Unit-3-CHPT07-Sec7-2PT1: Problem 1
(2 points)
Use the given data to find the $95 \%$ confidence interval estimate of the population mean $\mu$. Assume that the population has a normal distribution.
IQ scores of professional athletes:
Sample size $n=20$
Mean $\bar{x}=106$
Standard deviation $s=11$
$\square< \mu< \square$

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Step 1 ：Given the sample size \(n = 20\), the sample mean \(\overline{x} = 106\), and the standard deviation \(s = 11\).

Step 2 ：We are asked to find the 95% confidence interval estimate of the population mean \(\mu\). The z-score for a 95% confidence interval is approximately 1.96.

Step 3 ：First, calculate the margin of error using the formula: \(E = z \cdot \frac{s}{\sqrt{n}}\), where \(z\) is the z-score, \(s\) is the standard deviation, and \(n\) is the sample size.

Step 4 ：Substitute the given values into the formula: \(E = 1.96 \cdot \frac{11}{\sqrt{20}} \approx 4.82\).

Step 5 ：Next, calculate the confidence interval using the formula: \(\overline{x} - E < \mu < \overline{x} + E\).

Step 6 ：Substitute the given values into the formula: \(106 - 4.82 < \mu < 106 + 4.82\).

Step 7 ：Simplify to get the final answer: \(\boxed{101.18 < \mu < 110.82}\).