Answer to Question #173293 in Statistics and Probability for Denisse Bisuña

Question

The average public High School gas 468 students with a standard deviation of 87:

a. If a public school is selected, what is the probability that the number of students 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

(a) The number of students enrolled in a randomly selected public school is a random variable $X$, that has a normal probability distribution with parameters $\mu=468$ and $\sigma=87$. The value X=400 corresponds to z-value of a standard normal distribution (400-468)/87=-0.78. Then $P(X\geq 400)=1-\Phi(z)=\Phi(-z)=\Phi(0.78)=0.7823$

where $\Phi(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{x}e^{-t^2/2}dt$ is cumulative distribution function of standard normal distribution.

(b) If a random sample of n=38 public elementary schools is selected, then the average number of student enrolled would be a normally distributed random variable $Y$ with the mean $\mu=468$ and the standard deviation $\sigma/\sqrt{n}=87/\sqrt{38}=14.11$.

The value Y=445 corresponds to z-value of a standard normal distribution (445-468)/14.11=-1.63

The value Y=485 corresponds to z-value of a standard normal distribution (485-468)/14.11=1.20

Then

$P(445\leq Y\leq 485)=\Phi(1.20)-\Phi(-1.63)=0.8849-0.0516=0.8333$

Answer. (a) $P(X\geq 400)=0.7823$;

(b) $P(445\leq Y\leq 485)=0.8333$.