Answer to Question #341140 in Statistics and Probability for Hilka

Question

A courier service company has found that their delivery time of parcels to clients is approximately normally distributed with a mean delivery time of 30 minutes and a variance of 25 minutes (squared).

Required:

a) What is the probability that a randomly selected parcel will take longer than 33 minutes to deliver?

b) What is the probability that a randomly selected parcel will take less than 26 minutes to deliver?

c) What is the minimum delivery time (minutes) for the 2.5% of parcels with the longest time to

deliver?

d) What is the maximum delivery time (minutes) for the 10% of the parcels with the shortest time to deliver?

Answer 1

Let $X=$ delivery time: $X\sim N(\mu, \sigma^2).$

Given $\mu=30min, \sigma^2=25{min}^2$

a)

$$P(X>33)=1-P(Z\le \dfrac{33-30}{\sqrt{25}})$$

$$=1-P(Z\le0.6)\approx0.2743$$

b)

$$P(X<26)=P(Z<\dfrac{26-30}{\sqrt{25}})$$

$$=P(Z<-0.8)\approx0.2119$$

c)

$$P(Z>\dfrac{x-30}{\sqrt{25}})=0.025$$

$$\dfrac{x-30}{5}=1.96$$

$$x=30+5(1.96)$$

$$x=39.8\ min$$

d)

$$P(Z<\dfrac{x-30}{\sqrt{25}})=0.10$$

$$\dfrac{x-30}{5}=-1.28155$$

$$x=30+5(-1.28155)$$

$$x=23.6\ min$$