Question

A survey of 1,000 men and women asked, "Do you earn over $50,000 per year?" The table below shows the responses for males and females: Male Female Total Income over $50,000 475 375 850 Income below $50,000 75 75 150 Total 550 450 1,000 Based on these data, are "being female" and "earning over $50,000" independent events?

Answer 1

You are given the table

`\begin{array}{cccc} &\text{Male}&\text{Female}&\text{Total Income}\\\text{over \$50,000}&475&375&850\\\text{below \$50,000}&75&75&150\\\text{Total}&550&450&1,000\end{array}`

The events A and B are independent when

`Pr(A\cap B)=Pr(A)\cdot Pr(B).`

1. The probability "being female" is

`Pr(\text{Being Female})=(375+75)/(1,000)=(450)/(1,000)=0.45.`

2. The probability "earning over $50,000" is

`Pr(\text{Earning over \$50,000})=(475+375)/(1,000)=(850)/(1,000)=0.85.`

3. The probability of "being female and earning over $50,000" is

`Pr(\text{Being Female and Earning over \$50,000})=(375)/(1,000)=0.375.`

Since
`0.45\cdot 0.85=0.3825\\eq 0.375,` these events are dependent.