Question

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 69 inches and standard deviation 6 inches.What is the probability that an 18-year-old man selected at random is between 68 and 70 inches tall? (Round your answer to four decimal places.)

Answer 1

0.1350 = 13.50% probability that an 18-year-old man selected at random is between 68 and 70 inches tall

Explanation:

Problems of normally distributed samples can be 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 question, we have that:

`\mu = 69, \sigma = 6`

What is the probability that an 18-year-old man selected at random is between 68 and 70 inches tall?

This is the pvalue of Z when X = 70 subtracted by the pvalue of Z when X = 68. So

X = 70

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

`Z = (70 - 69)/(6)`

`Z = 0.17`

`Z = 0.17` has a pvalue of 0.5675

X = 68

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

`Z = (68 - 69)/(6)`

`Z = -0.17`

`Z = -0.17` has a pvalue of 0.4325

0.5675 - 0.4325 = 0.1350

0.1350 = 13.50% probability that an 18-year-old man selected at random is between 68 and 70 inches tall