Question

Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 2900 grams and a standard deviation of 800 grams while babies born after a gestation period of 40 weeks have a mean weight of 3200 grams and a standard deviation of 475 grams. If a 32​-week gestation period baby weighs 2700 grams and a 40​-week gestation period baby weighs 3000 ​grams, find the corresponding​ z-scores. Which baby weighs less relative to the gestation​ period?

Answer 1

The z-score of the 32 week gestation period baby is -0.25.

The z-score of the 40 week gestation period baby is -0.42.

The 40 week gestation period baby has the lower z-score, so he weighs less relative to the gestation​ period.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Which baby weighs less relative to the gestation​ period?

Whoever has the lower z-score.

32 week baby:

Mean weight of 2900 grams and a standard deviation of 800 grams. 32​-week gestation period baby weighs 2700 grams. We have to find Z when
`X = 2700, \mu = 2900, \sigma = 800`. So

`Z = (X - \mu)/(\sigma)`

`Z = (2700 - 2900)/(800)`

`Z = -0.25`

The z-score of the 32 week gestation period baby is -0.25.

40 week baby:

Mean weight of 3200 grams and a standard deviation of 475 grams. 40​-week gestation period baby weighs 3000 ​grams. We have to find Z when
`X = 3000, \mu = 3200, \sigma = 475`. So

`Z = (X - \mu)/(\sigma)`

`Z = (3000 - 3200)/(475)`

`Z = -0.42`

The z-score of the 40 week gestation period baby is -0.42.

The 40 week gestation period baby has the lower z-score, so he weighs less relative to the gestation​ period.