Question

the heights of fully grown trees of a specific species are normally distributed, with a mean of 57.0 feet and a standard deviation of 6.75 feet. random samples of size 14 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 $mu_{\bar{x}} = 57.0$.
the standard error of the sampling distribution is $sigma_{\bar{x}} = 1.80$. (round to two decimal places as needed.)
choose the correct graph of the sampling distribution below.
oa
ob.
oc.

Explanation:

Step1: Recall central limit theorem for mean

The mean of the sampling - distribution of the sample mean $\mu_{\bar{x}}$ is equal to the population mean $\mu$. Given $\mu = 57.0$, so $\mu_{\bar{x}}=57.0$.

Step2: Recall formula for standard error

The standard error of the sampling - distribution of the sample mean $\sigma_{\bar{x}}$ is given by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Here, $\sigma = 6.75$ and $n = 14$. Then $\sigma_{\bar{x}}=\frac{6.75}{\sqrt{14}}\approx\frac{6.75}{3.742}\approx1.80$.

Step3: Determine the range for the graph

For a normal distribution of the sample - mean, we can use the mean $\mu_{\bar{x}} = 57.0$ and standard error $\sigma_{\bar{x}}=1.80$. The values within 2 standard errors of the mean are $57.0 - 2\times1.80=53.4$ and $57.0 + 2\times1.80 = 60.6$.

Answer:

C.