Question

The lengths of pregnancy terms for a particular species of mammal are nearly normally distributed about a mean pregnancy length with a standard deviation of days. About what percentage of births would be expected to occur within days of the mean pregnancy​ length?
About what​% of births would be expected to occur within days of the mean pregnancy length.

Answer 1

Answer: About 99.74% of births would be expected to occur within 24 days of the mean pregnancy length.

Step-by-step explanation:

Complete question is attached below.

Given: The lengths of pregnancy terms for a particular species of mammal are nearly normally distributed about a mean pregnancy length with a standard deviation of 8 days.

i.e. $\sigma= 8$

let X denotes the random variable that represents the lengths of pregnancy.

The probability of births would be expected to occur within 24 days of the mean pregnancy​ length:

$P(\mu-24<X<\mu+24)=P(\dfrac{\mu-24-\mu}{8}<\dfrac{X-\mu}{\sigma}<\dfrac{\mu+24-\mu}{8})\\\\=P(\dfrac{-24}{8}<Z<\dfrac{24}{8})\ \ \ [\because Z=\dfrac{X-\mu}{\sigma}]\\\\=P(-3<Z<3)\\\\=P(Z<3)-P(Z<-3)\\\\=P(Z<3)-(1-P(Z<3))\\\\=2P(Z<3)-1$

$= 2(0.9987)-1\ \ \ [\text{ By z-table}]\\\\=0.9974$

=99.74%

Hence, about 99.74% of births would be expected to occur within 24 days of the mean pregnancy length.