Unit-3-CHPT07-Sec7-2PT1: Problem 1
(2 points)
Use the given data to find the $95\mathrm{\%}$ 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 $\overline{x}=106$
Standard deviation $s=11$
$◻<\mu <◻$

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Step 1 ：Given the sample size $n=20$, the sample mean $\bar{x}=106$, 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: $\bar{x}-E<\mu <\bar{x}+E$.

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

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