Question

In Exercises 13–16, use z scores to compare the given values.

Birth Weights Based on Data Set 4 “Births” in Appendix B,newbornmales have weights with a mean of 3272.8 g and a standard deviation of 660.2 g.Newbornfemales have weights with a mean of 3037.1 g and a standard deviation of 706.3 g. Who has the weight that is more extreme relative to the group from which they came: a male who weighs 1500 g or a female who weighs 1500 g?

Short answer

The male weight of 1500 grams is more extreme.

Step-by-step solution

Given information 

The weights ofnewbornsare given in grams.

The mean malenewbornweight is equal to 3272.8 grams, and the standard deviation of male weights is 660.2 grams.

The mean female newborn weight is equal to 3037.1 grams, and the standard deviation of female weights is 706.3 grams.

Z-score 

Z-score is frequently used to estimate the place of a value from the mean. It is expressed as shown below:

$z=\frac{x-\overline{x}}{s}$

A greater value of the z-score indicates a greater distance from the mean and a more extreme value.

The z-score for the male weightis:

$$\begin{gathered} z=\frac{x-\overline{x}}{s} \\ =\frac{1500-3272.8}{660.2} \\ =-2.69 \end{gathered}$$

Therefore, the z-score for the male weight equal to 1500 grams is -2.69.

The z-score forthe female weightis:

$$\begin{gathered} z=\frac{x-\overline{x}}{s} \\ =\frac{1500-3037.1}{706.3} \\ =-2.18 \end{gathered}$$

Therefore, the z-score for female weight equal to 1500 grams is -2.18.

Step 3: Comparison 

For a value of weight equal to 1500 grams, the female weight is 2.18 standard deviations below the mean, and the male weight is 2.69 standard deviations below the mean

As the male weight of 1500 grams is farther from the mean value than the female weight of 1500 grams; thus, the male weightof 1500 grams is more extreme.