Question #341140

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?


1
Expert's answer
2022-05-17T12:49:13-0400

Let X=X= delivery time: XN(μ,σ2).X\sim N(\mu, \sigma^2).

Given μ=30min,σ2=25min2\mu=30min, \sigma^2=25{min}^2

a)


P(X>33)=1P(Z333025)P(X>33)=1-P(Z\le \dfrac{33-30}{\sqrt{25}})

=1P(Z0.6)0.2743=1-P(Z\le0.6)\approx0.2743

b)


P(X<26)=P(Z<263025)P(X<26)=P(Z<\dfrac{26-30}{\sqrt{25}})

=P(Z<0.8)0.2119=P(Z<-0.8)\approx0.2119


c)


P(Z>x3025)=0.025P(Z>\dfrac{x-30}{\sqrt{25}})=0.025

x305=1.96\dfrac{x-30}{5}=1.96

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

x=39.8 minx=39.8\ min

d)


P(Z<x3025)=0.10P(Z<\dfrac{x-30}{\sqrt{25}})=0.10

x305=1.28155\dfrac{x-30}{5}=-1.28155

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

x=23.6 minx=23.6\ min


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