Question 2

A population has parameters $\mu=103.5$ and $\sigma=39.7$. You intend to draw a random sample of size $n=114$.

What is the mean of the distribution of sample means?
\[
\mu_{\bar{x}}=
\]

What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)
\[
\sigma_{x}=
\]
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Step 1 ：The mean of the distribution of sample means, often denoted as \(\mu_{\bar{x}}\), is equal to the population mean, \(\mu\). In this case, the population mean is given as 103.5, so \(\mu_{\bar{x}} = 103.5\).

Step 2 ：The standard deviation of the distribution of sample means, often denoted as \(\sigma_{\bar{x}}\), is equal to the population standard deviation, \(\sigma\), divided by the square root of the sample size, \(n\). In this case, the population standard deviation is given as 39.7 and the sample size is given as 114, so \(\sigma_{\bar{x}} = \frac{39.7}{\sqrt{114}}\).

Step 3 ：Let's calculate \(\sigma_{\bar{x}}\).

Step 4 ：\(\sigma_{\bar{x}} = 3.718245671979325\)

Step 5 ：The mean of the distribution of sample means is \(\mu_{\bar{x}} = \boxed{103.5}\).

Step 6 ：The standard deviation of the distribution of sample means is \(\sigma_{\bar{x}} = \boxed{3.72}\) (rounded to two decimal places).