AP Statistics 4.3 Mean CI Claims Overview

Use a confidence interval for a mean or mean difference to decide whether a claimed value remains plausible in the population context.

[Maximum number: 1]

Recent studies have determined with $94 \%$ confidence that the mean number of trees cut down every 24 hours to make toilet paper is between 25,000 and 29,000. What is meant by the confidence level in this context?

A

A confidence interval of the true mean number of trees cut down every 24 hours to make toilet paper was calculated using z-scores of ±1.881.

B

A confidence interval of the true mean number of trees cut down every 24 hours to make toilet paper was calculated using t-scores consistent with d f=n-1 and tail probabilities of 0.03 .

C

We are $94 \%$ confident that the true mean number of trees cut down every 24 hours to make toilet paper is between 25,000 and 29,000.

D

If all possible random samples of data are obtained by this method, approximately 94\% will yield confidence intervals that capture the true mean number of trees cut down every 24 hours to make toilet paper.

According to a 2017 national survey in Country B, the mean number of bedrooms in newly built

houses was 2.9. Rodney, a researcher, believes the mean number of bedrooms in newly built

houses in the country was different in 2024 than it was in 2017. To investigate his belief, he took

a large random sample of newly built houses in Country B in 2024 and recorded the number of

bedrooms in each house. The distribution of the number of bedrooms for the sampled houses is

summarized in the table.

Distribution of the Number of Bedrooms for the Houses Sampled in 2024

Table for Question 4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean question 2 — AP Statistics

A different researcher, Keisha, suggests using a confidence interval to investigate whether

the mean number of bedrooms in newly built houses in 2024 in Country B was different

from 2.9.

Assume the conditions for inference have been met. Using Rodney's data, Keisha

calculated a one-sample 97 percent confidence interval to estimate the population mean

as (3.01,3.19). Based on the confidence interval, what conclusion can be made for Rodney's

hypothesis test in part B at $\alpha=0.03$ ? Justify your answer.

	Model Solution	Scoring
C	Because the value 2.9 is not contained within the 97\% confidence interval, the null hypothesis should be rejected. Therefore, there is convincing statistical evidence, at the $\alpha=0.03$ level of significance, that the population mean number of bedrooms in newly built houses in 2024 from Country B is not equal to 2.9 (or is different than that in 2017).	Essentially correct (E) if the response satisfies the following two components: 1. States a conclusion consistent with and in terms of the alternative hypothesis using nondefinitive language 2. Provides a justification for the conclusion by indicating that 2.9 is not contained within the confidence interval Partially correct (P) if the response satisfies only one of the two components required for E. Incorrect (I) if the response does not meet the criteria for E or P.

Scoring Notes:

- A response that only provides an interpretation of the given confidence interval is scored I.

- A response may satisfy the conclusion aspect of component 1 by using words such as "there is evidence to

support the alternative," "there is statistical evidence that $\mathrm{H}_{\mathrm{a}}$ is true," "I am 97\% confident there is a

difference in means," or "because 2.9 is not included in our interval, it is not a plausible value for the

mean."

- If an explicit decision is stated and the conclusion is inconsistent with the decision, component 1 is not

satisfied. A response that incorrectly indicates that 2.9 is contained in the confidence interval and then

provides an otherwise correct conclusion based on 2.9 being contained in the interval satisfies component

1 but not component 2.

- A response that provides a conclusion that is consistent with an incorrect alternative hypothesis identified

in part B satisfies component 1.

- A definitive response that states that the average number of bedrooms is 3.10 (the value of the sample

mean or any other number) does not satisfy component 1, even if the response makes additional correct

statements about the alternative hypothesis.

- A response that draws a conclusion that is clearly about the sample mean (e.g., a statement about "average

number of bedrooms for newly built houses in the study") does not satisfy component 1.

- If the conclusion includes a definitive statement (e.g., "this proves that we have enough evidence that the

mean number of bedrooms in newly built homes in 2024 is 2.9" or "Rodney is correct; the mean number

of bedrooms in newly built homes in 2004 is 2.9"), then component 1 is not satisfied.

- A response that reverse engineers the given confidence interval to obtain a standard error using a

reasonable critical value for a 97\% confidence interval and then uses the standard error to compute a

t-statistic and p-value to reach a correct conclusion satisfies component 1 but does not satisfy

component 2.

[Maximum number: 1]

Two meteorologists use the same sample data to calculate the average temperature across the United States. The first meteorologist constructs a 90\% confidence interval, while the second constructs a 95\% confidence interval. Which of the following statements is true?

A

The midpoint of the first meteorologist's interval will be less than the midpoint of the second meteorologist's interval.

B

The midpoint of the first meteorologist's interval will be greater than the midpoint of the second meteorologist's interval.

C

The width of the first meteorologist's interval will be less than the width of the second meteorologist's interval.

D

The width of the first meteorologist's interval will be greater than the width of the second meteorologist's interval.

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Covered syllabus points

4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean question 1

4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean question 2

4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean question 3