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 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.

Answer 1

Answer:

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

Step-by-step 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:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

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

This means that $p = 0.32$

72 independent restaurants

This means that $n = 72$

Mean:

$E(X) = np = 72*0.32 = 23.04$

Standard deviation:

$\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{72*0.32*0.68} = 3.96$

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