Question

Length of pregnancies The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean of $266$days and standard deviation of$16$days. For each part, follow the four-step process.

(a) At what percentile is a pregnancy that lasts days (that’s about $8$months)?

(b) What percent of pregnancies last between $240$and $270$days (roughly between $8$months and $9$months)?

(c) How long do the longest $20$% of pregnancies last?

Short answer

From the given information,

a) About $5.2$% of pregnancies last less than $240$days

b) Approximately $55$%of pregnancies last between $240$and $270$ days.

c) The longest $20$% of pregnancies last approximately $279$or more days.

Step-by-step solution

Part (a) Step 1: Given Information

It is given in the question that,

$\mu =266daysand\sigma =16days$

At what percentile is a pregnancy that lasts $240$days (that’s about $8$months)?

Part (a) Step 2: Explanation

In the following equation, $\mathrm{x}$can be defined as a random variable representing the length of pregnancy.

Therefore, the percentage of pregnancies lasting less than $240$days will look like this:

Part (a) Step 3: Explanation

Find the corresponding $z$values for $x=240$

$$\begin{gathered} z=\frac{240-266}{16} \\ =-1.63 \end{gathered}$$

In the standard normal table, of observations are below $-1.63$.

Therefore, the $z-$scale area below $240$is the same as the area below $-1.63.$

Pregnancies of less than $240$days last about $5.2\%$of the cases.

Part (b) Step 1: Given Information

It is given in the question that,

$\mu =266daysand\sigma =16days$

What percent of pregnancies last between $240$and $270$days (roughly between $8$months and $9$months)?

Part (b) Step 2: Explanation

$X$is a random variable that represents the length of pregnancy.

Let $x$be a percentage representing the proportion of pregnancies lasting $240$to $270$days.

The percentage of pregnancies lasting between $240$and $270$days is represented by the following graph:

Part (b) step 3: Explanation

From part (a),$x=240,z=-1.63$

Find the $z-$value for $x=270$,

$$\begin{gathered} z=\frac{170-266}{16} \\ =0.25 \end{gathered}$$

In the standard normal table, the $z$value associated with $0.25$is $0.5987$:

$-1.63$corresponds to a $z$value of $0.052$in part (a).

In order to calculate the proportion of observations between $-1.63$and $0.25,0.5987-0.0516=0.5471$is to be used.

The average pregnancy lasts between $240$and $270$days.

Part (c) Step 1: Given Information

It is given in the question that, $\mu =266daysand\sigma =16days$

How long do the longest $20$% of pregnancies last?

Part (c) Step 2: Explanation

Let $x$represent the length of a human pregnancy.

Our goal is to determine the number of days that will result in $80\%$of pregnant women having shorter pregnancies than this number.

Standard normal tables give a $z$value of $0.84$for a pregnancy length of $80$weeks.

In other words, for a pregnancy length of localid="1649930129719" $80$weeks, the localid="1649930142698" ${80}^{th}$percentile is the value obtained by solving the equation below:

localid="1649997280411" $$\begin{gathered} 0.84=\frac{x-266}{16} \\ x=279.44 \end{gathered}$$

The graph of the localid="1649930247228" ${80}^{th}$percentile length of a human pregnancy is:

Hence, $20\%$of pregnancies last approximately $279$more days.