Error Probability: Can anyone share a detailed solution to the problem?

Question

Seventy data clerks at the Department of Motor Vehicles make an average of 18 errors per day, normally distributed with a standard deviation of 4. A field auditor can check the work of 15 clerks per day. What is the probability that the average number of errors in a group of 15 clerks checked on one day is (a) Fewer than 15.5? (b) Greater than 20?

Answer 1

Let $X \sim N(18, 16)$ be the random variable representing how many errors 70 clerks make in a day.

Then $Y=\frac{15}{70}X\sim N(\frac{15}{70} (18), (\frac{15}{70})^2 (16))$

So $Y\sim N(\frac{27}{7}, \frac{36}{49}$)

and $P(Y\le 15.5)$=$P\left(\frac{Y-E[Y]}{\sqrt{Var(Y)}}\le \frac{15.5-E[Y]}{\sqrt{Var(Y)}}\right)$,

where the expectation and variance are given above and where you'd just use your standard normal distribution tables to compute an answer.