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 32 % . She obtains a random sample of 72 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.

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:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

The probability that an independent restaurant will fail in the first year is 32%.

This means that [tex]p = 0.32[/tex]

72 independent restaurants

This means that [tex]n = 72[/tex]

Mean:

[tex]E(X) = np = 72*0.32 = 23.04[/tex]

Standard deviation:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{72*0.32*0.68} = 3.96[/tex]

The mean of the number of restaurants that failed within a year is 23.04 and the standard deviation is 3.96.