AP Statistics Unit 7 Inference for Means

Topic 7.3: Justifying a Claim About a Population Mean Based on a Confidence Interval

Interpretation, Confidence Level, Margin of Error, and Claims
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Plausible Value A parameter value that lies inside a confidence interval and is therefore consistent with the sample data.
Confidence Level The long-run proportion of intervals from repeated random samples that capture the true population parameter.
Repeated Random Sampling The process used to interpret confidence level by imagining many samples of the same size from the same population.
Capture When a confidence interval contains the true value of the population parameter.
Margin of Error The amount added to and subtracted from the point estimate to create the confidence interval.
Interval Width The total distance from the lower bound to the upper bound; it is twice the margin of error.
Directional Claim A claim that the population mean is greater than or less than a specific value.
Context The real-world description of the population and variable that must be included in statistical interpretations.

Video 1: Interpreting an Interval and Using It to Justify a Claim

[0:00 – 4:30]
1. [0:20] We are going to interpret a confidence interval for a population and justify a based on that interval.
Context — Sweet Sugary Goodness [0:38] A bakery owner uses the 95% confidence interval from Topic 7.2 to study the mean weight of powdered sugar bags from a new wholesaler.
2. [1:14] The interpretation template says that the interval captures the population .
3. [1:39] In the powdered sugar example, the 95% confidence interval goes from to grams.
4. [2:14] Because 907 grams is the interval, it is a value for μ.
5. [2:14] Therefore there is convincing evidence that the bags are being .
Using a Confidence Interval to Judge a Claim: [2:14 – 4:30]
  1. If the claimed mean is inside the interval, the claim is plausible.
  2. If the entire interval is above the comparison value, the data support a greater-than claim.
  3. If the entire interval is below the comparison value, the data support a less-than claim.
Practice — Fiddler Feeding Frenzy [3:08] The 90% confidence interval for the mean feeding rate is given as 67.65 ± 1.76 scoops per 30 seconds. The question asks whether the average feeding rate is faster than 2 scoops per second.
6. [3:08] In the crab example, the margin of error is scoops per 30 seconds.
7. [3:51] Rewriting the crab interval in lower-to-upper form gives to scoops per 30 seconds.
8. [4:30] Since 2 scoops per second equals scoops per 30 seconds, there convincing evidence the mean feeding rate is than 2 scoops per second.

Video 2: Interpreting Confidence Level and Margin of Error

[0:00 – 7:24]
9. [0:26] In this video, we are interpreting the confidence , not the confidence .
Context — Human Body Temperature [0:50 – 1:17] Suppose body temperatures are approximately normal with mean 98.6°F and standard deviation 0.8°F. We imagine taking many random samples of 10 people and building 95% confidence intervals for the population mean.
10. [3:00] In the simulation, about of the 1000 intervals captured the true mean.
11. [3:28] In repeated random sampling with the same sample size, approximately C% of C% intervals will capture the population .
12. [3:47] For the body temperature example, many random samples of people would produce intervals that capture μ about of the time.
13. [4:42] It is not correct to interpret 95% as the that one specific interval captures μ.
Margin of Error Reminders: [5:19 – 7:05]
confidence interval = point estimate ± margin of error
width = 2(margin of error)
margin of error for μ = t* (s / √n)
14. [5:27] The width of a confidence interval is the margin of error.
15. [5:51] One way to decrease margin of error is to use a larger sample .
16. [6:31] To cut the margin of error in half, you would have to the sample size.
17. [6:46] A lower confidence gives a smaller and therefore a smaller margin of .

Video 3: Full Process from Start to Finish

[0:00 – 4:29]
Practice — 2013 AP Exam Crows [0:31] An environmental group measured lead levels for a random sample of 23 crows in a region. A biologist says lead levels greater than 6.0 ppm are unhealthy.
2 | 8
3 | 0 1 1 5 8 8
4 | 1 1 2 3 3 3 4 7 9
5 | 0 1 2 2 3
6 | 1 2 4 6
Sample statistics for part (b): x̄ = 4.90 ppm, s = 1.12 ppm, n = 23.
18. [1:26] Four of the 23 crows were above 6.0 ppm, so the sample proportion classified as unhealthy is about .
19. [1:52] In part (b), we want a 95% confidence interval for μ, the mean lead level of all in the region.
20. [2:37] The correct procedure is a one-sample for μ.
21. [2:37] The conditions are met because the sample was , 23 is less than of all crows, and the stemplot shows no strong skewness or .
22. [3:23] With n = 23, the degrees of freedom are and the 95% critical value is .
23. [3:23] The margin of error is , so the interval runs from to ppm.
24. [4:11] We are 95% confident that this interval captures the mean lead level of all crows in the .

Key Takeaways

[Summary]
  1. Interpret a confidence interval by naming the confidence level, the interval bounds, and the population mean in context.
  2. Use interval location to justify claims: inside means plausible, all above supports a greater-than claim, and all below supports a less-than claim.
  3. Interpret confidence level with repeated random sampling, not as the probability for one completed interval.
  4. Increase sample size or decrease confidence level to reduce margin of error; then complete the full State-Plan-Do-Conclude process in context.

Reflection Questions

[AI-Graded]
R1. Using the powdered sugar and fiddler crab examples, explain how a confidence interval can be used to justify a claim about a population mean. Include what it means when a claimed value is inside the interval and what it means when the entire interval is above the comparison value.
R2. Explain how to interpret a 95% confidence level for the body temperature example and describe how increasing sample size or decreasing confidence level affects the margin of error for a confidence interval for a population mean.

Exit Ticket

An environmental group measured lead levels for a random sample of 23 crows in a region. The sample mean was 4.90 ppm, the sample standard deviation was 1.12 ppm, and a biologist classifies lead levels above 6.0 ppm as unhealthy.

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