Question

PROBLEM B: In the national achievement test, a school had a total mean of 59 with a standard deviation of 7. If 40 students who took the test are selected at random, what is the probability that the mean of their scores fall between 59 and 60?

Answer 1

We have a normal distribution, $μ=59,σ=7,n=40.$

Let's convert it to the standard normal distribution,

$\bar{z}=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}},\\ \bar{z}_1=\cfrac{59-59}{7/\sqrt{40}}=0,\\ \bar{z}_2=\cfrac{60-59}{7/\sqrt{40}}=0.90,\\ P(59<\bar{X}<60)=P(0<\bar{Z}<0.90)=\\ =P(\bar{Z}<0.90)-P(\bar{Z}<0)=\\ =0.8159-0.5000=0.3159\text{ (from z-table).}$