Answer :
Sure! Let's go through the solution for the problem step-by-step:
a. Mean of the Distribution of Sample Means
When dealing with a normal distribution, the mean of the distribution of sample means (also known as the expected value of the sample means) is the same as the population mean. This property holds for any sample size.
So, the mean of the distribution of sample means is:
[tex]\[
\mu_{\bar{x}} = \mu = 191.9
\][/tex]
b. Standard Deviation of the Distribution of Sample Means
The standard deviation of the distribution of sample means, also called the standard error, can be calculated using the formula:
[tex]\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- [tex]\(\sigma\)[/tex] is the standard deviation of the population, which is 97.9.
- [tex]\(n\)[/tex] is the sample size, which is 130.
Using the formula:
[tex]\[
\sigma_{\bar{x}} = \frac{97.9}{\sqrt{130}} \approx 8.59
\][/tex]
To two decimal places, the standard deviation of the distribution of sample means is:
[tex]\[
\sigma_{\bar{x}} \approx 8.59
\][/tex]
So, the answers are:
a. The mean of the distribution of sample means:
[tex]\[
\mu_{\bar{x}} = 191.9
\][/tex]
b. The standard deviation of the distribution of sample means, rounded to two decimal places:
[tex]\[
\sigma_{\bar{x}} \approx 8.59
\][/tex]
a. Mean of the Distribution of Sample Means
When dealing with a normal distribution, the mean of the distribution of sample means (also known as the expected value of the sample means) is the same as the population mean. This property holds for any sample size.
So, the mean of the distribution of sample means is:
[tex]\[
\mu_{\bar{x}} = \mu = 191.9
\][/tex]
b. Standard Deviation of the Distribution of Sample Means
The standard deviation of the distribution of sample means, also called the standard error, can be calculated using the formula:
[tex]\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- [tex]\(\sigma\)[/tex] is the standard deviation of the population, which is 97.9.
- [tex]\(n\)[/tex] is the sample size, which is 130.
Using the formula:
[tex]\[
\sigma_{\bar{x}} = \frac{97.9}{\sqrt{130}} \approx 8.59
\][/tex]
To two decimal places, the standard deviation of the distribution of sample means is:
[tex]\[
\sigma_{\bar{x}} \approx 8.59
\][/tex]
So, the answers are:
a. The mean of the distribution of sample means:
[tex]\[
\mu_{\bar{x}} = 191.9
\][/tex]
b. The standard deviation of the distribution of sample means, rounded to two decimal places:
[tex]\[
\sigma_{\bar{x}} \approx 8.59
\][/tex]
