A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of waiting time was found to be a random variable best approximated by an exponential distribution with a mean length of waiting time equal to 2.8 minutes (i.e. the mean number of calls answered in a minute is 1/2.8). What is the probability that a randomly selected caller is placed on hold fewer than 7 minutes?
A) 0.0009119
B) 0.082085
C) 0.917915
D) 0.9990881

Question

Quizplus · Accepted Answer

We know that the waiting time follows an exponential distribution with a mean of 2.8 minutes.
The probability distribution function of the exponential distribution is given by:
f(x) = (1/2.8) * e^(-x/2.8)
To find the probability that a caller is on hold for less than 7 minutes, we need to calculate the cumulative distribution function (CDF) up to x=7. This is given by:
F(7) = integral from 0 to 7 of f(x) dx
F(7) = integral from 0 to 7 of (1/2.8) * e^(-x/2.8) dx
F(7) = (-e^(-7/2.8) + 1)/2.8
F(7) = 0.9179
Therefore, the probability that a randomly selected caller is placed on hold for fewer than 7 minutes is 0.9179, which is closest to C) 0.917915.

A Catalog Company That Receives the Majority of Its Orders