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.)

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.)
Transcript text: 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 $z=$ $\square$ Find the $z$ score for the female is $z=$ $\square$ the $\square$ has the weight that is more extreme. (Round to two decimal places.)
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Solution

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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=Xμσ z = \frac{X - \mu}{\sigma}

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

Solution Approach
  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.
Step 1: Calculate the Z-score for the Male Newborn

Given:

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

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

Substituting the values: zmale=16003254.4858.41.9273 z_{\text{male}} = \frac{1600 - 3254.4}{858.4} \approx -1.9273

Step 2: Calculate the Z-score for the Female Newborn

Given:

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

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

Substituting the values: zfemale=16003077.2573.82.5744 z_{\text{female}} = \frac{1600 - 3077.2}{573.8} \approx -2.5744

Step 3: Compare the Absolute Values of the Z-scores

To determine which weight is more extreme, we compare the absolute values of the z-scores: zmale1.9273 |z_{\text{male}}| \approx 1.9273 zfemale2.5744 |z_{\text{female}}| \approx 2.5744

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

Final Answer

zmale1.93,zfemale2.57,female\boxed{z_{\text{male}} \approx -1.93, \, z_{\text{female}} \approx -2.57, \, \text{female}}

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