Question

the heights of fully grown trees of a specific species are normally distributed, with a mean of 70.5 feet and a standard deviation of 7.50 feet. random samples of size 16 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}}= \\) the standard error of the sampling distribution is \\( \sigma_{\overline{x}}= \\) (round to two decimal places as needed.) choose the correct graph of the sampling distribution below.

Explanation:

Step1: Find the mean of the sampling distribution

According to the central limit theorem, if the population has mean \(\mu\), then the mean of the sampling distribution of the sample mean \(\bar{x}\), denoted as \(\mu_{\bar{x}}\), is equal to the population mean. Given \(\mu = 70.5\) feet. So, \(\mu_{\bar{x}}=\mu\).

Step2: Find the standard error of the sampling distribution

The formula for the standard error (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 \(n = 16\). Then \(\sigma_{\bar{x}}=\frac{7.50}{\sqrt{16}}=\frac{7.50}{4}=1.875\approx1.88\) (rounded to two decimal places).

Answer:

The mean of the sampling distribution is \(\mu_{\bar{x}} = 70.5\).
The standard error of the sampling distribution is \(\sigma_{\bar{x}}\approx1.88\).