Question

the graph of the annual snowfall distribution (in feet) for a particular county is shown below on the left with its mean and standard deviation. assume that a sample size of 64 is drawn from the population. determine which of the graphs labeled (a)-(c) would most closely resemble the sampling distribution of the sample means. explain your reasoning.
choose the correct graph below.
a.
b.
c.

Explanation:

Step1: Recall the properties of the sampling distribution of the sample mean

According to the Central Limit Theorem, if we have a population with mean \(\mu\) and standard deviation \(\sigma\), and we take samples of size \(n\) from this population, the sampling distribution of the sample mean \(\bar{x}\) has mean \(\mu_{\bar{x}}=\mu\) and standard deviation \(\sigma_{\bar{x}}=\frac{\sigma}{n}\). Here, \(\mu = 4.3\), \(\sigma=2.8\), and \(n = 64\).

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

• For the mean of the sampling distribution of the sample mean: \(\mu_{\bar{x}}=\mu = 4.3\) (by the formula \(\mu_{\bar{x}}=\mu\) from the Central Limit Theorem).
• For the standard deviation of the sampling distribution of the sample mean: \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Substitute \(\sigma = 2.8\) and \(n = 64\) into the formula. Since \(\sqrt{64}=8\), then \(\sigma_{\bar{x}}=\frac{2.8}{8}=0.35\).

Answer:

B.