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. this graph most closely resembles the sampling distribution of the sample means, because $\mu_{\bar{x}}=\square, \sigma_{\bar{x}}=\square$, and the graph

Explanation:

Step1: Calculate the mean of the sampling distribution of the sample means

According to the Central Limit Theorem, the mean of the sampling distribution of the sample means \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Given \(\mu = 4.3\), so \(\mu_{\bar{x}}=4.3\)

Step2: Calculate the standard deviation of the sampling distribution of the sample means

The formula for the standard deviation of the sampling distribution of the sample means (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 2.8\) and \(n = 64\), then \(\sigma_{\bar{x}}=\frac{2.8}{\sqrt{64}}=\frac{2.8}{8}=0.35\)

Step3: Analyze the graphs

The graph with \(\mu_{\bar{x}} = 4.3\) and \(\sigma_{\bar{x}}=0.35\) (the second graph among (a)-(c)) is the correct one. Because the mean of the sampling distribution of the sample means is equal to the population mean, and the standard deviation of the sampling distribution of the sample means is \(\frac{\sigma}{\sqrt{n}}\)

Answer:

\(\mu_{\bar{x}} = 4.3\), \(\sigma_{\bar{x}}=0.35\)