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AP Statistics 7.7 Justifying a Claim About the Difference of Two Means Based on a Confidence Interval - MCQs - Exam Style Questions

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Question

The marketing director for an ice cream company investigated whether there was a difference in preference for two new ice cream flavors—cotton candy and mango. Each participant from a large group of people was randomly assigned to taste one of the two flavors. After tasting, each person rated the flavor on a numerical scale from \(1\) to \(5\), where \(1\) represented strongly dislike and \(5\) represented strongly like. A two-sample \(t\)-interval for a difference between means (cotton candy minus mango) was constructed. Based on the interval, there was convincing statistical evidence of a difference in population mean flavor ratings, with mango having the greater sample mean rating. Which of the following could be the constructed interval?
(A) \((-20,-15)\)
(B) \((-2.1,-1.3)\)
(C) \((-1.4, 2.6)\)
(D) \((1.5, 2.7)\)
(E) \((15, 20)\)
▶️ Answer/Explanation
Detailed solution

Interval is for \( \mu_{\text{cotton}} – \mu_{\text{mango}} \).
Since mango has the larger mean, the difference must be negative.
Ratings are between \(1\) and \(5\), so the difference must lie in \((-4,4)\).
An interval that shows convincing evidence must lie entirely below \(0\).
Only \(\boxed{(-2.1,-1.3)}\) is wholly negative and within \((-4,0)\); others are impossible, include \(0\), or are positive.
✅ Answer: (B)

Question

A two-sample t-test of the hypotheses \(H_{0}:\mu_{1}-\mu_{2}=0\) versus \(H_{a}:\mu_{1}-\mu_{2}>0\) produces a p-value of 0.08. Which of the following must be true?

I. A 90 percent confidence interval for the difference in means will contain the value 0.
II. A 95 percent confidence interval for the difference in means will contain the value 0.
III. A 99 percent confidence interval for the difference in means will contain the value 0.

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
▶️ Answer/Explanation
Detailed solution

A confidence interval will contain \(0\) if the corresponding two-sided hypothesis test fails to reject \(H_0\). A test fails to reject if its p-value is greater than \(\alpha\).

The given one-sided p-value is \(0.08\). The corresponding two-sided p-value is \(2 \times 0.08 = 0.16\).

– 90% CI (\(\alpha=0.10\)): Since \(0.16 > 0.10\), the test fails to reject. The interval contains \(0\). (Statement I is true).
– 95% CI (\(\alpha=0.05\)): Since \(0.16 > 0.05\), the test fails to reject. The interval contains \(0\). (Statement II is true).
– 99% CI (\(\alpha=0.01\)): Since \(0.16 > 0.01\), the test fails to reject. The interval contains \(0\). (Statement III is true).

All three statements must be true. (Note: This contradicts the provided answer key, which likely assumes the given p-value was from a two-sided test. Based on the question as written, all three are true).
✅ Answer: (E)

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