Chapter 4.6, Problem 13PPS

a.

To determine

Find an equation for the regression line.

Answer to Problem 13PPS

Equation for the regression line $y=84344.99655x+\mathrm{5003868.25862.}$

Explanation of Solution

Given information: The table shows the number of participant in high school athletics.

Year Since 1980x	0	10	20	25	30
Athletes y	5356913	5298671	6705223	7159904	7667955

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:

  $\begin{array}{l}\text{Sum}\text{of}x=85.\\ \text{Sum}\text{}\text{of}y=32188666.\\ \text{Mean}x=17.\\ \text{Mean}y=\mathrm{6437733.2.}\\ \text{Sum}\text{of}\text{squares}(S{S}_x)=580.\\ \text{Sum}\text{of}\text{products}(SP)=48920098.\\ \text{Regression}\text{equation}y=ax+b.\\ a=SP/S{S}_x=48920098/580=\mathrm{84344.99655.}\\ b=M_y-a{M}_x=6437733.2-(84345\times 17)=\mathrm{5003868.25862.}\\ y=84344.99655x+\mathrm{5003868.25862.}\end{array}$

b.

To determine

According to the equation, how many participated in 2008.

Answer to Problem 13PPS

Number of participants are in 2008 is $y\approx 73,65,528.$

Explanation of Solution

Given information: The table shows the number of participant in high school athletics.

Year Since 1980x	0	10	20	25	30
Athletesy	5356913	5298671	6705223	7159904	7667955

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:

  $\begin{array}{l}\text{Sum}\text{of}x=85.\\ \text{Sum}\text{}\text{of}y=32188666.\\ \text{Mean}x=17.\\ \text{Mean}y=\mathrm{6437733.2.}\\ \text{Sum}\text{of}\text{squares}(S{S}_x)=580.\\ \text{Sum}\text{of}\text{products}(SP)=48920098.\\ \text{Regression}\text{equation}y=ax+b.\\ a=SP/S{S}_x=48920098/580=\mathrm{84344.99655.}\\ b=M_y-a{M}_x=6437733.2-(84345\times 17)=\mathrm{5003868.25862.}\\ y=84344.99655x+\mathrm{5003868.25862.}\end{array}$

To estimate the number of participants in 2008, evaluate the regression equation for x = 28 because 2008 is 28 years after 1980.

  $\begin{array}{l}y=84345.99655\times 18+\mathrm{5003868.2562.}\\ y=2361659.903+\mathrm{5003868.2562.}\\ y=73,65,\mathrm{528.1598.}\\ y\approx 73,65,528.\end{array}$