Statistics homework question

Question

CNNBC recently reported that the mean annual cost of auto insurance is 1041 dollars. Assume the standard deviation is 300 dollars. You take a simple random sample of 91 auto insurance policies.

Find the probability that a sample of size n=91 is randomly selected with a mean less than 987 dollars.

P(M < 987) =

Answer 1

You can rewrite it as

$$P\left(\frac{\sqrt{91}(M-1041)}{\sqrt{91}300}<\frac{\sqrt{91}(987-1041)}{300}\right)$$

The quantity on the left size is the z-score and it has normal distribution (approximately, due the Central Limit Theorem). You can look up the resulting value using the z-table for the standard normal distribution in the back of your book.