Questions: Use z scores to compare the given values. Based on sample data, newborn males have weights with a mean of 3254.4 g and a standard deviation of 858.4 g. Newborn females have weights with a mean of 3077.2 g and a standard deviation of 573.8 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? Since the z score for the male is z= Find the z score for the female is z= the has the weight that is more extreme. (Round to two decimal places.)

Solution Steps

To compare the extremity of the weights of the male and female newborns relative to their respective groups, we need to calculate the z-scores for both weights. The z-score is calculated using the formula:

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

where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. We will then compare the absolute values of the z-scores to determine which weight is more extreme.

1. Calculate the z-score for the male newborn using the given mean and standard deviation for males.
2. Calculate the z-score for the female newborn using the given mean and standard deviation for females.
3. Compare the absolute values of the z-scores to determine which weight is more extreme.

Given:

• Mean weight of male newborns, \( \mu_{\text{male}} = 3254.4 \, \text{g} \)
• Standard deviation of male newborns, \( \sigma_{\text{male}} = 858.4 \, \text{g} \)
• Weight of the male newborn, \( X_{\text{male}} = 1600 \, \text{g} \)

The z-score for the male newborn is calculated using the formula: \[ z_{\text{male}} = \frac{X_{\text{male}} - \mu_{\text{male}}}{\sigma_{\text{male}}} \]

Substituting the values: \[ z_{\text{male}} = \frac{1600 - 3254.4}{858.4} \approx -1.9273 \]

Given:

• Mean weight of female newborns, \( \mu_{\text{female}} = 3077.2 \, \text{g} \)
• Standard deviation of female newborns, \( \sigma_{\text{female}} = 573.8 \, \text{g} \)
• Weight of the female newborn, \( X_{\text{female}} = 1600 \, \text{g} \)

The z-score for the female newborn is calculated using the formula: \[ z_{\text{female}} = \frac{X_{\text{female}} - \mu_{\text{female}}}{\sigma_{\text{female}}} \]

Substituting the values: \[ z_{\text{female}} = \frac{1600 - 3077.2}{573.8} \approx -2.5744 \]

To determine which weight is more extreme, we compare the absolute values of the z-scores: \[ |z_{\text{male}}| \approx 1.9273 \] \[ |z_{\text{female}}| \approx 2.5744 \]

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

Final Answer