Question

the heights of fully grown trees of a specific species are normally distributed, with a mean of 73.0 feet and a standard deviation of 5.75 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 μ_x̅ = . the standard error of the sampling distribution is σ_x̅ = . (round to two decimal places as needed.)

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 $\mu$.
$\mu_{\bar{x}}=\mu$
Given $\mu = 73.0$ feet, so $\mu_{\bar{x}}=73.0$ feet.

Step2: Recall formula for standard error

The standard error of the sampling distribution $\sigma_{\bar{x}}$ 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.
Given $\sigma = 5.75$ feet and $n = 16$.
$\sigma_{\bar{x}}=\frac{5.75}{\sqrt{16}}=\frac{5.75}{4}=1.44$ (rounded to two decimal places)

Answer:

The mean of the sampling distribution is $\mu_{\bar{x}} = 73.0$.
The standard error of the sampling distribution is $\sigma_{\bar{x}}=1.44$.