Certainly! Let's find the mean and standard deviation of a sampling distribution of sample means for the given population parameters.
#### Step 1: Identify the parameters of the population
We are given:
- The mean of the population, [tex]\(\mu = 82\)[/tex]
- The standard deviation of the population, [tex]\(\sigma = 28\)[/tex]
- The sample size, [tex]\(n = 232\)[/tex]
#### Step 2: Determine the mean of the sampling distribution of the sample means
The mean of the sampling distribution of the sample means (also known as the expected value of the sample mean) is equal to the mean of the population. Therefore:
[tex]\[
\mu_{\dot{x}} = \mu = 82
\][/tex]
#### Step 3: Determine the standard deviation of the sampling distribution of the sample means
The standard deviation of the sampling distribution of the sample means, also known as the standard error of the mean (SEM), is calculated using the formula:
[tex]\[
\sigma_{\dot{x}} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting in the given values:
[tex]\[
\sigma_{\dot{x}} = \frac{28}{\sqrt{232}}
\][/tex]
To find the numerical value, we simplify the expression:
[tex]\[
\sqrt{232} \approx 15.23
\][/tex]
And then:
[tex]\[
\sigma_{\dot{x}} \approx \frac{28}{15.23} \approx 1.838
\][/tex]
#### Step 4: Summarize the results
The mean of the sampling distribution is:
[tex]\[
\mu_{\dot{x}} = 82
\][/tex]
The standard deviation of the sampling distribution, rounded to three decimal places, is:
[tex]\[
\sigma_{\dot{x}} = 1.838
\][/tex]
Thus, the final answers are:
[tex]\[
\mu_{\dot{x}} = 82
\][/tex]
[tex]\[
\sigma_{\dot{x}} = 1.838
\][/tex]
