Question

Suppose Kristen is researching failures in the restaurant business. In the city where she lives, the probability that an independent restaurant will fail in the first year is 43 % . She obtains a random sample of 66 independent restaurants that opened in her city more than one year ago and determines if each one had closed within a year. What are the mean and standard deviation of the number of restaurants that failed within a year? Please give your answers precise to two decimal places.

Answer 1

The mean of the number of restaurants that failed within a year is 28.38 and the standard deviation is 4.02.

Explanation:

For each restaurant, there are only two possible outcomes. Either it fails during the first year, or it does not. The probability of a restaurant failling during the first year is independent of other restaurants. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

In the city where she lives, the probability that an independent restaurant will fail in the first year is 43 %.

This means that
`p = 0.43`

66 independent restaurants

This means that
`n = 66`

Mean:

`E(X) = np = 66*0.43 = 28.38`

Standard deviation:

`√(V(X)) = √(np(1-p)) = 4.02`

The mean of the number of restaurants that failed within a year is 28.38 and the standard deviation is 4.02.