Question

A random sample is selected from a population with mean \(\mu=200\) and standard deviation \(\sigma=15 .\) Determine the mean and standard deviation of the \(\bar{x}\) sampling distribution for each of the following sample sizes: a. \(n=12\) d. \(n=40\) b. \(n=20\) e. \(n=90\) c. \(n=25\) f. \(n=300\)

Short answer

The mean of the sampling distribution for all sample sizes is 200. The standard deviations are 15/\(\sqrt{12}\), 15/\(\sqrt{40}\), 15/\(\sqrt{20}\), 15/\(\sqrt{90}\), 15/\(\sqrt{25}\), and 15/\(\sqrt{300}\) respectively.

Step-by-step solution

Calculate the mean of the sampling distribution

As stated, the mean of the sampling distribution is equal to the mean of the population. Therefore, for all the given sample sizes (n), the mean, \(\mu_{\bar{x}}\) = \(\mu\) = 200.

Calculate the standard deviation for sample size \(n=12\)

The standard error is calculated using the formula \(\sigma_{\bar{x}} = \sigma/\sqrt{n}\). Substituting the given values, \(\sigma_{\bar{x}}\) = 15/\(\sqrt{12}\).

Calculate the standard deviation for the remaining sample sizes

Similarly, we can calculate the standard error for all sample sizes, with sample sizes \(n=40, 20, 90, 25,\) and \(300\), by substituting these values in the standard deviation formula.

Summary of results

We have now calculated the mean and standard deviation of the sampling distribution for each of the given sample sizes. The mean is constant at 200, and the standard deviations are calculated using the standard error formula for each sample size.