Birth Weights Babies born after a gestation period of 32–35 weeks have a mean weight of 2600 grams and a standard deviation of 660 grams. Babies born after a gestation period of 40 weeks have a mean weight of 3500 grams and a standard deviation of 470 grams. Suppose a 34-week gestation period baby weighs 3000 grams and a 40-week gestation period baby weighs 3900 grams. What is the z-score for the 34-week gestation period baby? What is the z-score for the 40-week gestation period baby? Which baby weighs less relative to the gestation period?

Verified step by step guidance

1. Identify the mean (\( \mu \)) and standard deviation (\( \sigma \)) for each gestation period group. For the 34-week baby (which falls in the 32–35 weeks group), use \( \mu = 2600 \) grams and \( \sigma = 660 \) grams. For the 40-week baby, use \( \mu = 3500 \) grams and \( \sigma = 470 \) grams.
2. Recall the formula for the z-score, which measures how many standard deviations a value is from the mean: \[ z = \frac{X - \mu}{\sigma} \] where \( X \) is the observed value (baby's weight), \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
3. Calculate the z-score for the 34-week baby by substituting \( X = 3000 \), \( \mu = 2600 \), and \( \sigma = 660 \) into the formula: \[ z_{34} = \frac{3000 - 2600}{660} \]
4. Calculate the z-score for the 40-week baby by substituting \( X = 3900 \), \( \mu = 3500 \), and \( \sigma = 470 \) into the formula: \[ z_{40} = \frac{3900 - 3500}{470} \]
5. Compare the two z-scores to determine which baby weighs less relative to their gestation period. The baby with the lower z-score (more negative or smaller value) weighs less compared to the average weight for that gestation period.

Key Concepts

Z-Score

A z-score measures how many standard deviations a data point is from the mean of its distribution. It is calculated by subtracting the mean from the value and dividing by the standard deviation. Z-scores allow comparison of values from different distributions by standardizing them.

Mean and Standard Deviation

The mean is the average value of a dataset, representing its central tendency. The standard deviation quantifies the amount of variation or dispersion around the mean. Together, they describe the distribution of data and are essential for calculating z-scores.

Comparing Relative Positions Across Different Groups

When comparing values from different groups with distinct means and standard deviations, z-scores help determine which value is relatively higher or lower within its group. This standardization enables meaningful comparisons despite differing scales or variability.