Question

According to the National Association of Theater Owners, the average price for a movie in the United States in 2012 was $7.96. Assume the population st. dev. is $0.50 and that a sample of 30 theaters was randomly selected. What is the probability that the sample mean will be between $7.75 and $8.20

Answer 1

0.985 = 98.5% probability that the sample mean will be between $7.75 and $8.20.

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The average price for a movie in the United States in 2012 was $7.96. Assume the population st. dev. is $0.50.

This means that
`\mu = 7.96, \sigma = 0.5`

Sample of 30:

This means that
`n = 30, s = (0.5)/(√(30))`

What is the probability that the sample mean will be between $7.75 and $8.20?

This is the p-value of Z when X = 8.2 subtracted by the p-value of Z when X = 7.75.

X = 8.2

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

By the Central Limit Theorem

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

`Z = (8.2 - 7.96)/((0.5)/(√(30)))`

`Z = 2.63`

`Z = 2.63` has a p-value of 0.9957

X = 7.75

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

`Z = (7.75 - 7.96)/((0.5)/(√(30)))`

`Z = -2.3`

`Z = -2.3` has a p-value of 0.0107.

0.9957 - 0.0157 = 0.985

0.985 = 98.5% probability that the sample mean will be between $7.75 and $8.20.