Question

use the central limit theorem to find the mean and standard error of the mean of the indicated sampling distribution. the monthly rents for studio apartments in a certain city have a mean of $1,100 and a standard deviation of $160. random samples of size 30 are drawn from the population and the mean of each sample is determined. round the answers to the nearest hundredth. a. $200.83, $29.21 b. $1,100.00, $29.21 c. $1,100.00, $5.33 d. $200.83, $160.00

Explanation:

Step1: Find the mean of the sampling distribution

According to the Central Limit Theorem, the mean of the sampling distribution of the sample mean ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$).
Given $\mu = 1100$, so $\mu_{\bar{x}}=\mu = 1100$.

Step2: Find the standard error of the sampling distribution

The formula for the standard error of the mean ($\sigma_{\bar{x}}$) is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
Given $\sigma = 160$ and $n = 30$, then $\sigma_{\bar{x}}=\frac{160}{\sqrt{30}}\approx\frac{160}{5.477}\approx29.21$.

Answer:

B. $\$1,100.00,\$29.21$