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kobusy [5.1K]
2 years ago
13

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 3.5 minutes.The ma

nager wants to advertise that anybody who isn't served within a certain number of minutes gets a free hamburger. But she doesn't want to give away free hamburgers to more than 1% of her customers. What number of minutes should the advertisement use
Mathematics
1 answer:
Lana71 [14]2 years ago
5 0

Answer:

The advertisement should use 16 minutes.

Step-by-step 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 = \frac{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}

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 3.5 minutes.

This means that m = 3.5, \mu = \frac{1}{3.5} = 0.2857

What number of minutes should the advertisement use?

The values of x for which:

P(X > x) = 0.01

So

e^{-\mu x} = 0.01

e^{-0.2857x} = 0.01

\ln{e^{-0.2857x}} = \ln{0.01}

-0.2857x = \ln{0.01}

x = -\frac{\ln{0.01}}{0.2857}

x = 16.12

Rounding to the nearest number, the advertisement should use 16 minutes.

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