Question

5) It is known that amounts of money spent on clothing in a year by students on a particular campus follow a normal distribution with a mean of $380 and a standard deviation of $50. a. What is the probability that a randomly chosen student will spend less than $400 on clothing in a year? b. What is the probability that a randomly chosen student will spend more than $360 on clothing in a year? c. What is the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year?

Answer 1

Solution:

$\mu=380,\sigma=50 \\X\sim N(\mu,\sigma)$

(a) $P(X<400)=P(z<\dfrac{400-380}{50})=P(z<0.4)=0.65542$

(b) $P(X>360)=1-P(X\le360)=1-P(z\le\dfrac{360-380}{50})$

$=1-P(z\le-0.4)=1-[P(z\ge0.4)]=1-[1-P(z\le0.4)] \\=P(z\le0.4)=0.65542$

(c) $P(300<X<400)=P(X<400)-P(X<300)$

$=P(z<\dfrac{400-380}{50})-P(z<\dfrac{300-380}{50}) \\=P(z<0.4)-P(z<-1.6) \\=P(z<0.4)-[1-P(z\le1.6)] \\=0.65542-1+0.94520 \\=0.60062$