Question

Assume that women's heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9 inches and 64.0 inches.

Answer 1

93.18% probability that they have a mean height between 62.9 inches and 64.0 inches.

Explanation:

To solve this question, 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`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 63.6, \sigma = 2.5, n = 90, s = (2.5)/(√(90)) = 0.2635`

If 90 women are randomly selected, find the probability that they have a mean height between 62.9 inches and 64.0 inches.

This is the pvalue of Z when X = 64 subtracted by the pvalue of Z when X = 62.9. So

X = 64

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

By the Central Limit Theorem

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

`Z = (64 - 63.6)/(0.2635)`

`Z = 1.52`

`Z = 1.52` has a pvalue of 0.9357

X = 62.9

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

`Z = (62.9 - 63.6)/(0.2635)`

`Z = -2.66`

`Z = -2.66` has a pvalue of 0.0039

0.9357 - 0.0039 = 0.9318

93.18% probability that they have a mean height between 62.9 inches and 64.0 inches.