Question

The heights of twelve month old boys are normally distributed with a mean of 29.8 inches and a standard deviation of 1.2 inches. About twenty-one percent of twelve month old boys are shorter than what height? Report your answer to the nearest tenths place.

Answer 1

About twenty-one percent of twelve month old boys are shorter than 28.8 inches.

Explanation:

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.

In this problem, we have that:

`\mu = 29.8, \sigma = 1.2`

About twenty-one percent of twelve month old boys are shorter than what height?

This is the value of X when Z has a pvalue of 0.21. So it is X when Z = -0.805.

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

`-0.805 = (X - 29.8)/(1.2)`

`X - 29.8 = -0.805*1.2`

`X = 28.8`

About twenty-one percent of twelve month old boys are shorter than 28.8 inches.