AP Statistics Unit 6 Inference for Proportions

Topic 6.9: Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions

Interpreting the Interval, Assessing 0, and Deciding Whether a Claim Is Supported
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Confidence Interval An interval of plausible values for a population parameter based on sample data.
Confidence Level The long-run percent of intervals from repeated sampling that would capture the true parameter.
Plausible Value A value for the parameter that is consistent with the confidence interval.
Difference in Proportions The parameter written as p1 - p2, with the direction clearly defined.
Consistent with the Claim An interval value that supports the direction or statement being claimed.
Convincing Evidence Support for a claim when all values in the interval agree with that claim.
Significantly Different from 0 A difference is significant when 0 is not a plausible value in the confidence interval.

Video 1: Interpreting a Confidence Interval and Justifying a Claim

[0:00 – 6:48]
Context — "Please Trees, Don't Leave!" [0:24] Random samples of trees from two large forests show that 36 of 240 trees at high elevation and 25 of 200 trees at low elevation died from a disease. From the previous lesson, the 90% confidence interval for high minus low is (-0.029, 0.079).
1. [0:08] One goal of this lesson is to interpret a confidence interval for a difference in .
2. [0:24] Another goal is to justify a based on a confidence interval for a difference in proportions.
Interpretation Template: [1:03] We are C% confident that the interval from lower bound to upper bound captures the value to be estimated.
3. [1:07] In the general template, we are C% that the interval captures the value to be .
4. [1:31] For the diseased-tree example, the 90% confidence interval is from to for high minus .
5. [1:33] We are 90% confident that this interval captures the high minus low in the of all trees in these forests that died from the disease.
6. [2:24] If the disease were equally lethal in both forests, then the difference high minus low would be .
7. [2:52] Because 0 is the interval, 0 is a value for the true difference.
8. [2:52] So there is convincing evidence that the disease is more at one elevation.
Context — "Who's a Good Dog?" [3:19] Dogs were randomly assigned to a new or old tick-repellent formula. The 95% confidence interval for new minus old in the true proportion that would get ticks is (-0.2907, -0.0343).
9. [3:53] In this problem, the parameter is written as new minus .
10. [4:27] We are 95% confident that the interval from to captures the true difference in the of dogs that would get .
11. [5:15] Because all the values in the interval are less than , there is convincing that the new formula is than the old formula.
12. [5:46] A negative interval here means the first population proportion is than the second.
Claim Rule: [6:31] If all values in the interval are consistent with the claim, there is convincing evidence for the claim. If one or more values are inconsistent with the claim, there is not convincing evidence.
13. [6:31] If all the values in a confidence interval are with the claim, then we have convincing evidence for the claim.
14. [6:31] If one or more values in the interval are with the claim, then there is not convincing evidence.

Video 2: Using an Interval to Decide Whether a Difference Is Significant

[0:00 – 9:00]
Context — Day Shift vs. Night Shift [0:30] A company manager compares the proportion of parts that meet specifications on the day shift and night shift. Random samples of 200 parts from each shift found 188 successes for day and 180 successes for night.
15. [1:56] We are trying to estimate the difference, minus , in the proportions of all parts produced within specifications.
16. [2:20] The procedure is a two-sample for a difference in .
17. [4:20] For a 96% confidence interval, the critical value is .
18. [4:40] The point estimate is and the margin of error is .
19. [4:48] So the 96% confidence interval goes from to .
20. [5:37] We are 96% confident that this interval captures the difference day minus night in the of all parts produced within .
21. [6:16] Because zero is the interval, zero is a value for the true difference.
22. [6:16] Therefore there is convincing evidence that the proportions are significantly different from .
Confidence Level vs. Confidence Interval: [6:51 – 7:51] Interpreting the interval tells what values are plausible for the parameter now. Interpreting the confidence level describes what would happen to many intervals in repeated random sampling.
23. [7:19] In repeated random sampling, about of the intervals built from sample pairs would the true difference in population proportions.
24. [8:11] When asked to construct and interpret an interval, be sure to the difference, identify the , and interpret the interval in context.

Key Takeaways

[Summary]
  1. Interpret a confidence interval by naming the confidence level, the bounds, the parameter, and the population context.
  2. If 0 is in the interval for a difference in proportions, then 0 is a plausible value and there is not convincing evidence of a difference.
  3. If all values in the interval support the claim, then the interval provides convincing evidence for that claim.
  4. The confidence level describes the long-run success rate of the method in repeated random sampling, not the chance that this one interval is correct.

Reflection Questions

[AI-Graded]
R1. For the tree-disease example, a 90% confidence interval for phigh - plow is (-0.029, 0.079). Interpret this interval in context, and explain whether it gives convincing evidence that the disease is more lethal at one elevation.
R2. For the dog tick-repellent experiment, a 95% confidence interval for pnew - pold is (-0.2907, -0.0343). Interpret this interval in context, and explain why it supports the claim that the new formula is better.

Exit Ticket

A large company compares the proportion of parts produced within specifications on its day shift and night shift. Random samples of 200 parts from each shift gave a 96% confidence interval of (-0.016, 0.096) for pday - pnight.

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