Problem
A sample of size $n=50$ has mean $\bar{x}=35.7$ and standard deviation $s=2.8$. Part 1 of 3 (a) Construct a $95 \%$ confidence interval for $\mu$. Round to 2 decimal places. Part: $1 / 3$ Part 2 of 3 (b) Estimate the sample size needed so that a $95 \%$ confidence interval for $\mu$ will have a margin of error equal to 2.5 . Round your answer to next whole number. Skip Part Check
Solution
Step 1 ：Given a sample size \(n = 50\), mean \(\overline{x} = 35.7\), and standard deviation \(s = 2.8\).
Step 2 ：We are asked to construct a 95% confidence interval for \(\mu\). The z-score for a 95% confidence interval is 1.96.
Step 3 ：The formula for a confidence interval is \(\overline{x} \pm z \cdot \frac{s}{\sqrt{n}}\).
Step 4 ：Substitute the given values into the formula to get the lower and upper bounds of the confidence interval.
Step 5 ：The lower bound is \(35.7 - 1.96 \cdot \frac{2.8}{\sqrt{50}} = 34.92\).
Step 6 ：The upper bound is \(35.7 + 1.96 \cdot \frac{2.8}{\sqrt{50}} = 36.48\).
Step 7 ：Final Answer: The 95% confidence interval for \(\mu\) is \(\boxed{(34.92, 36.48)}\).
