Home / AP® Exam / AP® Statistics / AP Statistics 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion- Exam Style Questions – MCQs

AP Statistics 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion- Exam Style Questions - MCQs - New Syllabus

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Question

A survey conducted by a national research center asked a random sample of \(920\) teenagers in the United States how often they use a video streaming service. From the sample, \(59\%\) answered that they use a video streaming service every day. A \(95\%\) confidence interval for the proportion of all teenagers in the United States who would respond that they use a video streaming service every day is \((0.558,0.622)\).
Based on the confidence interval, do the sample data provide convincing statistical evidence that the true proportion is not \(0.5\)?
(A) Yes, because \(0.5\) is not contained in the interval \((0.558,0.622)\).
(B) Yes, because \(0.59\) is the exact value of the population proportion.
(C) No, because \(0.5\) is less than the sample proportion \(0.59\).
(D) No, because a confidence interval cannot be used to make a conclusion about a claim.
▶️ Answer/Explanation

The confidence interval \((0.558,0.622)\) gives plausible values for the true proportion of all teenagers in the United States who would respond that they use a video streaming service every day.
The value \(0.5\) is not included in the interval. In fact, every value in the interval is greater than \(0.5\).
Therefore, the sample data provide convincing statistical evidence that the true proportion is not \(0.5\). More specifically, the evidence suggests the true proportion is greater than \(0.5\).

✅ Answer: (A)

Question

In 2021, information from the US Census indicated that 29% of US households were single person households. In 2022, a group of researchers surveyed 1000 random US households and found that 310 of these households were single person households. The researchers computed a 95% confidence interval for the percentage of single person US households to be (0.2813, 0.3387). What can we claim based on the calculations provided here?

(A) There is sufficient evidence at the 0.05 level of significance to indicate there is no difference in the percentage of single person US households between 2021 and 2022.
(B) There is sufficient significant evidence at the 0.05 level of significance to suggest there is a difference in the percentage of single person US households between 2021 and 2022.
(C) There is not sufficient evidence at the 0.05 level of significance to suggest a significant difference in the percentage of single person US households between 2021 and 2022.
(D) All new samples of 1000 US households in 2022 will show the percentage of single person households as between 28.13 and 33.87 percent.
(E) The percentage of single person households in 2021 and 2022 is the same.

▶️ Answer/Explanation

The 95% confidence interval for the 2022 proportion of single-person households is:

\((0.2813,\;0.3387)\)

The 2021 Census value was:

\(p=0.29\)

Notice that:

\(0.2813 < 0.29 < 0.3387\)

Since the 2021 value of 0.29 lies within the 95% confidence interval, it is a plausible value for the 2022 population proportion. Therefore, the data do not provide sufficient evidence that the proportion of single-person households changed between 2021 and 2022.

A confidence interval that contains the hypothesized value corresponds to failing to reject the null hypothesis in a two-sided significance test at the matching significance level.

Therefore, there is not sufficient evidence at the 0.05 significance level to conclude that a significant difference exists.

✅ Answer: (C)

Question

A 2015 survey of 400 randomly selected drivers reported that 20 percent of the drivers surveyed never use their cruise control. A 95% confidence interval is given by \((0.161, 0.239)\). Which of the following is a correct interpretation of the 95% confidence level?

(A) We are 95 percent confident that the true proportion of all drivers who never use their cruise control is between 0.161 and 0.239.
(B) There is a 0.95 probability that the true proportion of all drivers who never use their cruise control is between 0.161 and 0.239.
(C) 95 percent of all random samples of 400 drivers chosen from the population will result in confidence intervals which contain 0.20.
(D) 95 percent of all random samples of 400 drivers will result in confidence intervals which contain the true proportion of all drivers who never use their cruise control.
(E) 95 percent of all random samples of 400 drivers chosen from the population will have sample proportions between 0.161 and 0.239.

▶️ Answer/Explanation
The confidence level refers to the long-run performance of the confidence interval method, not to a specific interval.
Once an interval has been calculated, the true population proportion is fixed and is either inside or outside the interval.
A correct interpretation of a 95% confidence level is that if many random samples of the same size were taken and a confidence interval were constructed from each sample, approximately 95% of those intervals would contain the true population proportion.
Choice (A) is the correct interpretation typically used when reporting a specific confidence interval. Choice (D) correctly explains the meaning of the 95% confidence level itself.
✅ Answer: (D)
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