Question

13. (Bonus, 10 points) The service times for customers coming through a checkout counter in a retail store are independent random variable with a mean of 2.2 minutes and a standard deviation of 1.2 minutes. Is it reasonable to require that 100 customers to be served in less than 3 hours of total service time? Use a probability to explain your reasoning. Hint: This is not a hypothesis test problem because ju is known. Use the sampling distribution of the sample mean to help you find the probability

Answer 1

Let X denote the servie time (in minutes).

Using Central Limit theorem, we know,

Χ, ο Ν(ημ, ησ2) Ν(100(2.2), 100(1.242) Ν (220, 144)

Now,

$\\P(\sum X_{i}<3\times 60)=P(\sum X_{i}<180)=P(Z<\frac{180-220}{\sqrt{144}})\\ \\ =P(Z<-3.33)=P(Z>3.33)=1-P(Z<3.33)\\ \\ =1-0.99957=0.00043$

Since this probability is quite low, so we can say that it is unlikely to service 100 customers in less than 3 hours of total time.