AP Statistics 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means- Exam Style Questions - MCQs - New Syllabus
▶️ Answer/Explanation
The 95% confidence interval for the difference in population means is:
\(\mu_A-\mu_B=(-4.5,\;5.2)\)
Notice that the interval contains 0. This means that a difference of zero between the population means is a plausible value based on the sample data.
Because 0 lies within the confidence interval, we do not have convincing evidence that the population means are different. Therefore, we fail to conclude that \(\mu_A\) and \(\mu_B\) differ.
The conclusion must be about the population means, not the sample means, so options involving \(\overline{x}_A\) and \(\overline{x}_B\) are inappropriate.
✅ Answer: (D)
▶️ Answer/Explanation
The interval is for:
\(\mu_{\text{cotton candy}}-\mu_{\text{mango}}\)
Since mango has the greater sample mean rating, the difference (cotton candy − mango) should be negative. Also, because there is convincing statistical evidence of a difference, the confidence interval must not contain 0.
Choices (A) and (E) are impossible because ratings are on a scale from 1 to 5, so differences cannot be that large. Choice (C) contains 0, so it does not provide convincing evidence of a difference. Choice (D) is entirely positive, indicating cotton candy would have the greater mean rating. Only choice (B) is entirely negative and excludes 0.
✅ Answer: (B)
▶️ Answer/Explanation
1. Two-sample t-test Assumptions:
For small samples (\(n < 30\) per group), the normality assumption is crucial.
2. Evaluate Options:
(A) Not necessary for t-procedures
(B) Applies to proportions, not means
(C) Correct – necessary for t-procedures with small n
(D) Incorrect – this is an independent groups design
(E) Applies to chi-square tests, not t-tests
✅ Answer: (C)