Let $X$ represent the full height of a certain species of tree. Assume that $X$ has a normal probability distribution with $\mu=27.9 \mathrm{ft}$ and $\sigma=40.4 \mathrm{ft}$. You intend to measure a random sample of $n=139$ trees. What is the mean of the distribution of sample means? \[ \mu_{\bar{x}}= \] $\mathrm{ft}$ What is the standard deviation of the distribution of sample means (i.e., the standard error in estimating the mean)? (Report answer accurate to 2 decimal places.) \[ \sigma_{\bar{x}}= \] $\mathrm{ft}$ Question Help: D Post to forum

Step 1 ：Let $X$ represent the full height of a certain species of tree. Assume that $X$ has a normal probability distribution with $\mu=27.9 \mathrm{ft}$ and $\sigma=40.4 \mathrm{ft}$. You intend to measure a random sample of $n=139$ trees.

Step 2 ：The mean of the distribution of sample means, also known as the expected value of the sample mean, is equal to the population mean. So, in this case, the mean of the distribution of sample means is $\mu_{\bar{x}}=27.9 \mathrm{ft}$.

Step 3 ：The standard deviation of the distribution of sample means, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size. So, in this case, the standard error is $\sigma_{\bar{x}}=\frac{40.4}{\sqrt{139}} \mathrm{ft}$.

Step 4 ：Calculate the standard error to get $\sigma_{\bar{x}}=3.43 \mathrm{ft}$ (rounded to two decimal places).

Step 5 ：Final Answer: The mean of the distribution of sample means is $\boxed{27.9}$ ft and the standard deviation of the distribution of sample means is $\boxed{3.43}$ ft.