Question

Suppose the head circumferences of adult males have a bell-shaped distribution with a mean of 55 cm and a standard deviation of 3 cm. (a) Explain whether or not it would be unusual for an adult male to have a 49-cm head circumference.

Answer 1

Z scores of -2 or lower are considered unusually low. Since the z-score of a 49-cm head circunference is -2, it is an unusual measure.

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.

Z scores of -2 or lower are considered unusually low, and zscores of 2 or higher are considered unusually high.

In this problem, we have that:

`\mu = 55, \sigma = 3`

49cm head circunference unusual?

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

`Z = (49 - 55)/(3)`

`Z = -2`

Z scores of -2 or lower are considered unusually low. Since the z-score of a 49-cm head circunference is -2, it is an unusual measure.