Question

assuming the normal model can be used, describe the sampling distribution x. choose the correct answer below.
a. normal, with μ_x = 62 and σ_x = 19 / √15
b. normal, with μ_x = 62 and σ_x = 15 / √19
c. normal, with μ_x = 62 and σ_x = 19
d. the sampling distribution must be assumed to be normal.

Explanation:

Step1: Recall the formula for the mean and standard deviation of the sampling distribution of the sample mean

If \(X\) is a random variable with mean \(\mu\) and standard deviation \(\sigma\), and we take a sample of size \(n\) from the population, the sampling distribution of the sample mean \(\bar{X}\) has mean \(\mu_{\bar{X}}=\mu\) and standard deviation \(\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}\). Here, assume the population mean \(\mu = 62\) and the population standard deviation \(\sigma=19\), and the sample size \(n = 15\).

Step2: Calculate the mean and standard deviation of the sampling distribution of the sample mean

The mean of the sampling distribution of the sample mean \(\mu_{\bar{X}}=\mu = 62\). The standard deviation of the sampling distribution of the sample mean \(\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{19}{\sqrt{15}}\)

Answer:

A. Normal, with \(\mu_{\bar{X}} = 62\) and \(\sigma_{\bar{X}}=\frac{19}{\sqrt{15}}\)