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

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

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

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

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

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

Since $0.45\cdot 0.85=0.3825\neq 0.375,$ these events are dependent.