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)?
To solve these questions, we need to use the exponential distribution formula:
f(x) = λ * e^(-λx)
Where λ is the rate parameter, which is equal to 1/mean.
Given that the average service time for Wendy's is 2.2 minutes, we can find the rate parameter:
λ = 1/2.2 = 0.4545
a. To find the probability that a service time is less than or equal to one minute:
P(X ≤ 1) = 1 - e^(-λx)
P(X ≤ 1) = 1 - e^(-0.4545 * 1)
P(X ≤ 1) ≈ 0.3951
Therefore, the probability that a service time is less than or equal to one minute is approximately 0.3951.
b. To find the probability that a service time is between 30 seconds and one minute:
P(0.5 ≤ X ≤ 1) = e^(-λ * 0.5) - e^(-λ * 1)
P(0.5 ≤ X ≤ 1) = e^(-0.4545 * 0.5) - e^(-0.4545 * 1)
P(0.5 ≤ X ≤ 1) ≈ 0.2651
Therefore, the probability that a service time is between 30 seconds and one minute is approximately 0.2651.
c. To find the probability that the time it takes to serve you exceeds five minutes:
P(X > 5) = 1 - P(X ≤ 5)
P(X > 5) = 1 - (1 - e^(-0.4545 * 5))
P(X > 5) ≈ 0.029
Therefore, the probability that you will get your food for free (if the time exceeds five minutes) is approximately 0.029.
To answer these questions, we can use the exponential distribution formula. In this case, the average service time for Wendy's is given as 2.2 minutes.
The exponential distribution is defined by the parameter λ, which represents the rate at which events occur. In this case, λ is equal to 1 divided by the average service time, so λ = 1/2.2.
a. To find the probability that a service time is less than or equal to one minute, we need to calculate the cumulative distribution function (CDF) at one minute or 60 seconds.
CDF(x) = 1 - e^(-λx)
Substituting the values, we get:
CDF(60s) = 1 - e^(-(1/2.2) * 60)
Calculating this, we find that the probability is approximately 0.6928 (rounded to 4 decimals).
b. To find the probability that a service time is between 30 seconds and one minute, we subtract the cumulative distribution function (CDF) at 30 seconds from the CDF at one minute.
CDF(30s) = 1 - e^(-(1/2.2) * 30)
CDF(60s) = 1 - e^(-(1/2.2) * 60)
Probability = CDF(60s) - CDF(30s)
Calculating this, we find that the probability is approximately 0.3392 (rounded to 4 decimals).
c. To find the probability that the time it takes to serve you exceeds five minutes (300 seconds), we need to calculate the survival function or complementary cumulative distribution function (CCDF).
CCDF(x) = 1 - CDF(x)
CCDF(300s) = 1 - CDF(300s) = 1 - (1 - e^(-(1/2.2) * 300))
Calculating this, we find that the probability is approximately 0.0270 (rounded to 4 decimals).
Therefore, the probability that you will get your food for free if the service time exceeds five minutes is approximately 0.0270.