Q. 5.12 - A First Course in Probability

Question

The following table uses $1992$ data concerning the percentages of male and female full-time workers whose

annual salaries fall into different ranges:

Suppose that random samples of 200 male and 200 female full-time workers are chosen. Approximate the probability

that

(a) at least $70$ of the women earn $\backslash (25,000$ or more;

(b) at most $60$ percent of the men earn $\backslash )25,000$ or more;

(c) at least three-fourths of the men and at least half the women earn $\$20,000$ or more.

Step-by-Step Solution

Answer

 (A) $70$ of the women earning is $0.4407$

(B) $60$ of the men earning is $0.6456$

(C) At least three fourth of men and women earning is $0.3974$

Step :1 Women earning (part a)

Construct the poisson distribution $N$ to symbolize the number of women earning $\$25,000$ or more. $N$ sim operator name in $0 m (200,p)$, where $p$ is the chance that a random woman earns $\$25,000$ or more. As can be seen from the table, $p=0.34$. We'll use the Normal approximation method. Obtain

$$\begin{gathered} E\left(N\right)=200\times 0.34 \\ =68 \\ \mathrm{Var}\left(N\right)=200\times 0.34\times 0.66 \\ =44.88 \end{gathered}$$

$$\begin{gathered} P(N\ge 70)=1-P(N<69) \\ =1-P\left(\frac{N-68}{\sqrt{44.88}}<\frac{69-68}{\sqrt{44.88}}\right) \end{gathered}$$

$$\begin{gathered} \approx 1-\mathrm{\Phi }\left(0.1493\right) \\ P(N\ge 70)=0.4407 \end{gathered}$$

Step :2 Men earning (part b)

Create a dependent vector. The quantity of males earning $\$25,000$ or more is denoted by the letter $M$. We have $M~\mathrm{Binom}(200,p)$ , where $p$ is the chance of random men earning  $p=0.587$ $\$25,000$equals $N\le 120$. We'll use the Average interpolation technique. Obtain

$$\begin{gathered} E\left(N\right)=200\times 0.587 \\ =117.4 \\ \mathrm{Var}\left(N\right)=200\times 0.587\times 0.413 \\ =48.49 \end{gathered}$$

which implies, 

$$\begin{gathered} P(M\le 120) \\ =P\left(\frac{M-117.4}{\sqrt{48.49}}<\frac{120-117.4}{\sqrt{48.49}}\right) \end{gathered}$$

$$\begin{gathered} \approx \mathrm{\Phi }\left(0.3734\right) \\ P(M\le 120)=0.6456 \end{gathered}$$

Step :3 Three fourth earning (part c)

We're going to assume that men's and women's salaries are equal. Establish the random variables $X$ and $Y$, which represent the number of women and men in the sample who have a paycheck minus sign.

$$\begin{gathered} 20,000\text{ ormore. Wehavethat} \\ X~\mathrm{Binom}(200,0.534), \\ Y~\mathrm{Binom}(200,0.745) \end{gathered}$$

$\begin{array}{r}P(X\ge 100,Y\ge 150)=\left(1-P\left(\frac{X-106.8}{\sqrt{49.77}}\le \frac{99.5-106.8}{\sqrt{49.77}}\right)\right)\\ \left(1-P\left(\frac{Y-149}{\sqrt{38}}\le \frac{149.5-149}{\sqrt{38}}\right)\right)\end{array}$

$=(1-\mathrm{\Phi }(-1.035\left)\right)(1-\mathrm{\Phi }(0.0811\left)\right)$

$$\begin{gathered} =0.8497\times 0.4677 \\ =0.3974 \end{gathered}$$