The pregnancy length in days for a population of new mothers can be approximated by a normal distribution with a mean of 270 days and a standard deviation of 8 days. (a) What is the minimum pregnancy length that can be in the top 8% of pregnancy lengths? (b) What is the maximum pregnancy length that can be in the bottom 3% of pregnancy lengths?

a) 281 days.

b) 255 days

Step-by-step explanation:

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

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

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.

In this problem, we have that:

(a) What is the minimum pregnancy length that can be in the top 8% of pregnancy lengths?

100 - 8 = 92th percentile.

X when Z has a pvalue of 0.92. So X when Z = 1.405.

(b) What is the maximum pregnancy length that can be in the bottom 3% of pregnancy lengths?

3rd percentile.

X when Z has a pvalue of 0.03. So X when Z = -1.88