Question

A film distribution manager calculates that 7% of the films released are flops. If the manager is correct, what is the probability that the proportion of flops in a sample of 404 released films would be greater than 4%? Round your answer to four decimal places.

Answer 1

0.9909 = 99.09% probability that the proportion of flops in a sample of 404 released films would be greater than 4%

Explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

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

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

In this problem, we have that:

`p = 0.07, n = 404`. So

`\mu = E(X) = np = 404*0.07 = 28.28`

`\sigma = √(V(X)) = √(np(1-p)) = √(404*0.07*0.93) = 5.13`

If the manager is correct, what is the probability that the proportion of flops in a sample of 404 released films would be greater than 4%?

This is the pvlaue of Z when X = 0.04*404 = 16.16. So

`Z = (X - \mu)/(\sigma)`

`Z = (16.16 - 28.28)/(5.13)`

`Z = -2.36`

`Z = -2.36` has a pvalue of 0.0091

1 - 0.0091 = 0.9909

0.9909 = 99.09% probability that the proportion of flops in a sample of 404 released films would be greater than 4%