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 time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes. Find the waiting time at which only 10% of the customers will continue to hold.
A) 2.3 minutes
B) 3.3 minutes
C) 6.9 minutes
D) 13.8 minutes

Question

Quizplus · Accepted Answer

We know that the waiting time, X, is exponentially distributed with a mean of 3 minutes, which implies that the parameter λ = 1/3 (since λ = 1/mean for an exponential distribution).

We want to find the waiting time at which only 10% of the customers will continue to hold, which means we are looking for the value of x such that P(X > x) = 0.10.

Using the cumulative distribution function (CDF) for an exponential distribution, we have:

P(X > x) = e^(-λx)

Setting this equal to 0.10 and solving for x, we get:

e^(-λx) = 0.10
-ln(0.10) = -λx
x = (-ln(0.10))/λ

Substituting in λ = 1/3, we get:

x = (-ln(0.10))/(1/3) ≈ 6.9 minutes

Therefore, the waiting time at which only 10% of the customers will continue to hold is approximately 6.9 minutes, which corresponds to choice C.

A Catalog Company That Receives the Majority of Its Orders