Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 3000 grams and a standard deviation of 1000 grams while babies born after a gestation period of 40 weeks have a mean weight of 3400 grams and a standard deviation of 470 grams. If a 35​-week gestation period baby weighs 2900 grams and a 41​-week gestation period baby weighs 3300 ​grams, find the corresponding​ z-scores. Which baby weighs less relative to the gestation​ period?

To find the corresponding z-scores for the two babies, we use the formula for the z-score:

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

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

### Step 1: Z-score for the 35-week baby

- Mean weight (\(\mu\)) for babies born after 32 to 35 weeks: 3000 grams
- Standard deviation (\(\sigma\)): 1000 grams
- Baby's weight (\(x\)): 2900 grams

\[
z_{35} = \frac{2900 - 3000}{1000} = \frac{-100}{1000} = -0.1
\]

### Step 2: Z-score for the 41-week baby

- Mean weight (\(\mu\)) for babies born after 40 weeks: 3400 grams
- Standard deviation (\(\sigma\)): 470 grams
- Baby's weight (\(x\)): 3300 grams

\[
z_{41} = \frac{3300 - 3400}{470} = \frac{-100}{470} \approx -0.213
\]

### Step 3: Interpretation

The z-score for the 35-week baby is \(-0.1\), and the z-score for the 41-week baby is approximately \(-0.213\).

Since z-scores indicate how far a data point is from the mean in terms of standard deviations, the 41-week baby has a lower z-score. This means that the 41-week baby weighs less **relative to the gestation period** compared to the 35-week baby.

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Do you want any more details or clarifications?

Here are 5 related questions to expand on this:
1. What does a z-score of 0 mean in the context of baby weights?
2. How can z-scores be used to compare two distributions with different means and standard deviations?
3. What would the z-score be for a 35-week baby weighing exactly 3000 grams?
4. How can z-scores help in identifying outliers in data?
5. What is the probability that a 41-week baby weighs less than 3300 grams?

**Tip:** Z-scores are especially useful when comparing values from different distributions with different means and standard deviations!

Calculate the z-scores for a 35-week gestation baby weighing 2900 grams and a 41-week gestation baby weighing 3300 grams. Learn how to apply the z-score formula to compare baby weights relative to their gestation period. This problem involves statistics concepts such as normal distribution and z-scores, ideal for students in Grades 10-12.

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