakes": [ "Using Ha: beta > 0 instead of beta < 0 (the question says slower growth with higher temperature)", "Using Ha: beta != 0 when the question implies a directional claim", "Forgetting to define beta in the context of coral reefs", "Listing conditions for a different test such as a t-test for a mean", "Not mentioning the 10% condition for independence" ], "worksheet": "u9_lesson4_live.html" }, "WS-U9L5-reflect1": { "elements": [ { "id": "assume-null", "description": "Explains that a p-value is calculated under the assumption that the null hypothesis is true" }, { "id": "extreme-language", "description": "Explains that the p-value includes results as extreme or more extreme than the observed sample result" }, { "id": "chance-alone", "description": "Mentions that the p-value describes what could happen by chance or random sampling variation alone" }, { "id": "tail-direction", "description": "May note that the meaning of extreme depends on whether the test is one-sided or two-sided" } ], "commonMistakes": [ "Describing the p-value as the probability that the null hypothesis is true", "Leaving out the assumption that H0 is true", "Forgetting to mention outcomes as extreme or more extreme than the observed result", "Confusing p-value interpretation with the final conclusion" ], "worksheet": "u9_lesson5_live.html" }, "WS-U9L5-reflect2": { "elements": [ { "id": "full-setup", "description": "Lists the core pieces of a complete test such as hypotheses, parameter definition, significance level, procedure, conditions, test statistic, p-value, and conclusion" }, { "id": "decision-rule", "description": "Explains that the decision is based on comparing the p-value to alpha" }, { "id": "reject-vs-fail", "description": "States that p-value <= alpha leads to reject H0, while p-value > alpha leads to fail to reject H0" }, { "id": "context-conclusion", "description": "May mention that the final conclusion must be written in context of the variables" } ], "commonMistakes": [ "Skipping major pieces such as hypotheses, conditions, or the conclusion", "Reversing the decision rule for p-value and alpha", "Saying to accept H0 instead of fail to reject H0", "Giving a conclusion without writing it in context" ], "worksheet": "u9_lesson5_live.html" }, "WS-U9L5-exitTicket": { "elements": [ { "id": "null-hypothesis", "description": "States H0: beta = 0" }, { "id": "alternative-hypothesis", "description": "States Ha: beta > 0 because the question asks whether more training hours are associated with greater improvement" }, { "id": "define-beta", "description": "Defines beta as the slope of the population regression line for predicting VO2 max improvement from weekly training hours for cyclists in the population of interest" }, { "id": "test-statistic", "description": "States the test statistic is t = 4.00" }, { "id": "degrees-freedom", "description": "States df = 14 - 2 = 12" }, { "id": "one-sided-pvalue", "description": "Explains that because the output p-value 0.002 is for a two-sided test, the one-sided p-value is 0.001" }, { "id": "interpret-pvalue", "description": "Interprets the p-value in context assuming H0 is true and describing a sample slope as large as or larger than the observed slope by chance alone" }, { "id": "decision-conclusion", "description": "Concludes that because 0.001 < 0.05, reject H0 and there is convincing statistical evidence that greater training hours are associated with greater VO2 max improvement" } ], "commonMistakes": [ "Using Ha: beta != 0 instead of beta > 0 even though the question is directional", "Forgetting that computer output usually gives a two-sided p-value and not halving 0.002", "Using the wrong degrees of freedom instead of n - 2", "Interpreting the p-value as the probability that H0 is true", "Writing the conclusion without comparing the p-value to alpha" ], "worksheet": "u9_lesson5_live.html" } }, "items": { "WS-U1L10-reflect1": { "percentile-definition": [], "zscore-definition": [ "zscore-percent" ], "sign-meaning": [], "two-tools": [], "context": [], "example-values": [], "any-distribution": [] }, "WS-U1L10-reflect2": { "table-a-left": [], "area-right": [], "area-between": [], "backward-process": [], "zscore-language": [], "context": [], "example-values": [] }, "WS-U1L10-exitTicket": { "shape-parameters": [], "within-one-sd": [], "above-two-sd": [], "backward-left-area": [], "backward-zscore": [], "context": [], "notation": [] }, "WS-U1L1-reflect1": { "needs-context": [], "units-or-variable": [], "links-to-question": [], "threshold-idea": [] }, "WS-U1L1-reflect2": { "changed-percent": [], "changed-conclusion": [], "responsible-use": [], "real-consequences": [] }, "WS-U1L1-exitTicket": { "question-in-context": [], "variable": [], "variation": [], "context-for-30": [], "statistics-process": [] }, "WS-U1L2-reflect1": { "individuals-are-properties": [], "variable-definition": [], "gives-example-variable": [], "rows-vs-columns": [] }, "WS-U1L2-reflect2": { "numbers-as-labels": [], "zip-code-location": [], "grade-level-categories": [], "not-average": [] }, "WS-U1L2-exitTicket": { "individuals": [], "two-variables": [], "sport-played-categorical": [], "heart-rate-quantitative": [], "id-not-variable": [ "numeric-label" ] }, "WS-U1L3-reflect1": { "frequency-counts": [], "relative-frequency-proportion": [ "counts-vs-percents" ], "superpower-context": [], "divide-by-total": [] }, "WS-U1L3-reflect2": { "use-counts-or-percentages": [], "specific-example": [], "claim-judgment": [], "majority-threshold": [] }, "WS-U1L3-exitTicket": { "frequency-definition": [], "chips-relative-frequency": [], "yogurt-relative-frequency": [], "no-majority-chips": [], "context-justification": [] }, "WS-U1L4-reflect1": { "frequency-counts": [], "relative-frequency-proportions": [ "counts-vs-percents" ], "same-superpower-data": [], "superpower-example": [], "divide-by-total": [] }, "WS-U1L4-reflect2": { "different-group-sizes": [], "use-proportions": [ "counts-vs-percents" ], "lesson-example": [], "comparison-conclusion": [], "graph-or-table-link": [] }, "WS-U1L4-exitTicket": { "different-grade-sizes": [], "ninth-jersey-frequency": [], "tenth-jersey-frequency": [], "claim-not-supported": [], "context-justification": [] }, "WS-U1L5-reflect1": { "discrete-definition": [], "continuous-definition": [], "counting-measuring": [], "lesson-example": [], "rounded-values-note": [] }, "WS-U1L5-reflect2": { "large-data-set": [], "shows-individual-values": [], "histogram-intervals": [], "lose-exact-values": [ "histogram-exact-values" ], "shape-idea": [] }, "WS-U1L5-exitTicket": { "discrete-classification": [], "countable-gaps-explanation": [], "best-display-choice": [], "chosen-display-advantage": [], "histogram-limitation": [] }, "WS-U1L6-reflect1": { "shape-description": [], "center-description": [], "variability-description": [], "unusual-features": [], "context": [], "single-peak-detail": [] }, "WS-U1L6-reflect2": { "two-unusual-features": [], "flint-connection": [], "distribution-description-role": [], "gaps-clusters-together": [] }, "WS-U1L6-exitTicket": { "shape": [], "center": [], "variability": [], "unusual-features": [], "context": [] }, "WS-U1L7-reflect1": { "mean-interpretation": [], "median-interpretation": [], "iqr-interpretation": [ "iqr-vs-range" ], "context": [], "center-vs-variability": [], "numerical-detail": [] }, "WS-U1L7-reflect2": { "shape-or-outliers": [], "median-iqr-choice": [], "resistant-language": [ "mean-resistant" ], "nonresistant-language": [ "mean-resistant" ], "context": [], "removal-example": [] }, "WS-U1L7-exitTicket": { "best-center": [], "best-variability": [ "iqr-vs-range" ], "outlier-identification": [], "resistant-justification": [ "mean-resistant" ], "context": [], "skew-language": [] }, "WS-U1L8-reflect1": { "five-number-summary": [], "boxplot-feature": [ "boxplot-whiskers" ], "flint-context": [], "numerical-details": [], "quartile-meaning": [] }, "WS-U1L8-reflect2": { "skewed-right": [], "skewed-left": [], "symmetric": [], "outlier-effect": [ "mean-resistant" ], "flint-connection": [], "median-resistant": [ "mean-resistant" ] }, "WS-U1L8-exitTicket": { "box-and-median": [], "whiskers-and-outlier": [], "middle-50": [], "mean-vs-median": [], "context": [], "quartile-language": [] }, "WS-U1L9-reflect1": { "shape-comparison": [], "center-comparison": [], "variability-comparison": [], "unusual-features": [], "comparative-language": [], "context": [], "boxplot-caution": [] }, "WS-U1L9-reflect2": { "reason-for-a": [], "a-feature": [], "reason-for-b": [], "b-feature": [], "context": [], "based-on-box-plots": [] }, "WS-U1L9-exitTicket": { "center": [], "variability": [], "unusual-features": [], "shape": [], "comparative-context": [], "numerical-support": [] }, "WS-U2L1-reflect1": { "identify-type-first": [], "categorical-tools": [], "quantitative-tools": [], "match-method-to-data": [] }, "WS-U2L1-reflect2": { "random-variation": [], "not-automatic-proof": [], "need-analysis": [], "uncertain-conclusions": [] }, "WS-U2L1-exitTicket": { "study-a-types": [], "study-a-tools": [], "study-b-types": [], "study-b-tools": [], "pattern-study-a": [], "pattern-study-b": [], "random-variation": [], "careful-conclusion": [] }, "WS-U2L2-reflect1": { "within-group-percent": [], "fair-comparison": [], "compare-distributions": [], "sum-to-100": [] }, "WS-U2L2-reflect2": { "compare-distributions": [], "not-same-means-association": [], "predictive-language": [], "mosaic-widths": [] }, "WS-U2L2-exitTicket": { "identify-variables": [], "two-way-table": [], "ninth-grade-distribution": [], "eleventh-grade-distribution": [], "association-conclusion": [], "mosaic-advantage": [ "counts-vs-percents" ] }, "WS-U2L3-reflect1": { "marginal-definition": [], "conditional-definition": [], "overall-vs-within-group": [], "denominator-language": [] }, "WS-U2L3-reflect2": { "within-group-distributions": [], "compare-groups": [], "different-means-association": [], "supports-graphical-conclusion": [] }, "WS-U2L3-exitTicket": { "identify-variables": [], "two-way-table": [], "joint-relative-frequency": [], "marginal-relative-frequency": [], "conditional-ninth": [], "conditional-eleventh": [], "association-conclusion": [] }, "WS-U2L4-reflect1": { "explanatory-role": [], "x-axis": [], "y-axis": [], "paired-values": [] }, "WS-U2L4-reflect2": { "direction-only": [], "form": [], "strength": [], "features-context": [] }, "WS-U2L4-exitTicket": { "identify-variables": [], "explanatory-response": [], "scatterplot-appropriate": [], "positive-direction": [], "linear-form": [], "strong-strength": [], "construction-features": [], "no-obvious-unusual-features": [] }, "WS-U2L5-reflect1": { "sign-direction": [], "positive-negative": [], "magnitude-strength": [], "close-values": [], "linear-focus": [] }, "WS-U2L5-reflect2": { "no-causation": [ "correlation-causation" ], "no-form": [], "no-unusual-features": [], "need-graph-context": [], "attendance-example": [] }, "WS-U2L5-exitTicket": { "positive-direction": [], "strong-strength": [], "r-alone-limit": [], "no-causation": [ "correlation-causation" ], "technology-use": [], "unit-free-range": [] }, "WS-U2L6-reflect1": { "plug-in-x": [], "slope-and-intercept": [], "predicted-response": [], "decimal-possible": [], "variability-idea": [] }, "WS-U2L6-reflect2": { "define-extrapolation": [ "extrapolation" ], "trend-may-not-continue": [ "extrapolation" ], "less-reliable": [ "extrapolation" ], "compare-to-interval": [], "example-link": [] }, "WS-U2L6-exitTicket": { "variables-context": [], "slope-context": [], "intercept-context": [], "correct-prediction": [], "prediction-not-exact": [], "extrapolation-unreliable": [ "extrapolation" ], "further-less-reliable": [] }, "WS-U2L7-reflect1": { "actual-minus-predicted": [], "single-point-difference": [], "positive-underestimate": [ "residual-sign" ], "negative-overestimate": [ "residual-sign" ], "context-example": [] }, "WS-U2L7-reflect2": { "apparent-randomness": [], "centered-at-zero": [], "no-clear-pattern": [], "pattern-warning": [], "nonlinear-next-step": [] }, "WS-U2L7-exitTicket": { "correct-residual": [], "context-interpretation": [], "underpredicted": [ "residual-sign" ], "good-fit-randomness": [], "bad-fit-pattern": [], "residual-plot-role": [] }, "WS-U2L8-reflect1": { "canceling-problem": [], "least-squares-definition": [ "least-squares" ], "mean-point-property": [], "slope-formula-property": [], "numeric-example": [] }, "WS-U2L8-reflect2": { "slope-change": [], "attendance-context": [], "intercept-definition": [ "intercept-context" ], "intercept-not-meaningful": [ "intercept-context" ], "example-illogical-value": [] }, "WS-U2L8-reflect3": { "r-squared-meaning": [], "context-interpretation": [ "rsquared-points" ], "r-squared-equals-r-squared": [], "output-slope": [], "output-intercept": [], "strength-note": [] }, "WS-U2L8-exitTicket": { "squared-residuals": [ "least-squares" ], "slope-interpretation": [], "intercept-definition": [], "intercept-not-meaningful": [ "intercept-context" ], "r-squared-context": [ "rsquared-points" ], "mean-point": [], "slope-formula-note": [] }, "WS-U2L9-reflect1": { "high-leverage-definition": [], "outlier-definition": [], "influential-definition": [], "high-leverage-effect": [], "outlier-effect": [], "both-type": [] }, "WS-U2L9-reflect2": { "nonlinear-trigger": [], "purpose-of-transform": [], "log-skew": [], "residual-evidence": [], "r-squared-evidence": [], "numeric-example": [] }, "WS-U2L9-exitTicket": { "high-leverage-identification": [], "outlier-identification": [], "influential-definition": [], "different-effects": [], "log-transform-purpose": [], "better-fit-evidence": [], "both-point-type": [] }, "WS-U3L1-reflect1": { "returned-planes": [], "missing-planes": [], "vulnerable-areas": [], "analysis-depends": [] }, "WS-U3L1-reflect2": { "who-sampled": [], "who-missed": [], "chance-method": [], "representative": [], "untrustworthy": [] }, "WS-U3L1-exitTicket": { "population": [], "sample": [], "nonchance": [], "difference": [], "better-method": [], "trust-conclusion": [] }, "WS-U3L2-reflect1": { "returned-only": [], "shot-down-excluded": [], "not-representative": [], "generalization-risk": [], "tail-insight": [] }, "WS-U3L2-reflect2": { "observational-no-treatment": [], "observational-no-cause": [ "observational-causation" ], "experiment-treatment": [], "experiment-causal": [], "random-assignment": [] }, "WS-U3L2-exitTicket": { "population": [], "sample": [], "observational": [], "generalize-yes": [], "no-causation": [], "confounding": [] }, "WS-U3L3-reflect1": { "both-group": [], "cluster-select-groups": [], "cluster-all-members": [], "stratified-sample-within": [], "stratified-combine": [], "hetero-homo": [] }, "WS-U3L3-reflect2": { "segregation-pattern": [], "sample-each-region": [], 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