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AP Statistics 7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval - MCQs - Exam Style Questions

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Question

A researcher will investigate the following question by selecting 50 people with pain in their knees: “For people with pain in their knees, is there a difference in the mean level of pain between people who use a supplement daily and people who use a placebo pill daily?” The researcher’s assistant will randomly assign half of the participants to receive the supplement in the form of a daily pill and the other half to receive a daily placebo pill. Neither the researcher nor the participants will know which type of pill each participant will receive. At the end of one week, the participants will record their level of pain on a scale from 1 to 10, where 1 represents no pain and 10 represents extreme pain.
Which of the following confidence intervals should the researcher use for the investigation?
(A) A one-sample t interval for estimating a sample mean
(B) A one-sample t interval for estimating a population mean
(C) A matched-pairs t interval for estimating a mean difference
(D) A two-sample t interval for estimating a difference between sample means
(E) A two-sample t interval for estimating a difference between population means
▶️ Answer/Explanation
Detailed solution
Two independent groups are created by random assignment (supplement vs. placebo).
We compare the difference in population means in pain level between the two groups.
Therefore, the appropriate interval is a two-sample t interval for the difference in population means (\(\mu_{\text{supp}}-\mu_{\text{plac}}\)).
✅ Answer: (E)

Question

A national survey asked \(1{,}501\) randomly selected employed adults how many hours they work per week. Based on the collected data, a \(95\%\) confidence interval for the mean number of hours worked per week for all employed adults was given as \((41.18,\; 42.63)\). Which of the following statements is a correct interpretation of the interval?
(A) Ninety-five percent of all employed adults work between \(41.18\) hours and \(42.63\) hours per week.
(B) The probability is \(0.95\) that a sample of size \(1{,}501\) will produce a mean between \(41.18\) hours and \(42.63\) hours.
(C) Of all samples of size \(1{,}501\) taken from the population, \(95\%\) of the samples will have a mean between \(41.18\) hours and \(42.63\) hours.
(D) We are \(95\%\) confident that the mean number of hours worked per week for employed adults in the sample is between \(41.18\) hours and \(42.63\) hours.
(E) We are \(95\%\) confident that the mean number of hours worked per week for all employed adults is between \(41.18\) hours and \(42.63\) hours.
▶️ Answer/Explanation
Detailed solution

A \(95\%\) confidence interval for a population mean \(\mu\) is a method that, in repeated sampling, captures \(\mu\) in about \(95\%\) of such intervals constructed the same way.
The correct interpretation is a statement about the population mean, not individual data, not the sample mean, and not a probability about this already-computed interval.
Thus we say: “We are \(95\%\) confident that the true mean hours worked per week for all employed adults lies between \(41.18\) and \(42.63\).”
Options (A), (B), and (C) misinterpret the percentage (individuals, probability of a specific sample result, or proportion of sample means). Option (D) incorrectly refers to the sample mean rather than the population mean.
✅ Answer: (E)

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