Question

the heights of fully grown trees of a specific species are normally distributed, with a mean of 72.5 feet and a standard deviation of 7.50 feet. random samples of size 20 are drawn from the population. use the central limit theorem to find the mean and standard error of the sampling distribution. then sketch a graph of the sampling distribution
the mean of the sampling distribution is \\( \mu _ { \overline { x } } = 72.5 \\)
the standard error of the sampling distribution is \\( \sigma _ { \overline { x } } = 1.68 \\)
(round to two decimal places as needed.)
choose the correct graph of the sampling distribution below.
oa.
ob.
oc.

Explanation:

Step1: Calculate the mean of the sampling distribution

According to the central limit theorem, for a sampling distribution of sample means, the mean of the sampling distribution \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Given \(\mu = 72.5\) feet, so \(\mu_{\bar{x}}=\mu = 72.5\)

Step2: Calculate the standard error of the sampling distribution

The formula for the standard error of the sampling distribution (also known as the standard deviation of the sampling distribution of the sample mean) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. Given \(\sigma = 7.50\) feet and \(n = 20\), then \(\sigma_{\bar{x}}=\frac{7.50}{\sqrt{20}}\approx\frac{7.50}{4.472}\approx1.68\)

Step3: Analyze the graph

The mean of the sampling distribution is \(72.5\). The standard error is \(1.68\). For a normal sampling distribution, we can find the range \(\mu_{\bar{x}}\pm3\sigma_{\bar{x}}\). \(72.5 - 3\times1.68=72.5 - 5.04 = 67.46\approx69.1\) (approximate due to rounding of standard error), \(72.5+3\times1.68 = 72.5+5.04=77.54\approx75.9\) (approximate due to rounding of standard error). So the graph with mean \(72.5\) and range approximately from \(69.1\) to \(75.9\) is correct.

Answer:

C.