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 $μ$ . 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 $μ$ will have a margin of error equal to 2.5 . Round your answer to next whole number.
Skip Part
Check

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The 95% confidence interval for $μ$ is $(34.92, 36.48)$ .

Steps

Step 1 ：Given a sample size $n = 50$ , mean $\overset{―}{x} = 35.7$ , and standard deviation $s = 2.8$ .

Step 2 ：We are asked to construct a 95% confidence interval for $μ$ . The z-score for a 95% confidence interval is 1.96.

Step 3 ：The formula for a confidence interval is $\overset{―}{x} \pm z \cdot \frac{s}{\sqrt{n}}$ .

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 $μ$ is $(34.92, 36.48)$ .