Question

for the following situation, find the mean and standard deviation of the population. list all samples (with replacement) of the given size from that population and find the mean of each. find the mean and standard deviation of the sampling distribution and compare them with the mean and standard deviation of the population. the scores of three students in a study group on a test are 91, 90, 92. use a sample size of 3. g. 90,92,92 $\bar{x}=91.33$ j. 92,91,91 $\bar{x}=91.33$ m. 92,91,92 $\bar{x}=91.67$ p. 92,92,92 $\bar{x}=92$ s. 90,91,91 $\bar{x}=90.67$ v. 92,92,91 $\bar{x}=91.67$ y. 90,90,90 $\bar{x}=90$ h. 90,90,91 $\bar{x}=90.33$ k. 92,90,92 $\bar{x}=91.33$ n. 90,91,90 $\bar{x}=90.33$ q. 91,90,90 $\bar{x}=90.33$ t. 91,91,91 $\bar{x}=91$ w. 91,90,92 $\bar{x}=91$ z. 91,91,92 $\bar{x}=91.33$ i. 91,92,91 $\bar{x}=91.33$ l. 92,91,90 $\bar{x}=91$ o. 92,92,90 $\bar{x}=91.33$ r. 90,92,90 $\bar{x}=90.67$ u. 91,92,92 $\bar{x}=91.67$ x. 90,90,92 $\bar{x}=90.67$ . 90,91,92 $\bar{x}=91$ the mean of the sampling distribution is 91.00 (round to two decimal places as needed.) the standard deviation of the sampling distribution is (round to two decimal places as needed.)

Explanation:

Step1: Calculate population mean

The population data is \(91,90,92\). The population mean \(\mu\) is calculated as \(\mu=\frac{91 + 90+92}{3}=\frac{273}{3}=91\).

Step2: Calculate population variance

The population variance \(\sigma^{2}=\frac{(91 - 91)^{2}+(90 - 91)^{2}+(92 - 91)^{2}}{3}=\frac{0 + 1+1}{3}=\frac{2}{3}\). Then the population standard - deviation \(\sigma=\sqrt{\frac{2}{3}}\approx0.82\).

Step3: List all samples of size \(n = 3\) with replacement and their means

The samples and their means are already given. There are \(3\times3\times3=27\) samples.

Step4: Calculate the mean of the sampling distribution

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean. We are given that \(\mu_{\bar{x}} = 91.00\) (which is consistent with the theoretical result \(\mu_{\bar{x}}=\mu\)).

Step5: Calculate the standard deviation of the sampling distribution

The formula for the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard - deviation and \(n\) is the sample size. Since \(\sigma=\sqrt{\frac{2}{3}}\) and \(n = 3\), then \(\sigma_{\bar{x}}=\frac{\sqrt{\frac{2}{3}}}{\sqrt{3}}=\sqrt{\frac{2}{9}}\approx0.47\).

Answer:

\(0.47\)