Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 3000 grams and a standard deviation of 900 grams while babies born after a gestation period of 40 weeks have a mean weight of 3600 grams and a standard deviation of 590 grams. If a 33​-week gestation period baby weighs 3375 grams and a 41​-week gestation period baby weighs 3975 ​grams, find the corresponding​ z-scores. Which baby weighs more relative to the gestation​ period?

To find the \( z \)-scores of the babies, we will use the \( z \)-score formula:

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

Where:
- \( x \) is the baby's weight,
- \( \mu \) is the mean weight for the gestation period,
- \( \sigma \) is the standard deviation of the weights for that gestation period.

### For the 33-week gestation baby:
- \( x = 3375 \) grams,
- \( \mu = 3000 \) grams (mean for 32–35 weeks),
- \( \sigma = 900 \) grams (standard deviation for 32–35 weeks).

Substitute these values into the formula:

\[
z = \frac{3375 - 3000}{900} = \frac{375}{900} = 0.417
\]

### For the 41-week gestation baby:
- \( x = 3975 \) grams,
- \( \mu = 3600 \) grams (mean for 40 weeks),
- \( \sigma = 590 \) grams (standard deviation for 40 weeks).

Substitute these values into the formula:

\[
z = \frac{3975 - 3600}{590} = \frac{375}{590} = 0.636
\]

### Interpretation:
- The \( z \)-score of the 33-week baby is 0.417.
- The \( z \)-score of the 41-week baby is 0.636.

Since the 41-week baby has a higher \( z \)-score, this baby weighs more **relative to its gestation period** compared to the 33-week baby.

Would you like further details on how to interpret the \( z \)-scores, or do you have other questions?

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Here are 5 related questions to expand your understanding:

1. What does a negative \( z \)-score indicate in this context?
2. How would the \( z \)-score change if the 33-week baby weighed 2900 grams?
3. Can you calculate the probability of a baby weighing less than 3375 grams at 33 weeks using the normal distribution?
4. How would a larger standard deviation affect the \( z \)-scores of these babies?
5. What percentage of babies fall within one standard deviation of the mean for a given gestation period?

**Tip:** The \( z \)-score tells you how many standard deviations a data point is from the mean, helping you compare different data points across different distributions.

Learn how to calculate the z-scores for a 33-week gestation baby weighing 3375 grams and a 41-week gestation baby weighing 3975 grams using the formula z = (x - μ) / σ. This problem demonstrates the concept of z-scores and normal distribution, making it ideal for high school students in Grades 10-12.

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