Chapter 4.6, Problem 23PFA

a

To determine

To find: An equation for the regression line. Letx represents the number of years since 2010.

Answer to Problem 23PFA

The equation of regression line is $y=250.4x+1002.6.$ .

Explanation of Solution

Given information: The table shows U.S. online sales, in millions of dollars ,on Cyber .

Monday since 2010.

  

Let x be the number of years since 2010.

Years x	0	1	2	3	4
Sales y	1028	1251	1465	1735	2038

Calculation:

Use a graphing calculator for above table ,

Equation of regression line is,

  $\begin{array}{l}\text{Sum}\text{of}X=10.\\ \text{Sum}\text{}\text{of}Y=7515.\\ \text{Mean}X=2.\\ \text{Mean}Y=\mathrm{1503.4.}\\ \text{Sum}\text{of}\text{squares}(S{S}_X)=10.\\ \text{Sum}\text{of}\text{products}(SP)=2504.\\ \text{Regression}\text{equation}y=aX+b.\\ a=SP/S{S}_X=2504/10=\mathrm{250.4.}\\ b=M_Y-a{M}_X=1503.4-(250.4\times 2)=1002.6\\ y=250.4x+1002.6.\end{array}$

The equation of regression line is $y=250.4x+1002.6.$

b

To determine

To predict: The Cyber Monday sales in 2025.

Answer to Problem 23PFA

Cyber Monday sales in 2025 is 4758.6.

Explanation of Solution

Given information: The table shows U.S. online sales, in millions of dollars ,on Cyber .

Monday since 2010.

  

Let x be the number of years since 2010.

Years x	0	1	2	3	4
Sales y	1028	1251	1465	1735	2038

Calculation:

Use a graphing calculator for above table ,

Equation of regression line is,

  $\begin{array}{l}\text{Sum}\text{of}X=10.\\ \text{Sum}\text{}\text{of}Y=7515.\\ \text{Mean}X=2.\\ \text{Mean}Y=\mathrm{1503.4.}\\ \text{Sum}\text{of}\text{squares}(S{S}_X)=10.\\ \text{Sum}\text{of}\text{products}(SP)=2504.\\ \text{Regression}\text{equation}y=aX+b.\\ a=SP/S{S}_X=2504/10=\mathrm{250.4.}\\ b=M_Y-a{M}_X=1503.4-(250.4\times 2)=1002.6\\ y=250.4x+1002.6\mathrm{.........}(1).\end{array}$

The equation of regression line is $y=250.4x+1002.6.$ .

For calculating Cyber Monday sales in 2025, substitutingx =2025-2010=15 in equation (1),

  $\begin{array}{l}y=250.4x+1002.6.\\ x=15.\\ y=250.4\times 15+\mathrm{1002.6.}\\ y=3756+1002.6=\mathrm{4758.6.}\end{array}$

Cyber Monday sales in 2025 is 4758.6.

c

To determine

To find: The correlation −coefficient.

Answer to Problem 23PFA

Correlation −coefficient is 0.89010509790.

Explanation of Solution

Given information: The table shows U.S. online sales, in millions of dollars ,on Cyber .

Monday since 2010.

  

Let x be the number of years since 2010.

Years x	0	1	2	3	4
Sales y	1028	1251	1465	1735	2038

Calculation:

Use a graphing calculator for above table ,

Equation of regression line is,

  $\begin{array}{l}\text{Sum}\text{of}X=10.\\ \text{Sum}\text{}\text{of}Y=7515.\\ \text{Mean}X=2.\\ \text{Mean}Y=\mathrm{1503.4.}\\ \text{Sum}\text{of}\text{squares}(S{S}_X)=10.\\ \text{Sum}\text{of}\text{products}(SP)=2504.\\ \text{Regression}\text{equation}y=aX+b.\\ a=SP/S{S}_X=2504/10=\mathrm{250.4.}\\ b=M_Y-a{M}_X=1503.4-(250.4\times 2)=1002.6\\ y=250.4x+1002.6\mathrm{.........}(1).\end{array}$

The equation of regression line is $y=250.4x+1002.6.$ .

For calculating Cyber Monday sales in 2025, substitutingx =2025-2010=15 in equation (1),

  $\begin{array}{l}y=250.4x+1002.6.\\ x=15.\\ y=250.4\times 15+\mathrm{1002.6.}\\ y=3756+1002.6=\mathrm{4758.6.}\end{array}$

Cyber Monday sales in 2025 is 4758.6.

Use a graphing calculator for above table ,

The correlation −coefficient is 0.89010509790.

d

To determine

A classmate says that the correlation coefficient in not close to 0,so the equation in part a does not modal the data well .Do you agree?

Answer to Problem 23PFA

No, not agree with your class mate because correlation coefficient is 0.845 which is close to 1 not close to 0, indicates that’s the best-fit line model the data well.

Explanation of Solution

Given information: The table shows U.S. online sales, in millions of dollars ,on Cyber .

Monday since 2010.

  

Let x be the number of years since 2010.

Years x	0	1	2	3	4
Sales y	1028	1251	1465	1735	2038

Calculation:

Use a graphing calculator for above table ,

Equation of regression line is,

  $\begin{array}{l}\text{Sum}\text{of}X=10.\\ \text{Sum}\text{}\text{of}Y=7515.\\ \text{Mean}X=2.\\ \text{Mean}Y=\mathrm{1503.4.}\\ \text{Sum}\text{of}\text{squares}(S{S}_X)=10.\\ \text{Sum}\text{of}\text{products}(SP)=2504.\\ \text{Regression}\text{equation}y=aX+b.\\ a=SP/S{S}_X=2504/10=\mathrm{250.4.}\\ b=M_Y-a{M}_X=1503.4-(250.4\times 2)=1002.6\\ y=250.4x+1002.6\mathrm{.........}(1).\end{array}$

The equation of regression line is $y=250.4x+1002.6.$ .

For calculating Cyber Monday sales in 2025, substitutingx =2025-2010=15 in equation (1),

  $\begin{array}{l}y=250.4x+1002.6.\\ x=15.\\ y=250.4\times 15+\mathrm{1002.6.}\\ y=3756+1002.6=\mathrm{4758.6.}\end{array}$

Cyber Monday sales in 2025 is 4758.6.

Use a graphing calculator for above table ,

The correlation −coefficient is 0.89010509790.

The correlation coefficient is 0.845 which is close to 1 not close to 0, indicates that’s the best-fit line model the data well.