Question

The average public high school has 468 students with a standard deviation of 87.

a. If a public school is selected what is the probability that the number of student enrolled is greater than 400?

b. If a random sample of 38:public elementary schools is selected what is the probability that the number of students enrolled is between 445 and 485?

Answer 1

We have a normal distribution, $μ=468,σ=87.$

Let's convert it to the standard normal distribution.

$\text{a. }z=\cfrac{x-\mu}{\sigma}=\cfrac{400-468}{87}=-0.78,\\ P(X>400)=P(Z>-0.78)=\\ =1-P(Z<-0.78)=1-0.2177=0.7823\\\text{ (from z-table).}$

$\text{b. }\bar{z}=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}},\\ \bar{z}_1=\cfrac{445-468}{87/\sqrt{38}}=-1.63,\\ \bar{z}_2=\cfrac{485-468}{87/\sqrt{38}}=1.20,\\ P(445<\bar{X}<485)=P(-1.63<\bar{Z}<1.20)=\\ =P(\bar{Z}<1.20)-P(\bar{Z}<-1.63)=\\ =0.8849-0.0516=0.8333\text{ (from z-table).}$