Question

The number of times that students go to the movies per year has mean is a normal distribution with a mean of 17 with standard deviation of 8. What is the probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times?

Answer 1

53.84% probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times.

Explanation:

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

Normal probability distribution

Problems of normally distributed samples are 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.

Central limit Theorem

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

In this problem, we have that:

`\mu = 17, \sigma = 8, n = 10, s = (8)/(√(10)) = 2.53`

What is the probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times?

This probability is the pvalue of Z when X = 18 subtracted by the pvalue of Z when X = 14. So

X = 18

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

By the Central Limit Theorem

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

`Z = (18 - 17)/(2.53)`

`Z = 0.4`

`Z = 0.4` has a pvalue of 0.6554

X = 14

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

`Z = (14 - 17)/(2.53)`

`Z = -1.19`

`Z = -1.19` has a pvalue of 0.1170

0.6554 - 0.1170 = 0.5384

53.84% probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times.