Answer to Question #300652 in Statistics and Probability for gds

Question

 A certain population has a mean of 15.4 and a standard deviation of 5.6. if random samples of size 5 is taken from this population, which of the following statements is true? *

1 point

The standard deviation of the sampling distribution of the sample mean is 5.6.

The mean of the sampling distribution of the sample means is equal to 15.4.

The mean of the sampling distribution of the sample means is less than 15.4.

The standard deviation of the sampling distribution of the sample mean is 15.4.

Answer 1

By the Central Limit Theorem (CLT)

If the population is normal to begin with then the sample mean also has a normal distribution, regardless of the sample size:

$\bar{x}$ has approximately a normal distribution with $\mu_{\bar{x}}=\mu$ and $\sigma_{\bar{x}}^2=\sigma^2/n.$

Given $\mu=15.4, \sigma=5.6, n=5.$

Then $\mu_{\bar{x}}=\mu=15.4, \sigma_{\bar{x}}=\sigma/\sqrt{n}=5.6/\sqrt{5}.$

a. The standard deviation of the sampling distribution of the sample mean is 5.6. False

b. The mean of the sampling distribution of the sample means is equal to 15.4. True

c. The mean of the sampling distribution of the sample means is less than 15.4. False

d. The standard deviation of the sampling distribution of the sample mean is 15.4. False

The mean of the sampling distribution of the sample means is equal to 15.4. True