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 minutes was less than those of Burger King, Chick-fil-A, Krystal, McDonald's, Taco Bell, and Taco John's (QSR Magazine website, December ). Assume that the service time for Wendy's has an exponential distribution. Do not round intermediate calculations.a. What is the probability that a service time is less than or equal to minute (to 4 decimals)

Answer 1

0.3935 = 39.35% probability that a service time is less than or equal to minute.

Explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

`f(x) = \mu e^(-\mu x)`

In which
`\mu = (1)/(m)` is the decay parameter.

The probability that x is lower or equal to a is given by:

`P(X \leq x) = \int\limits^a_0 {f(x)} \, dx`

Which has the following solution:

`P(X \leq x) = 1 - e^(-\mu x)`

The probability of finding a value higher than x is:

`P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^(-\mu x)) = e^(-\mu x)`

In this question:

The mean time for the service at Wendy's is not given, so i will use
`m = 2, \mu = (1)/(2) = 0.5`

What is the probability that a service time is less than or equal to minute?

`P(X \leq x) = 1 - e^(-\mu x)`

`P(X \leq 1) = 1 - e^(-0.5*1) = 0.3935`

0.3935 = 39.35% probability that a service time is less than or equal to minute.