Chapter 4.6, Problem 15HP

To determine

To find an equation for the regression line for each (men, women) and graph them. Compare and contrast men’s and women’s graph.

Answer to Problem 15HP

Equation for the regression line for Men $y=-2.901x+\mathrm{99.967.}$

Equation for the regression line for Women $y=-2.666x+93.664$ .

The slop for women’s scores is steeper than men’s scores.

Explanation of Solution

Given information: Below are the results of the W championships in2013.

Men	Score	Rank	Woman	Score
S W	98.00	1	K C	90.33
A H	92.33	2	E H	90.00
M M	91.33	3	A G	85.00
S L	90.00	4	T B	82.00
G B	84.66	5	H T	81.00

Formula used:

The equation for the regression line is given below.

  $\begin{array}{c}y=ax+b.\\ \text{Where},a=\frac{n{\displaystyle \sum xy-({\displaystyle \sum x)({\displaystyle \sum y)}}}}{n{\displaystyle \sum x^2-({\displaystyle \sum x{)}^2}}}.\\ b=\frac{{\displaystyle \sum y-a{\displaystyle \sum x}}}{n}.\\ \text{Where, }\text{n=number}\text{of}\text{observations}.\end{array}$

Calculation:

For Men,

  $\begin{array}{l}\text{Sum}\text{of}x=15.\\ \text{Sum}\text{}\text{of}y=456.32.\\ \text{Mean}x=3.\\ \text{Mean}y=\mathrm{91.264.}\\ \text{Sum}\text{of}\text{squares}(S{S}_x)=10.\\ \text{Sum}\text{of}\text{products}(SP)=-\mathrm{29.01.}\\ \text{Regression}\text{equation}y=ax+b.\\ a=SP/S{S}_x=-29.01/10=-\mathrm{2.901.}\\ b=M_y-a{M}_x=91.26-(-2.9\times 3)=\mathrm{99.967.}\\ y=-2.901x+\mathrm{99.967.}\end{array}$

The graph for Men’s regression line is given below.

  

Where x- axis represents Rank and y −axis represents Scores.

For Women,

  $\begin{array}{l}\text{Sum}\text{of}x=15.\\ \text{Sum}\text{}\text{of}y=428.33.\\ \text{Mean}x=3.\\ \text{Mean}y=\mathrm{85.666.}\\ \text{Sum}\text{of}\text{squares}(S{S}_x)=10.\\ \text{Sum}\text{of}\text{products}(SP)=-\mathrm{26.66.}\\ \text{Regression}\text{equation}y=ax+b.\\ a=SP/S{S}_x=-26.66/10=-\mathrm{2.666.}\\ b=M_y-a{M}_x=85.67-(-2.67\times 3)=\mathrm{93.664.}\\ y=-2.666x+\mathrm{93.664.}\end{array}$

The graph for Women’s regression line is given below.

  

Where x- axis represents Rank and y −axis represents Scores.

So, it is clear from both graphs, the slop for women’s scores is steeper than men’s scores.