Question

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with a mean of 266 days and standard deviation of 16 days.What percent of pregnancies last less than 240 days?What percent of pregnancies last between 240 days and 270 days?How long do the longest 20% of pregnancies last?

Answer 1

P(X<240)=0.0521

P(240<X<270)=0.5466

280 days

Explanation:

we are given the following information

mean=μ=266 days

standard deviation=σ=16 days

P(X<240)=P(X-μ/σ<240-266/16)=P(z<-1.625)

This probability can either be determined by utilizing normal distribution table or using Excel.

Using Excel function NORM.S.DIST(-1.625,TRUE) we get

P(X<240)=0.0521

P(240<X<270)=P((240-266)/16<X-μ/σ<(270-266)/16)=P(-1.625<z<0.25)

P(-1.625<z<0.25)=P(z<0.25)-P(z<-1625)

Using Excel function

NORM.DIST(270,266,16,TRUE)-NORM.DIST(240,266,16,TRUE) we get

P(240<X<270)=0.5466

P(X>x)=0.20

P(z>(x-266)/16)=0.20

Using Excel function NORM.INV(0.8,266,16) we get

x=280

OR

P(X>x)=0.2

The area to the right of x is 0.2 and the area between mean and x is

0.5-0.2=0.3

Looking for 0.3 in the normal distribution table we don't find 0.3 but the closest value is .3023 and it corresponds to z-value 0.85.

z=0.85

z=x-μ/σ

σ*z+μ=x

x=16*0.85+266

x=279.6≅280

The longest 20% of pregnancies lasts 280 days