AP Statistics Unit 7 Inference for Means

Topic 7.6: Confidence Intervals for the Difference of Two Means

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

2-Sample t-Interval A confidence interval procedure used to estimate the difference between two population means from independent samples.
Independent Samples Two samples collected so that knowing the outcomes in one group does not determine the outcomes in the other group.
Difference in Means The parameter μ1 − μ2 or the statistic x̄1 − x̄2.
Point Estimate The sample statistic at the center of the interval; here it is the difference in sample means.
Margin of Error The amount added and subtracted from the point estimate to create the interval.
Critical Value The t* value determined by the confidence level and the degrees of freedom.
Standard Error A measure of how much the difference in sample means typically varies from the difference in population means.
Degrees of Freedom For a two-sample t-interval, a value usually found with technology and used to determine the critical value.
Pooled: No The default calculator setting for a 2-SampTInt because pooling requires a different procedure and extra conditions.
Interpretation A contextual statement about plausible values of the true difference in population means.

Video 1: Identifying the Procedure and Checking Conditions

[0:00 – 7:00]
1. [0:11] In this video, we learn how to identify an appropriate confidence procedure and verify the for a difference of two population means.
Context — Argiope Spiders [0:50 – 1:09] Researchers collected a random sample of 14 adult female spiders and a random sample of 14 adult male spiders from the genus Argiope. The goal was to calculate and interpret a 95% confidence interval for the difference in the population mean body lengths of female and male spiders.
2. [1:21] The spider example compares groups, uses data, and asks for a 95% confidence interval.
3. [1:48] That makes the correct procedure a for the difference in population .
Procedure Check: [1:21 – 1:48]
  1. Count the number of groups. Here there are two groups: female spiders and male spiders.
  2. Identify the type of data. Body lengths are quantitative measurements.
  3. Notice the goal. We are estimating a difference in population means with a confidence interval.
4. [2:01] For independence, the data should come from two independent random samples or a randomized .
5. [2:01] When sampling without replacement, each sample should be less than or equal to of its respective population.
6. [2:23] For the normal condition, both sample sizes should be greater than or both samples should come from approximately populations.
7. [2:23] If both samples are small, the sample distributions should be free from extreme or .
8. [2:51 – 3:04] In the original spider study, both samples were randomly , and 14 in each group is safely under the condition.
9. [3:28] The histograms showed both samples were roughly and , so it was reasonable to assume approximately normal populations.
10. [4:06 – 4:32] If the researchers instead had 35 females and 35 males, condition 3 would be easier because both sample sizes are greater than , making the sampling distribution approximately .
Practice — Slow Internet and Stress [5:03 – 6:41] A teacher randomly assigned 18 students to work with slow internet and 18 students to work with fast internet, then measured diastolic blood pressure. The boxplots showed clear skewness and an outlier.
11. [5:27 – 6:41] In the slow-internet experiment, random satisfies the first condition, but strong skewness and an mean the teacher should not proceed with the interval.

Video 2: Margin of Error and Calculating the Interval

[0:00 – 8:14]
12. [0:11] In this video, we learn how to determine the and calculate a confidence for a difference in means.
13. [0:22 – 0:58] In general, a confidence interval is the statistic plus or minus a times the of the statistic.

2-Sample t-Interval Formulas

Confidence interval = statistic ± (critical value)(standard error of statistic)
(x̄1 − x̄2) ± t*√((s12/n1) + (s22/n2))
Margin of error = t* × standard error
14. [1:08 – 1:21] The point estimate is another name for the and is the of the interval.
15. [1:32 – 1:55] The margin of error combines the and the , and a higher confidence level gives a critical value.
Return to the Spider Example [2:14 – 2:33] The calculator gave the following summary statistics for body lengths in millimeters.
Group n x̄ s
Female1414.153.54
Male144.1070.92
16. [2:24 – 2:42] For the spiders, the point estimate for female minus male is 14.15 − 4.107 = millimeters.
17. [2:59 – 3:34] For a two-sample t-interval, the degrees of freedom are best found using , and the calculator setting should be pooled: .
18. [4:00] Using 2-SampTInt gives a 95% interval of (, ) with degrees of freedom about .
19. [4:24 – 4:37] To find the critical value for this 95% interval, use invT with bottom-tail area and df = 14.73, which gives t* about .
20. [4:54 – 5:20] The margin of error for the spider interval is millimeters, so the interval can be written as 10.043 ± .
21. [6:20 – 7:30] In the practice multiple-choice problem, answer choice A was wrong because it used a critical value, and the correct answer was choice with t* about for a 90% interval.

Key Takeaways

[Summary]
  1. Use a 2-sample t-interval when you have two independent groups, quantitative data, and want to estimate a difference in population means.
  2. Check independence, the 10% condition when sampling without replacement, and either both sample sizes greater than 30 or roughly normal sample distributions with no extreme skewness or outliers.
  3. For two means, the confidence interval is (x̄1 − x̄2) ± t*√((s12/n1) + (s22/n2)).
  4. Degrees of freedom and the final interval are typically found with technology, and the same interval can be written with endpoints or as point estimate ± margin of error.

Reflection Questions

[AI-Graded]
R1. Explain how to identify the correct confidence interval procedure for the difference of two population means and how to verify the conditions. Use the spider example and the slow-internet experiment to discuss the roles of two groups, quantitative data, independence, the 10% condition, and the shape or sample-size requirement.
R2. Explain how to calculate and interpret a 2-sample t-interval for a difference in means. Use the spider example to describe the point estimate, margin of error, how technology gives the degrees of freedom, why pooled should be set to no, and what the interval (7.956, 12.129) means in context.

Exit Ticket

Random samples of 14 adult female Argiope spiders and 14 adult male Argiope spiders were collected. Histograms for both groups were roughly unimodal and symmetric. The summary statistics were x̄F = 14.15 mm, sF = 3.54 mm, x̄M = 4.107 mm, and sM = 0.92 mm. A calculator produced the 95% confidence interval (7.956, 12.129) for (female − male).

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