Use z scores to compare the given values.
Based on sample​ data, newborn males have weights with a mean of 3242.6 g and a standard deviation of 898.6 g. Newborn females have weights with a mean of 3044.2 g and a standard deviation of 518.6 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?

To determine who has the more extreme weight relative to their group, we can use **z-scores**. A z-score tells us how many standard deviations a particular value is from the mean of the distribution. The formula for calculating a z-score is:

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

Where:
- \(X\) is the value of the data point (the weight of the newborn in this case),
- \(\mu\) is the mean of the data,
- \(\sigma\) is the standard deviation.

### Step 1: Calculate the z-score for the male newborn
For the male newborn:
- \(X = 1600 \, \text{g}\)
- \(\mu = 3242.6 \, \text{g}\)
- \(\sigma = 898.6 \, \text{g}\)

\[
z_{\text{male}} = \frac{1600 - 3242.6}{898.6} = \frac{-1642.6}{898.6} \approx -1.83
\]

### Step 2: Calculate the z-score for the female newborn
For the female newborn:
- \(X = 1600 \, \text{g}\)
- \(\mu = 3044.2 \, \text{g}\)
- \(\sigma = 518.6 \, \text{g}\)

\[
z_{\text{female}} = \frac{1600 - 3044.2}{518.6} = \frac{-1444.2}{518.6} \approx -2.78
\]

### Step 3: Interpret the results
- The z-score for the male is approximately \(-1.83\).
- The z-score for the female is approximately \(-2.78\).

Since the absolute value of the female's z-score (2.78) is larger than the male's z-score (1.83), the female's weight is more extreme relative to the group from which she came. 

### Conclusion:
The female who weighs 1600 g has a weight that is more extreme relative to her group than the male who weighs 1600 g.

Would you like more details on any part of the solution? 

Here are 5 related questions to explore further:
1. How are z-scores used to compare different distributions?
2. What does a negative z-score signify in this context?
3. How could we interpret z-scores in terms of probability?
4. What is considered an "extreme" value in terms of z-scores?
5. How can z-scores be applied in hypothesis testing?

**Tip**: When comparing values from different groups, using z-scores allows you to make fair comparisons by accounting for the differences in means and standard deviations.

Based on sample data, newborn males have weights with a mean of 3242.6 g and a standard deviation of 898.6 g. Newborn females have weights with a mean of 3044.2 g and a standard deviation of 518.6 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?

Learn how to compare z-scores for newborn weights. Using a mean of 3242.6 g and a standard deviation of 898.6 g for males, and a mean of 3044.2 g with a standard deviation of 518.6 g for females, we calculate who has the more extreme weight: a male or female both weighing 1600 g. This problem demonstrates how to use the z-score formula to compare values across different distributions.

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