AP Statistics 7.1 Introducing Statistics: Should I Worry About Error? - MCQs - Exam Style Questions

Question

Based on previous research, the standard deviation of the distribution of the age at which children begin to walk is estimated to be 1.5 months. A random sample of children will be selected, and the age at which each child begins to walk will be recorded. A 99 percent confidence interval for the average age at which children begin to walk will be constructed using the data obtained from the sample of children. Of the following, which is the smallest sample size that will result in a margin of error of 0.1 month or less for the confidence interval?

(A) 400
(B) 900
(C) 1,300
(D) 1,600
(E) 2,100

▶️ Answer/Explanation

Detailed solution

1. Margin of Error Formula:
\(ME = z^* \times \frac{\sigma}{\sqrt{n}}\) where \(z^*\) for 99% CI ≈ 2.576

2. Set Up Inequality:
\(2.576 \times \frac{1.5}{\sqrt{n}} \leq 0.1\)

3. Solve for \(n\):
\(\frac{2.576 \times 1.5}{\sqrt{n}} \leq 0.1\)
\(\frac{3.864}{\sqrt{n}} \leq 0.1\)
\(\sqrt{n} \geq \frac{3.864}{0.1} = 38.64\)
\(n \geq (38.64)^2 \approx 1493\)

4. Find Smallest Option:
Smallest \(n\) ≥ 1493 is 1,600

✅ Answer: (D) 1,600

AP Statistics 7.1 Introducing Statistics: Should I Worry About Error? - MCQs - Exam Style Questions

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