AP Statistics Unit 7 Inference for Means

Topic 7.7: Justifying a Claim About the Difference of Two Means Based on a Confidence Interval

Interpreting the Interval, Avoiding Common Errors, and Using 0 to Evaluate Claims
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Interpretation Template A sentence structure for confidence intervals: we are C% confident that an interval captures the parameter being estimated.
Direction of Subtraction The order of the groups in the difference, such as female − male or northern − southern.
Parameter The true population value being estimated, such as μ1 − μ2.
Plausible Value A value for the parameter that is consistent with the confidence interval.
Zero in the Interval Means no difference is plausible, so the interval does not support a claim that the means are different.
Zero not in the Interval Means no difference is not plausible, so the interval provides evidence that the population means differ.
Convincing Evidence Evidence strong enough to support a claim about a population parameter in context.
Sample Statistic A value computed from the sample, such as x̄1 − x̄2, which should not be the focus of the interval interpretation.
Confidence Level In repeated sampling, the percent of intervals that would capture the true parameter.
Context The real situation being described, which must be included in the interpretation and conclusion.

Video 1: Interpreting the Spider Interval

[0:00 – 6:18]
1. [0:13] In this lesson, we learn how to interpret a confidence interval for a difference in means and how to justify a based on a confidence interval.
Context — Argiope Spiders [0:23 – 0:35] From Topic 7.6, the 95% confidence interval for the difference in mean body lengths (female − male) was (7.956, 12.129) millimeters. In this lesson, the focus is how to interpret that interval and use it to evaluate a claim.
2. [0:48] In general, we are C% that the interval from the lower bound to the upper bound captures the value to be .
3. [1:20] For the spider example, the interval from 7.956 to 12.129 mm captures the difference, minus , in the true mean body lengths of all Argiope spiders.
4. [1:39] A correct interpretation must include the direction that you .
5. [1:46] Another correct interpretation is that female Argiope spiders are between 7.956 and 12.129 mm than male Argiope spiders.
6. [2:05 – 2:29] If we subtract male minus female instead, the confidence interval values are the same but .
7. [2:46 – 3:14] Saying the interval describes "these spiders" is wrong because it refers to the instead of the population parameter.
8. [3:22 – 3:47] Saying there is a 95% the interval captures the parameter is incorrect for a fixed interval.
Interpretation Rules: [0:48 – 3:47]
  1. State the confidence level and the interval endpoints.
  2. Name the parameter being estimated, not the sample statistic.
  3. Include the direction of subtraction in words or symbols.
  4. Do not describe the interval as a probability statement about this one fixed interval.

Video 2: Using the Interval to Justify a Claim

[0:00 – 7:03]
Claim Question — Follow-Up Spider Study [3:55 – 4:19] Researchers want to know whether Argiope spiders should be included in a follow-up study of spider species where females are larger than males on average.
9. [4:25] If female and male Argiope spiders were equal in size, the difference in their mean body lengths would be .
10. [4:35] Because 0 is not on the 95% interval from 7.956 to 12.129, 0 is not a value for the difference in population mean body lengths.
11. [4:50] Therefore, there is convincing evidence that female Argiope spiders have significantly body lengths than males on average.
12. [5:31 – 5:44] In the restaurant example, the correct conclusion must be about the true difference in population , not the eight sample containers.
13. [6:18 – 6:42] For the interval (−9.3, 3.2), 0 is the interval, so there is not convincing evidence of a between foam and plastic containers.
Claim Rule: [4:25 – 6:42]
  1. If 0 is not in the interval, then no difference is not plausible, so the interval supports a claim of a difference.
  2. If every value in the interval is positive, then group 1 tends to have the larger mean.
  3. If 0 is in the interval, then no difference is plausible, so the interval does not support a claim that the means differ.

Video 3: Full Procedure with the Fire Station Problem

[0:00 – 6:53]
AP Problem Pointers: [0:16 – 0:50]
  1. Read the entire question first for understanding, then reread for details.
  2. Annotate important facts such as sample sizes, means, standard deviations, and the claim.
  3. Answer the exact question that is asked.
  4. Define subscripts and keep the context in every part of the response.
AP Practice — Fire Stations [1:15 – 1:53] A council member wants to know whether the northern and southern fire stations have different mean response times. Random samples of 50 calls were taken from each station.
Station n x̄ s
Northern504.33.7
Southern505.33.2
14. [0:50] On AP problems, you must communicate with clear .
15. [2:02 – 2:14] Let μN and μS be the mean response times for the northern and southern fire stations. The correct procedure is a 2-sample t-interval for a difference in population .
16. [2:47 – 3:07] Each sample of 50 calls should satisfy the condition, and because both sample sizes are greater than 30, the sampling distribution is approximately .
17. [3:21 – 3:36] On the calculator, use 2-SampTInt with input and keep pooled set to .
18. [4:00 – 4:08] Using as the bottom-tail area with df about 96 gives a critical value of about .
19. [4:17 – 4:34] The 95% confidence interval for northern minus southern is between −2.37 and 0.37 .

Video 4: Drawing the Conclusion from the Interval

[0:00 – 6:54]
Fire Station Conclusion [4:34 – 5:06] Use the interval from part (a), (−2.37, 0.37), to decide whether the confidence interval supports the council member's belief that the two stations have different mean response times.
20. [4:45] Since 0 is the interval (−2.37, 0.37), it is a plausible value for the difference in mean response times.
21. [4:58] Therefore, the confidence interval does not the council member's belief that the two fire stations have different mean response times.
22. [5:06] Do not say the council member is ; instead say the interval does not provide convincing evidence of a difference.
23. [6:00 – 6:14] If many random samples were taken and 95% intervals were constructed, about 95% of those intervals would the true difference in population mean response times.

Key Takeaways

[Summary]
  1. Interpret a confidence interval as a range of plausible values for the true difference in population means, not for the sample statistics.
  2. Always include the direction of subtraction; reversing the order negates the interval and changes the wording of the conclusion.
  3. If 0 is not in the interval, then the interval supports a claim of a difference in the direction shown by the interval.
  4. If 0 is in the interval, then no difference is plausible, so the interval does not support a claim that the means differ.
  5. On a full AP response, define the parameter, identify the 2-sample t-interval, check conditions, report the interval, and conclude in context.

Reflection Questions

[AI-Graded]
R1. Explain how to correctly interpret a confidence interval for the difference of two population means. Use the spider example to discuss the general interpretation template, the direction of subtraction, why the interval must describe a parameter instead of a sample, and two incorrect interpretations students should avoid.
R2. Explain how to use a confidence interval to justify or not justify a claim about two population means. Use the spider interval, the restaurant interval (−9.3, 3.2), and the fire-station interval (−2.37, 0.37) to discuss what it means when 0 is or is not in the interval and how to state the conclusion correctly.

Exit Ticket

A town council member believes the two fire stations have different mean response times. Random samples of 50 calls from the northern and southern fire stations were selected. The summary statistics were x̄N = 4.3 minutes, sN = 3.7 minutes, x̄S = 5.3 minutes, and sS = 3.2 minutes. A calculator produced the 95% confidence interval (−2.37, 0.37) for (northern − southern).

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