Question

Assume that the heights of men are normally distributed with a mean of "71.3" inches and a standard deviation of 2.1 inches. If 36 men are randomly​ selected, find the probability that they have a mean height greater than 72.3 inches. Round to four decimal places.

Answer 1

0.0021 = 0.21% probability that they have a mean height greater than 72.3 inches.

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 estabilishes 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.

Assume that the heights of men are normally distributed with a mean of "71.3" inches and a standard deviation of 2.1 inches.

This means that
`\mu = 71.3, \sigma = 2.1`

Sample of 36:

This means that
`n = 36, s = (2.1)/(√(36)) = 0.35`

Find the probability that they have a mean height greater than 72.3 inches.

This is 1 subtracted by the pvalue of Z when X = 72.3. So

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

By the Central Limit Theorem

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

`Z = (72.3 - 71.3)/(0.35)`

`Z = 2.86`

`Z = 2.86` has a pvalue of 0.9979

1 - 0.9979 = 0.0021

0.0021 = 0.21% probability that they have a mean height greater than 72.3 inches.