Question

Redo Example$5b$under the assumption that the number of man-woman pairs is (approximately) normally distributed. Does this seem like a reasonable supposition?

Short answer

Therefore,

$P\{X\le 30\}\approx 0$.

Step-by-step solution

Given Information.

the number of man-woman pairs is (approximately) normally distributed.

Explanation.

Consider Example$5b$(textbook). Now, assume that the number of man-woman pairs

$X=\sum _{i=1}^{100}{X}_i,\text{where}{X}_i=\left\{\begin{array}{l}1,\text{man}i\text{is paired with a woman}\\ 0,\text{otherwise}\end{array}\right.$

is approximately normally distributed with a mean$E\left[X\right]\approx 50.25$and a variance $Var\left(X\right)\approx 25.126$

The probability that most $30$pairs will consist of a man and a woman is$P\{X\le 30\}$. To approximate this probability we can use the central limit theorem and in that case, we get:

localid="1649832005870" $P\{X\le 30\}$$\approx P\left\{\frac{X-E\left[X\right]}{\sqrt{Var\left(X\right)}}\le \frac{30-E\left[X\right]}{\sqrt{Var\left(X\right)}}\right\}$localid="1649831889184" $=P\left\{\frac{X-50.25}{\sqrt{25.126}}\le \frac{30-50.25}{\sqrt{25.126}}\right\}$

$\approx 0$.