The lengths of pregnancies are normally distributed with a mean of 264 days and a standard deviation of 15 days. if 36 women are randomly selected, find the probability that they have a mean pregnancy between 264 days and 266 days.

Given that a sample of 36 women is selected, the probability that the mean is between 264 days and 266 days will be found as follows:
the std deviation of the sample will be:
σ/√n
plugging the values we obtain:
15/√36
=15/6
=2.5
hence the z-score when x=264 will be:
z=(264-264)/2.5
z=0
thus:
P(x<264)=P(z<0)=0.5

the z-score when x=266 will be:
z=(266-264)/2.5
z=0.8
hence
P(x<266)=P(z<0.8)=0.7881
thus
P(264<x<266)=P(0.5<z<0.7881)
=0.7881-0.5
=0.2881

0.2881