AP Statistics Unit 1 Introducing Statistics

Topic 1.10: The Normal Distribution

Percentiles, z-Scores, and the Normal Distribution
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:
🖩 Practice on the TI-84 →

Key Vocabulary

PercentileThe percent of data values less than or equal to a given value.
Relative PositionHow a value compares to the rest of the values in a distribution.
Standardized ScoreA score that measures how far a value is from the mean in standard deviation units.
z-ScoreA standardized score found by subtracting the mean and dividing by the standard deviation.
Normal DistributionA mound-shaped, symmetric distribution used to model many quantitative variables.
MeanThe center of a normal distribution, written with the symbol mu.
Standard DeviationThe spread of a normal distribution, written with the symbol sigma.
Empirical RuleThe 68-95-99.7 rule for data within 1, 2, and 3 standard deviations of the mean.
Table AThe standard normal table that gives area to the left of a z-score.
Area to the LeftThe proportion of values less than a boundary value on a normal curve.
Area to the RightThe proportion of values greater than a boundary value on a normal curve.
Cutoff ValueA data value found by working backward from a given area or percentile.

Video 1: Relative Position with Percentiles and z-Scores

[0:00 – 7:58]
1. [0:15 – 0:20] One way to describe the relative position of a value in a quantitative data set is with its .
2. [0:23 – 0:27] Another way to describe relative position is with .
3. [0:41 – 0:57] Percentile is the percent of data values that are than or to a given value.
4. [1:33 – 2:20] In AP Statistics, the standardized score we most often calculate is the , which tells how many standard deviations a value is or the mean.
Relative Position Tools [0:41 – 2:20]
5. [2:22 – 2:52] Percentiles and z-scores can be calculated for distribution, not just a distribution.
Flint Water Crisis Context [2:54 – 3:25] The example uses lead levels from 71 water samples collected from Flint residents in 2015. The lesson finds the percentile and z-score for samples of 20 parts per billion and 2 parts per billion.
6. [4:14 – 4:31] The water sample of 20 parts per billion is at the percentile.
7. [5:20 – 5:37] The value of 20 parts per billion is 0.88 standard deviations the mean.
8. [6:12 – 6:27] The water sample of 2 parts per billion is at the percentile.
9. [7:04 – 7:19] The z-score for 2 parts per billion is negative because the value is the mean.
10. Reflection: Explain how percentiles and z-scores each describe the relative position of a Flint water-sample value. Include what percentile counts and what a positive or negative z-score means.

Video 2: The Normal Distribution and the Empirical Rule

[0:00 – 8:57]
11. [0:44 – 0:56] A normal distribution is and .
12. [0:58 – 1:04] Many real-world quantitative variables can be closely modeled by a .
13. [1:15 – 1:23] A normal distribution is determined by the population and the population .
Empirical Rule [2:22 – 3:35]
Blood Pressure Context [1:43 – 2:19] Adult systolic blood pressure is modeled with a normal distribution with mean 110 and standard deviation 10. The lesson uses this model to estimate percents from shaded regions.
14. [2:49 – 2:57] About of the data lie within standard deviation of the mean.
15. [3:00 – 3:12] About of the data lie within standard deviations of the mean.
16. [3:15 – 3:21] About of the data lie within standard deviations of the mean.
17. [4:10 – 4:35] The percent of adults with systolic blood pressure below 100 is about .
18. [4:47 – 5:21] The percent of adults with systolic blood pressure below 130 is about .
19. [5:40 – 7:47] If a boundary value is not exactly 1, 2, or 3 standard deviations from the mean, we need to calculate a and use .

Video 3: Table A, Areas, and Working Backward

[0:00 – 8:42]
Review from the Last Video [1:01 – 1:57] The lesson reviews that a blood pressure of 125 has z = 1.50, and Table A gives an area of 0.9332 to the left of that z-score.
20. [2:42 – 3:14] To find the area to the right of a value, subtract the Table A area to the from 1.
21. [2:54 – 3:07] The proportion of adults with systolic blood pressure above 125 is about .
22. [3:45 – 5:09] To find the area between two values in a normal distribution, calculate two z-scores and subtract the two Table A .
23. [4:46 – 4:54] The proportion of adults with systolic blood pressure between 120 and 129 is about .
24. [6:00 – 6:12] If the highest 10% of the distribution are to the right, then the area to the left must be .
25. [6:44 – 6:54] The z-score closest to a left-tail area of 0.90 is about .
26. [7:19 – 7:31] The cutoff for the highest 10% of systolic blood pressures is about .
Table A Strategy [7:41 – 8:34]
27. Reflection: Explain how Table A helps you solve three kinds of normal-distribution problems: area to the right, area between two values, and working backward from an area to a cutoff value. Use z-score language in your response.

Exit Ticket

A bakery models loaf weights with a normal distribution with mean 16 ounces and standard deviation 0.5 ounce.

(a) Describe the shape of this distribution and identify its two parameters.

(b) Use the empirical rule to estimate the percent of loaves between 15.5 and 16.5 ounces and the percent of loaves above 17 ounces.

(c) The bakery gives a special label to the heaviest 10% of loaves. Explain how to use Table A and a z-score to estimate the cutoff weight.

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