Question

The length of human pregnancies from conception to birth approximates a normal distribution with a mean of 266 days and a standard deviation of 16 days. What proportion of all pregnancies will last between 240 and 270 days (roughly between 8 and 9 months)?

Answer 1

Applications of the Normal Distribution Solutions

Rick Gumina STCC201 NormalDist1_sol.doc

Figure 3: Accompanies problem 3

Figure 4: Accompanies problem 4

3) What is the probability that a woman’s

pregnancy will last more than 258 or

less than 222 days?

Here we can use the fact that the total

area under the normal “density” is 1.

We’ve already computed the area

(probability) for pregnancies between

222 and 258 days in problem 2. To

get the corresponding probability that

a pregnancy will last less than 222 days or more than 258 days we can,

simply subtract from 1. Hence,

Pr{x < 222 or x > 258} = 1 – 0.3055 = 0.6945

4) What proportion of human pregnancies

is shorter than 282 days?

This is a type I problem!

A) Convert to z

00.1

16

282 266

=

−

z =

B) Look up probability in Table A

Pr{x < 282} = Pr{z<1.00} = 0.8413

5) What two values enclose the middle 95% of human gestation periods? (Hint:

for a “close enough” answer use the empirical rule)

According to the empirical rule, 95% of the data in a BSD will lie between ± 2

σ of the mean. Since gestation length is normally distributed (bell shaped)

with a mean of 266 and a standard deviation of 16, we would expect 95% of

the data to lie between 266 ± 2(16) days. So, 95% of all pregnancies will last

between 234 and 298 days.

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Answer 2

Given:

$\[\mu = 266\,days\]$

$\[\sigma = 16\,days\]$

To Find: Probability of pregnancies lasting between 240 and 270 days

Solution:

First, convert 240 and 270 into $z$ scores as below:

$\[\begin{gathered} z(240) = \frac{{(240 - 266)}}{{16}} \hfill \\ z(240) = \frac{{ - 26}}{{16}} = \frac{{ - 13}}{8} \hfill \\ \end{gathered} \]$

and

$\[\begin{gathered} z(270) = \frac{{(270 - 266)}}{{16}} \hfill \\ z(270) = \frac{4}{{16}} = \frac{1}{4} \hfill \\ \end{gathered} \]$

Now, find the probability of the pregnancy, say ‘$x$’, lying between 240 to 270 as follows:

$\[\begin{gathered} P(240 < x < 270) = P\left( {\frac{{ - 13}}{8} < z < \frac{1}{4}} \right) \hfill \\ P(240 < x < 270) = normal\,cdf\left( {\frac{{ - 13}}{8},\frac{1}{4}} \right) \hfill \\ P(240 < x < 270) = 0.5466 \hfill \\ \end{gathered} \]$

Hence,  the probability of a human pregnancy lying between $240$ and $270$ days is $\[54.66\% \]$.

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