Answer to Question #285835 in Statistics and Probability for jane

Question

The heights of male college students are normally distributed with

mean of 68 inches and standard deviation of 3 inches. If 80 samples consisting of 25 students each are drawn from the population, what would be the expected mean and standard deviation of the resulting sampling distribution of the means?

Answer 1

The population is normally distributed with mean $68 \ in$ and standard deviation $3\ in.$ There were a total of $80$ samples of size $25.$ Average height was collected for each sample and these averages comprise the sampling distribution of means.

The mean of the sampling distribution is equal to the mean of the population

$$\mu_{\bar{x}}=\mu=68\ in$$

$N=80$ is the population size, and $n=25$ is the sample size.

Use the finite population correction (fpc) factor $\sqrt{\dfrac{N-n}{N-1}}$

If the population is of size $N ,$ if sampling is without replacement, and if the sample size is $n\leq N,$ then the standard error of the sampling distribution  is

$$\sigma_{\bar{x}}=\dfrac{\sigma}{\sqrt{n}}\cdot\sqrt{\dfrac{N-n}{N-1}}$$

$$=\dfrac{3\ in}{\sqrt{25}}\cdot\sqrt{\dfrac{80-25}{80-1}}=0.5\ in$$

The expected mean would be $68$ inches, and standard deviation would be $0.5$ inches.