Questions: Use z scores to compare the given values. Based on sample data, newborn males have weights with a mean of 3287.6 g and a standard deviation of 663.8 g. Newborn females have weights with a mean of 3067.8 g and a standard deviation of 798.6 g. Who has the weight that is more extreme relative to the group from which they came: a male who weighs 1600 g or a female who weighs 1600 g?

Solution Steps

To determine which newborn has a more extreme weight relative to their group, we need to calculate the z-scores for both the male and female newborns. The z-score is calculated using the formula:

\[ z = \frac{(X - \mu)}{\sigma} \]

where \( X \) is the observed value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. We will calculate the z-score for both the male and female newborns and compare their absolute values to determine which is more extreme.

To find the z-score for the male newborn, we use the formula:

\[ z_{\text{male}} = \frac{X_{\text{male}} - \mu_{\text{male}}}{\sigma_{\text{male}}} \]

Substituting the given values:

\[ z_{\text{male}} = \frac{1600 - 3287.6}{663.8} \approx -2.542 \]

Similarly, we calculate the z-score for the female newborn:

\[ z_{\text{female}} = \frac{X_{\text{female}} - \mu_{\text{female}}}{\sigma_{\text{female}}} \]

Substituting the given values:

\[ z_{\text{female}} = \frac{1600 - 3067.8}{798.6} \approx -1.838 \]

To determine which weight is more extreme, we compare the absolute values of the z-scores:

\[ |z_{\text{male}}| \approx 2.542, \quad |z_{\text{female}}| \approx 1.838 \]

Since \( |z_{\text{male}}| > |z_{\text{female}}| \), the male newborn's weight is more extreme relative to their group.

Final Answer