Answer to Question #199990 in Statistics and Probability for Amorette

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 more than 26 minutes to deliver? (2) b) What is the minimum delivery time (minutes) for the 2.5% of parcels with the longest time to deliver? (4

Answer 1

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

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

a)

$$P(X>26)=1-P(X\leq 26)$$

$$=1-P(Z\leq\dfrac{26-30}{5})=1-P(Z\leq-0.8)$$

$$\approx0.788147$$

b)

$$P(X<x)=P(Z<\dfrac{x-30}{5})=0.025$$

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

$$x\approx 20.2\ min$$