Random variable that is normally distributed

Question

For a random variable that is normally distributed with a population mean of 80 and a population standard deviation =10 the probability that is simple random sample of 25 items will have a mean that is between 79 and 85

Answer 1

Hint:

• If $X_i$'s are normal random variables with mean $\mu$ and variance $\sigma^2$ then the sample mean, $Y = \frac{\sum_{i=1}^{n} X_i}{n}$, is a normal random variable with mean $\mu$ and variance $\sigma^2/n$.
• $\Pr{(y_1< Y \le y_2)}=F_Y(y_2)-F_Y(y_1).$