Wendy\'s restaurant has been recognized for having the fastest average service t

Question

Wendy's restaurant has been recognized for having the fastest average service time among fast food restaurants. In a benchmark study, Wendy's average service time of 2.2 minutes was less than those of Burger King, Chick-fil-A, Krystal, McDonald's, Taco Bell, and Taco John's (QSR Magazine website, December 2014). Assume that the service time for Wendy's has an exponential distribution.

a. What is the probability that a service time is less than or equal to one minute (to 4 decimals)?

b. What is the probability that a service time is between 30 seconds and one minute (to 4 decimals)?

c. Suppose a manager of a Wendy's is considering instituting a policy such that if the time it takes to serve you exceeds five minutes, your food is free. What is the probability that you will get your food for free (to 4 decimals)?

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Explanation / Answer

Let the random variable X denotes the service time and it is exponentially distributed with parameter . The average service time is 2.2 minutes. If X follows the exponential distribution with parameter , then mean is 1/.

The parameter is, 1/ = 2.2; so = 1/2.2

The distribution is, f(x) = (1/2.2) x e-(1/2.2) x, x 0

0, otherweise

The cumulative probability distribution is given as F (x) = 1 - e-(1/2/2)x

(a) Probability that the service time is less than 1 minute, P(X 1) = 1 - e-(1/2/2)*1 = 1 - 0.6347 = 0.3653

(b) P (0.5 X 1) = P (X 1) - P (X 0.5) = 1 - e-(1/2/2)*1 - 1 - e-(1/2/2)*0.5 = 0.7967 - 0.6347 = 0.162

(c) P (X 5) = 1 - P (X 5 ) = 1 - [1 - e-(1/2/2)*5] = e-(1/2/2)*5 = 0.103

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