Answer to Question #169754 in Statistics and Probability for Rachel Lim

Question

7.The daily delivery of mail at a large city firm follows a time pattern conforming to the Normal

Distribution, with a mean time of arrival of 9.40 am and with a standard deviation of 20 minutes.

Estimate the number of occasions during the 250 working days in the year when the mail arrives

a. before the main gates open at 9 am

b. after the arrival of the office staff at 9.20 am

c. between 10 am and 10.20 am

Answer 1

Let $X=$ difference in time of arrival with respect to 9.40 am in minutes.

$X\sim N(\mu, \sigma^2).$ Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0, 1)$

Given $\mu=0, \sigma=20\ min$

a.

$$P(X<-40)=P(Z<\dfrac{-40-0}{20})$$

$$=P(Z<-2)\approx0.02275$$

$0.02275(250)=6(occasions)$

b.

$$P(X>-20)=1-P(X\leq -20)$$

$$=1-P(Z\leq\dfrac{-20-0}{20})=1-P(Z\leq-1)$$ $$\approx0.84134$$

$0.84134(250)=210(occasions)$

c.

$$P(20<X<40)=P(X<40)-P(X\leq 20)$$

$$=P(Z<\dfrac{40-0}{20})-P(Z\leq\dfrac{20-0}{20})$$

$$=P(Z<2)-P(Z\leq1)$$

$$\approx0.97725-0.84134\approx0.13591$$

$0.13591(250)=34(occasions)$