AP Statistics Unit 6 Inference for Proportions

Topic 6.8: Confidence Intervals for the Difference of Two Proportions

Choosing the Procedure, Checking Conditions, and Calculating the Interval
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:
🖩 Practice on the TI-84 →

Key Vocabulary

Two-Sample z-Interval A confidence interval procedure used to estimate the difference between two population proportions.
Margin of Error The amount a sample statistic is likely to vary from the population parameter for a chosen confidence level.
Critical Value The z* multiplier that captures the middle C% of the standard normal distribution.
Standard Error An estimate of the standard deviation of the sampling distribution of p̂1 - p̂2.
Independent Random Samples Two separate random samples taken from two populations being compared.
Randomized Experiment A study in which subjects are randomly assigned to treatment groups.
Point Estimate The sample difference p̂1 - p̂2, used as the center of the interval.

Video 1: Choosing the Procedure and Checking Conditions

[0:00 – 8:04]
Context — "Please Trees, Don't Leave!" [0:30] Random samples of trees from two large forests reveal that 36 of 240 trees at high elevation and 25 of 200 trees at low elevation died from a disease. We want a 90% confidence interval for the difference, high minus low, in the population proportions.
1. [1:25] This procedure is called a z-interval for a difference in .
2. [1:45] This interval is appropriate when we have two random samples from two populations or subjects randomly to two treatment groups in a randomized .
3. [1:59] In general, we check for and that the sampling distribution has the correct .
Condition Checklist: [2:22 – 3:21]
4. [2:29] When sampling without replacement, each sample size should be less than or equal to of the population size.
5. [2:49] To justify the z-interval, the observed numbers of and in each sample must all be at least .
6. [4:09] In the tree example, let subscript 1 mean elevation and subscript 2 mean elevation.
7. [4:52] In the high-elevation sample, trees died from the disease and did not.
8. [5:08] In the low-elevation sample, trees died from the disease and did not.
9. [5:08] Because all four counts are at least 10, the sampling distribution is approximately , so the conditions are .
Context — "Practice Is a Drag" [5:39] In a randomized experiment, 23 swimmers wore drag suits and 24 wore regular suits. A success was a swimmer going slower than their average time. The drag-suit group had 13 successes, and the regular-suit group had 8 successes.
10. [6:23] In the swimmer study, swimmers were randomly to treatment groups, but the regular-suit group had only successes, so the conditions are .

Video 2: Margin of Error and Calculating the Interval

[0:00 – 8:56]
11. [1:06] Confidence intervals are written as point estimate plus or minus .
12. [1:36] Margin of error equals times the of the statistic.
13. [2:09] The standard error is an estimate of the of the sampling distribution, so we replace the unknown population proportions with the .
14. [3:26] The middle C% of the standard normal distribution is the , and for a 90% interval the positive critical value is .
Confidence Interval Formula: [4:18]
Margin of error = z* √[(p̂1(1 - p̂1))/n1 + (p̂2(1 - p̂2))/n2]
CI = (p̂1 - p̂2) ± z* √[(p̂1(1 - p̂1))/n1 + (p̂2(1 - p̂2))/n2]
Context — Tree Example Revisited [4:36] For the diseased-tree problem, let 1 = high elevation and 2 = low elevation. Use the sample proportions to build a 90% confidence interval for phigh - plow.
15. [4:47] In the tree example, p̂1 = and p̂2 = .
16. [5:23] The point estimate for high minus low is , and the margin of error is .
17. [5:34] The 90% confidence interval goes from to .
18. [5:43] You do not have to the full formula because you can piece it together from the formula .
Context — "Who's a Good Dog?" [6:33] Dogs were randomly assigned to a new tick-repellent formula or an old one. Of the 80 dogs with the new formula, 12 got ticks; of the 80 dogs with the old formula, 25 got ticks. We want a 95% confidence interval for new minus old.
19. [7:52] For the dog example, the 95% critical value is , and the point estimate for new minus old is .
20. [8:08] That gives a 95% confidence interval from to .

Key Takeaways

[Summary]
  1. Use a two-sample z-interval for a difference in proportions when comparing two population proportions.
  2. Check for two independent random samples or random assignment, apply the 10% condition when sampling without replacement, and verify all success/failure counts are at least 10.
  3. Margin of error = critical value times standard error.
  4. The confidence interval is centered at p̂1 - p̂2 and gives plausible values for the true difference p1 - p2 in context.

Reflection Questions

[AI-Graded]
R1. A botanist takes two independent random samples from two large parks to compare the proportion of trees with leaf damage. In Park A, 42 of 210 sampled trees had leaf damage. In Park B, 30 of 190 sampled trees had leaf damage. Identify the appropriate confidence interval procedure and verify whether the conditions are met. Justify each condition in context.
R2. In the dog tick experiment, a 95% confidence interval for pnew - pold is (-0.2907, -0.0343), where the parameters are the true proportions of similar dogs that would get ticks with each formula. Interpret this interval in context, and explain what the negative values suggest about the new formula.

Exit Ticket

A school district wants to compare the proportion of students who buy lunch at two high schools. Random samples of 150 students from High School A and 150 students from High School B found that 60 students from A and 39 students from B bought lunch that day. Assume each school has far more than 1,500 students. Use pA - pB as the parameter and use z* = 1.96 for a 95% confidence interval.

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