Answer to Question #336389 in Statistics and Probability for June

Question

The number of violent crimes committed in a day possesses a distribution with a mean of 2.8 crimes per day and a standard deviation of 4 crimes per day. A random sample of 100 days was observed, and the sample mean number of crimes for the sample was calculated. Describe the sampling distribution of the sample means.

a.) shape unknown with mean of 2.8 and a standard deviation of 0.4

b.) approximately normal with mean of 2.8 and standard deviation of 4

c.) shape unknown with mean of 2.8 and standard deviation of 4

d.) approximately normal with mean of 2.8 and standard deviation of 0.4.

Answer 1

By the Central Limit Theorem: if $n$ is sufficiently large, $\bar{X}$ has approximately a normal distribution with $\mu_{\bar{X}}=\mu$ and $\sigma_{\bar{X}}=\sigma^2/n.$

The normal approximation for $\bar{X}$ will generally be good if $n\ge 30.$

Given $n=100>30, \mu=2.8, \sigma=4.$

Then the normal approximation for $\bar{X}$ will generally be good and

$$\mu_{\bar{X}}=\mu=2.8, \sigma_{\bar{X}}=\sigma/\sqrt{n}=4/\sqrt{100}=0.4$$

d.) approximately normal with mean of 2.8 and standard deviation of 0.4.