Regression Analysis Question with Solution Sample Assignment

PROBLEM

1. Assume you have noted the following prices for books and the number of pages that each book contains.

Book

Pages (x)

Price (y)

A

500

$7.00

B

700

7.50

C

750

9.00

D

590

6.50

E

540

7.50

F

650

7.00

G

480

4.50

a.

Develop a least-squares estimated regression line.

b.

Compute the coefficient of determination and explain its meaning.

c.

Compute the correlation coefficient between the price and the number of pages. Test to see if x and y are related. Use a = 0.10.

ANS: 

a.

y= 1.0416 + 0.0099x

b.

r 2 = .5629; the regression equation has accounted for 56.29% of the total sum of squares

c.

rxy = 0.75

t = 2.54 > 2.015 (df = 5); p-value is between .05 and 0.1; (Excel’s results: p-value of 0.052); reject Ho, and conclude x and y are related

2. Assume you have noted the following prices for books and the number of pages that each book contains.

Book

Pages (x)

Price (y)

A

500

$7.00

B

700

7.50

C

750

9.00

D

590

6.50

E

540

7.50

F

650

7.00

G

480

4.50

a.

Perform an F test and determine if the price and the number of pages of the books are related. Let a = 0.01.

b.

Perform a t test and determine if the price and the number of pages of the books are related. Let a = 0.01.

c.

Develop a 90% confidence interval for estimating the average price of books that contain 800 pages.

d.

Develop a 90% confidence interval to estimate the price of a specific book that has 800 pages.

ANS: 

a.

F = 6.439 < 16.26; p-value is between 0.1 and 0.2

(Excel’s result: p-value = .052); do not reject Ho; conclude x and y are not related

b.

t = 2.5376 < 4.032; p-value is between 0.1 and 0.2.

(Excel’s result: p-value = .052); do not reject Ho; conclude x and y are not related

c.

$7.29 to $10.63 (rounded)

d.

$5.62 to $12.31 (rounded)

3. The following data represent the number of flash drives sold per day at a local computer shop and their prices.

Price (x)

Units Sold (y)

$34

3

36

4

32

6

35

5

30

9

38

2

40

1

a.

Develop a least-squares regression line and explain what the slope of the line indicates.

b.

Compute the coefficient of determination and comment on the strength of relationship between x and y.

c.

Compute the sample correlation coefficient between the price and the number of flash drives sold. Use a= 0.01 to test the relationship between x and y.

ANS: 

a.

y= 29.7857 - 0.7286x

The slope indicates that as the price goes up by $1, the number of units sold goes down by 0.7286 units.

b.

r 2 = .8556; the regression equation has accounted for 85.56% of the total sum of squares

c.

rxy = -0.92

t = -5.44 < -4.032 (df = 5); p-value < .01; (Excel’s result: p-value = .0028); reject Ho, and conclude x and y are related

4. The following data represent the number of flash drives sold per day at a local computer shop and their prices.

Price (x)

Units Sold (y)

$34

3

36

4

32

6

35

5

30

9

38

2

40

1

a.

Perform an F test and determine if the price and the number of flash drives sold are related. Let a = 0.01.

b.

Perform a t test and determine if the price and the number of flash drives sold are related. Let a = 0.01.

ANS: 

a.

F = 29.624 > 16.26; p-value < .01; (Excel’s result: p-value = .0028); reject Ho, x and y are related

b.

t = -5.4428 < -4.032; p-value < .01; (Excel’s result: p-value = .0028); reject Ho, x and y are related

5. Shown below is a portion of an Excel output for regression analysis relating Y (dependent variable) and X (independent variable).

ANOVA

df

SS

Regression

1

110

Residual

8

74

Total

9

184

Coefficients

Standard Error

Intercept

39.222  

5.943  

x

 -0.5556

0.1611

a.

What has been the sample size for the above?

b.

Perform a t test and determine whether or not X and Y are related. Let a = 0.05.

c.

Perform an F test and determine whether or not X and Y are related. Let a = 0.05.

d.

Compute the coefficient of determination.

e.

Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific.

ANS: 

a through d

Summary Output

Regression Statistics

Multiple R

0.7732

R Square

0.5978

Adjusted R Square

0.5476

Standard Error

3.0414

Observations

10

ANOVA

df

SS

MS

F

Significance F

Regression

1

110

110     

11.892

0.009

Residual

8

  74

    9.25

Total

9

184

Coefficients

Standard Error

t Stat

P-value

Intercept

39.222

5.942

 6.600

0.000

x

 -0.556

0.161

-3.448

0.009

e.

59.783% of the variability in Y is explained by the variability in X.

6. Shown below is a portion of a computer output for regression analysis relating Y (dependent variable) and X (independent variable).

ANOVA

df

SS

Regression

1

24.011

Residual

8

67.989

Coefficients

Standard Error

Intercept

11.065

2.043

x

-0.511

0.304

a.

What has been the sample size for the above?

b.

Perform a t test and determine whether or not X and Y are related. Let a = 0.05.

c.

Perform an F test and determine whether or not X and Y are related. Let a = 0.05.

d.

Compute the coefficient of determination.

e.

Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific.

ANS: 

a through d

Summary Output

Regression Statistics

Multiple R

0.511

R Square

0.261

Adjusted R Square

0.169

Standard Error

2.915

Observations

10

ANOVA

df

SS

MS

F

Significance F

Regression

1

24.011

24.011

2.825

0.131

Residual

8

67.989

  8.499

Total

9

92       

Coefficients

Standard Error

t Stat

P-value

Intercept

11.065

2.043

 5.415

0.001

x

 -0.511

0.304

-1.681

0.131

e.

26.1% of the variability in Y is explained by the variability in X.

7. Part of an Excel output relating X (independent variable) and Y (dependent variable) is shown below. Fill in all the blanks marked with "?".

Summary Output

Regression Statistics

Multiple R

0.1347

R Square

?

Adjusted R Square

?

Standard Error

3.3838

Observations

?

ANOVA

df

SS

MS

F

Significance F

Regression

?

2.7500

?

?

0.632

Residual

?

?

11.45

Total

14

?

Coefficients

Standard Error

t Stat

P-value

Intercept

8.6  

2.2197

?

0.0019

x

0.25

0.5101

?

0.632  

ANS: 

Summary Output

Regression Statistics

Multiple R

0.1347

R Square

0.0181

Adjusted R Square

-0.0574

Standard Error

3.384

Observations

15

ANOVA

 

df

SS

MS

F

Significance F

Regression

  1

    2.750

  2.75

0.2402

0.6322

Residual

13

148.850

11.45

Total

14

151.600

Coefficients

Standard Error

t Stat

p-value

Intercept

8.6  

2.2197

3.8744

0.0019

x

0.25

0.5101

0.4901

0.6322

8. Shown below is a portion of a computer output for a regression analysis relating Y (dependent variable) and X (independent variable).

ANOVA

df

SS

Regression

  1

115.064

Residual

13

  82.936

Total

Coefficients

Standard Error

Intercept

15.532

1.457

x

 -1.106

0.261

a.

Perform a t test using the p-value approach and determine whether or not Y and X are related. Let a = 0.05.

b.

Using the p-value approach, perform an F test and determine whether or not X and Y are related.

c.

Compute the coefficient of determination and fully interpret its meaning. Be very specific.

ANS: 

a and b

Summary Output

Regression Statistics

Multiple R

0.7623

R Square

0.5811

Adjusted R Square

0.5489

Standard Error

2.5258

Observations

15

ANOVA

df

SS

MS

F

Significance F

Regression

  1

115.064

115.064

18.036

0.001

Residual

13

  82.936

    6.380

Total

14

198       

Coefficients

Standard Error

t Stat

P-value

Intercept

15.532

1.457

10.662

0.000

x

 -1.106

0.261

 -4.247

0.001

c.

58.11% of the variability in Y is explained by the variability in X.

9. Part of an Excel output relating X (independent variable) and Y (dependent variable) is shown below. Fill in all the blanks marked with "?".

Summary Output

Regression Statistics

Multiple R

?

R Square

0.5149

Adjusted R Square

?

Standard Error

7.3413

Observations

11

ANOVA

df

SS

MS

F

Significance F

Regression

?

?

?

?

0.0129

Residual

?

?

?

Total

?

1000

Coefficients

Standard Error

t Stat

P-value

Intercept

?

29.4818

 3.7946

0.0043

x

?

  0.7000

-3.0911

0.0129

ANS: 

Summary Output

Regression Statistics

Multiple R

0.7176

R Square

0.5149

Adjusted R Square

0.4611

Standard Error

7.3413

Observations

11

ANOVA

df

SS

MS

F

Significance F

Regression

  1

  514.9455

514.9455

9.5546

0.0129

Residual

  9

  485.0545

  53.8949

Total

10

1000.0000

Coefficients

Standard Error

t Stat

P-value

Intercept

111.8727

29.4818

 3.7946

0.0043

x

   -2.1636

  0.7000

-3.0911

0.0129

10. Shown below is a portion of a computer output for a regression analysis relating Y (demand) and X (unit price).

ANOVA

df

SS

Regression

  1

5048.818

Residual

46

3132.661

Total

47

8181.479

Coefficients

Standard Error

Intercept

80.390

3.102

X

 -2.137

0.248

a.

Perform a t test and determine whether or not demand and unit price are related. Let a = 0.05.

b.

Perform an F test and determine whether or not demand and unit price are related. Let a = 0.05.

c.

Compute the coefficient of determination and fully interpret its meaning. Be very specific.

d.

Compute the coefficient of correlation and explain the relationship between demand and unit price.

ANS: 

a and b

Summary Output

Regression Statistics

Multiple R

0.786

R Square

0.617

Adjusted R Square

0.609

Standard Error

8.252

Observations

48

ANOVA

df

SS

MS

F

Significance F

Regression

  1

5048.818

5048.818

74.137

0.000

Residual

46

3132.661

    68.101

Total

47

8181.479

Coefficients

Standard Error

t Stat

P-value

Intercept

80.390

3.102

25.916

0.000

X

 -2.137

0.248

 -8.610

0.000

c.

R2 = 0.617; 61.7% of the variability in demand is explained by the variability in price.

d.

R = -0.786; since the slope is negative, the coefficient of correlation is also negative, indicating that as unit price increases demand decreases.

11. Shown below is a portion of a computer output for a regression analysis relating supply (Y in thousands of units) and unit price (X in thousands of dollars).

ANOVA

df

SS

Regression

  1

  354.689

Residual

39

7035.262

Coefficients

Standard Error

Intercept

54.076

2.358

X

  0.029

0.021

a.

What has been the sample size for this problem?

b.

Perform a t test and determine whether or not supply and unit price are related. Let a = 0.05.

c.

Perform and F test and determine whether or not supply and unit price are related. Let a = 0.05.

d.

Compute the coefficient of determination and fully interpret its meaning. Be very specific.

e.

Compute the coefficient of correlation and explain the relationship between supply and unit price.

f.

Predict the supply (in units) when the unit price is $50,000.

ANS: 

a through c

Regression Statistics

Multiple R

0.219

R Square

0.048

Adjusted R Square

0.024

Standard Error

13.431

Observations

41

ANOVA

df

SS

MS

F

Significance F

Regression

  1

  354.689

354.689

1.966

0.169

Residual

39

7035.262

180.391

Total

40

7389.951

Coefficients

Standard Error

t Stat

P-value

Intercept

54.076

2.358

22.938

0.000

X

  0.029

0.021

  1.402

0.169

d.

R2 = 0.048; 4.8% of the variability in supply is explained by the variability in price.

e.

R = 0.219; since the slope is positive, as unit price increases so does supply.

f.

supply = 54.076 + .029(50) = 55.526 (55,526 units)

12. Given below are four observations collected in a regression study on two variables x (independent variable) and y (dependent variable).

x

y

2

4

6

7

9

8

9

9

a.

Develop the least squares estimated regression equation.

b.

At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero.

c.

Perform an F test to determine whether or not the model is significant. Let a = 0.05.

d.

Compute the coefficient of determination.

ANS: 

Regression Statistics

Multiple R

0.977

R Square

0.955

Adjusted R Square

0.932

Standard Error

0.564

Observations

4

ANOVA

df

SS

MS

F

Significance F

Regression

1

13.364

13.364

42.000

0.023

Residual

2

  0.636

  0.318

Total

3

14       

Coefficients

Standard Error

t Stat

P-value

Intercept

2.864

0.698

4.104

0.055

X

0.636

0.098

6.481

0.023

a.

 = 2.864 + 0.636x

b.

p-value < .05; reject Ho

c.

p-value < .05; reject Ho

d.

0.955

13. Given below are five observations collected in a regression study on two variables, x (independent variable) and y (dependent variable).

x

y

2

4

3

4

4

3

5

2

6

1

a.

Develop the least squares estimated regression equation.

b.

At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero.

c.

Perform an F test to determine whether or not the model is significant. Let a = 0.05.

d.

Compute the coefficient of determination.

e.

Compute the coefficient of correlation.

ANS: 

Regression Statistics

Multiple R

0.970

R Square

0.941

Adjusted R Square

0.922

Standard Error

0.365

Observations

5

ANOVA

df

SS

MS

F

Significance F

Regression

1

6.4

6.400

48.000

0.006

Residual

3

0.4

0.133

Total

4

6.8

Coefficients

Standard Error

t Stat

P-value

Intercept

 6.000

0.490

12.247

0.001

X

-0.800

0.115

 -6.928

0.006

a.

 = 6 - 0.8 x

b.

p-value < .05; reject Ho

c.

p-value < .05; reject Ho

d.

0.941

e.

-0.970

14. Below you are given a partial computer output based on a sample of 8 observations, relating an independent variable (x) and a dependent variable (y).

Coefficient

Standard Error

Intercept

13.251

10.77

X

  0.803

  0.385

Analysis of Variance

 

SOURCE

SS

Regression

Error (Residual)

41.674

Total

71.875

a.

Develop the estimated regression line.

b.

At a = 0.05, test for the significance of the slope.

c.

At a = 0.05, perform an F test.

d.

Determine the coefficient of determination.

ANS: 

a.

 = 13.251 + 0.803x

b.

t = 2.086; p-value is between .05 and .1 (critical t = 2.447); do not reject Ho

c.

F = 4.348; p-value is between .05 and .1 (critical F = 5.99); do not reject Ho

d.

0.42

15. Below you are given a partial computer output based on a sample of 8 observations, relating an independent variable (x) and a dependent variable (y).

Coefficient

Standard Error

Intercept

-9.462

7.032

x

 0.769

0.184

Analysis of Variance

 

SOURCE

SS

Regression

400

Error (Residual)

138

a.

Develop the estimated regression line.

b.

At a = 0.05, test for the significance of the slope.

c.

At a = 0.05, perform an F test.

d.

Determine the coefficient of determination.

ANS: 

a.

y= -9.462 + 0.769x

b.

t = 4.17; p-value (actual p-value using Excel = 0.0059) < .05; reject Ho

c.

F = 17.39; p-value (actual p-value using Excel = 0.0059) < .05; reject Ho

d.

0.743

16. The following data represent a company's yearly sales volume and its advertising expenditure over a period of 8 years.

(Y)

Sales in Millions of Dollars

(X)

Advertising in ($10,000)

15

32

16

33

18

35

17

34

16

36

19

37

19

39

24

42

a.

Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising.

b.

Use the method of least squares to compute an estimated regression line between sales and advertising.

c.

If the company's advertising expenditure is $400,000, what are the predicted sales? Give the answer in dollars.

d.

What does the slope of the estimated regression line indicate?

e.

Compute the coefficient of determination and fully interpret its meaning.

f.

Use the F test to determine whether or not the regression model is significant at a = 0.05.

g.

Use the t test to determine whether the slope of the regression model is significant at a = 0.05.

h.

Develop a 95% confidence interval for predicting the average sales for the years when $400,000 was spent on advertising.

i.

Compute the correlation coefficient.

ANS: 

a.

The scatter diagram shows a positive relation between sales and advertising.

b.

 = -10.42 + 0.7895X

c.

$21,160,000

d.

As advertising is increased by $10,000, sales are expected to increase by $789,500.

e.

0.8459; 84.59% of variation in sales is explained by variation in advertising

f.

F = 32.93; p-value (actual p-value using Excel = 0.0012) < .05; reject Ho; it is significant (critical F = 5.99)

g.

t = 5.74; p-value (actual p-value using Excel = 0.0012) < .05; reject Ho; significant (critical t = 2.447)

h.

$19,460,000 to $22,860,000

i.

0.9197

17. Given below are five observations collected in a regression study on two variables x (independent variable) and y (dependent variable).

x

y

10

7

20

5

30

4

40

2

50

1

a.

Develop the least squares estimated regression equation

b.

At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero.

c.

Perform an F test to determine whether or not the model is significant. Let a = 0.05.

d.

Compute the coefficient of determination.

e.

Compute the coefficient of correlation.

ANS: 

a.

y= 8.3 - 0.15x

b.

t = -15; p-value (actual p-value using Excel = 0.0001) < .05; reject Ho (critical t = 3.18)

c.

F = 225; p-value (actual p-value using Excel = 0.0001) < .05; reject Ho (critical F = 10.13)

d.

0.9868

e.

0.9934

18. Below you are given a partial computer output based on a sample of 14 observations, relating an independent variable (x) and a dependent variable (y).

Predictor

Coefficient

Standard Error

Constant

6.428

1.202

X

0.470

0.035

Analysis of Variance

 

SOURCE

SS

Regression

  958.584

Error (Residual)

Total

1021.429

a.

Develop the estimated regression line.

b.

At a = 0.05, test for the significance of the slope.

c.

At a = 0.05, perform an F test.

d.

Determine the coefficient of determination.

e.

Determine the coefficient of correlation.

ANS: 

a.

 = 6.428 + 0.47x

b.

t = 13.529; p-value (actual p-value using Excel = 0.0000) < .05; reject Ho (critical t = 2.179)

c.

F = 183.04; p-value (actual p-value using Excel = 0.0000) < .05; reject Ho (critical F = 4.75)

d.

0.938

e.

0.968

19. Below you are given a partial computer output based on a sample of 21 observations, relating an independent variable (x) and a dependent variable (y).

Predictor

Coefficient

Standard Error

Constant

30.139

1.181

X

 -0.252

0.022

Analysis of Variance

 

SOURCE

SS

Regression

1,759.481

Error

   259.186

a.

Develop the estimated regression line.

b.

At a = 0.05, test for the significance of the slope.

c.

At a = 0.05, perform an F test.

d.

Determine the coefficient of determination.

e.

Determine the coefficient of correlation.

ANS: 

a.

y= 30.139 - 0.252X

b.

t = -11.357; p-value (almost zero) < a = .05; reject Ho (critical t = 2.093)

c.

F = 128.982; p-value (almost zero) < a = .05; reject Ho (critical F = 4.38)

d.

0.872

e.

-0.934

20. An automobile dealer wants to see if there is a relationship between monthly sales and the interest rate. A random sample of 4 months was taken. The results of the sample are presented below. The estimated least squares regression equation is

y= 75.061 - 6.254X

Y

X

Monthly Sales

Interest Rate (In Percent)

22

9.2

20

7.6

10

10.4

45

5.3

a.

Obtain a measure of how well the estimated regression line fits the data.

b.

You want to test to see if there is a significant relationship between the interest rate and monthly sales at the 1% level of significance. State the null and alternative hypotheses.

c.

At 99% confidence, test the hypotheses.

d.

Construct a 99% confidence interval for the average monthly sales for all months with a 10% interest rate.

e.

Construct a 99% confidence interval for the monthly sales of one month with a 10% interest rate.

ANS: 

a.

R2 = 0.8687

b.

H0: b1 = 0

Ha: b1  0

c.

test statistic t = -3.64; p-value is between .05 and .10 (critical t = 9.925); do not reject H0

d.

-33.151 to 58.199; therefore, 0 to 58.199

e.

-67.068 to 92.116; therefore, 0 to 92.116

21. Jason believes that the sales of coffee at his coffee shop depend upon the weather. He has taken a sample of 6 days. Below you are given the results of the sample.

Cups of Coffee Sold

Temperature

350

  50

200

  60

210

  70

100

  80

  60

  90

  40

100

a.

Which variable is the dependent variable?

b.

Compute the least squares estimated line.

c.

Compute the correlation coefficient between temperature and the sales of coffee.

d.

Is there a significant relationship between the sales of coffee and temperature? Use a .05 level of significance. Be sure to state the null and alternative hypotheses.

e.

Predict sales of a 90 degree day.

ANS: 

a.

Cups of coffee sold

b.

y= 605.714 - 5.943X

c.

0.95197

d.

H0: b1 = 0

Ha: b1  0

t = -6.218; p-value (actual p-value using Excel = 0.0034) < a = .05; reject Ho (critical t = 2.776)

e.

70.8 or 71 cups

22. Researchers have collected data on the hours of television watched in a day and the age of a person. You are given the data below.

Hours of Television

Age

1

45

3

30

4

22

3

25

6

  5

a.

Determine which variable is the dependent variable.

b.

Compute the least squares estimated line.

c.

Is there a significant relationship between the two variables? Use a .05 level of significance. Be sure to state the null and alternative hypotheses.

d.

Compute the coefficient of determination. How would you interpret this value?

ANS: 

a.

Hours of Television

b.

 = 6.564 - 0.1246X

c.

H0: b1 = 0

Ha: b1  0

t = -12.018; p-value (actual p-value using Excel = 0.0002) < a = .05; reject H0 (critical t = 3.18)

d.

0.98 (rounded); 98 % of variation in hours of watching television is explained by variation in age.

23. Given below are seven observations collected in a regression study on two variables, X (independent variable) and Y (dependent variable).

X

Y

2

12

3

  9

6

  8

7

  7

8

  6

7

  5

9

  2

a.

Develop the least squares estimated regression equation.

b.

At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero.

c.

Perform an F test to determine whether or not the model is significant. Let a = 0.05.

d.

Compute the coefficient of determination.

ANS: 

a.

 = 13.75 -1.125X

b.

t = -5.196; p-value (actual p-value using Excel = 0.0001) < a = .05; reject Ho (critical t = 2.571)

c.

F = 27; p-value (actual p-value using Excel = 0.0001) < a = .05; reject Ho (critical F = 6.61)

d.

0.844

24. The owner of a retail store randomly selected the following weekly data on profits and advertising cost.

Week

Advertising Cost ($)

Profit ($)

1

    0

200

2

50

270

3

250

420

4

150

300

5

125

325

a.

Write down the appropriate linear relationship between advertising cost and profits. Which is the dependent variable? Which is the independent variable?

b.

Calculate the least squares estimated regression line.

c.

Predict the profits for a week when $200 is spent on advertising.

d.

At 95% confidence, test to determine if the relationship between advertising costs and profits is statistically significant.

e.

Calculate the coefficient of determination.

ANS: 

a.

E(Y) = b0 + b1X, where Y is profit and X is advertising cost

b.

 = 210.0676 + 0.80811X

c.

$371.69

d.

t = 6.496; p-value (actual p-value using Excel = 0.0013) < a = .05; reject Ho; relationship is significant (critical t = 3.182)

e.

0.9336

25. The owner of a bakery wants to analyze the relationship between the expenditure of a customer and the customer's income. A sample of 5 customers is taken and the following information was obtained.

Y

X

Expenditure

Income (In Thousands)

    .45

20

10.75

19

  5.40

22

  7.80

25

  5.60

14

The least squares estimated line is  = 4.348 + 0.0826 X.

a.

Obtain a measure of how well the estimated regression line fits the data.

b.

You want to test to see if there is a significant relationship between expenditure and income at the 5% level of significance. Be sure to state the null and alternative hypotheses.

c.

Construct a 95% confidence interval estimate for the average expenditure for all customers with an income of $20,000.

d.

Construct a 95% confidence interval estimate for the expenditure of one customer whose income is $20,000.

ANS: 

a.

R2 = 0.0079

b.

H0: b1 = 0

Ha: b1  0

t = 0.154; p-value (actual p-value using Excel = 0.8871) > a = .05; do not reject H0; (critical t = 3.182)

c.

0.185 to 12.185

d.

-9.151 to 21.151

26. Below you are given information on annual income and years of college education.

Income (In Thousands)

Years of College

28

0

40

3

36

2

28

1

48

4

a.

Develop the least squares regression equation.

b.

Estimate the yearly income of an individual with 6 years of college education.

c.

Compute the coefficient of determination.

d.

Use a t test to determine whether the slope is significantly different from zero. Let a = 0.05.

e.

At 95% confidence, perform an F test and determine whether or not the model is significant.

ANS: 

a.

y= 25.6 + 5.2X

b.

$56,800

c.

0.939

d.

t = 6.789; p-value (actual p-value using Excel = 0.0008) < a = .05; reject Ho; significant (critical t = 3.182

e.

F = 46.091; p-value (actual p-value using Excel = 0.0008) < a = .05; reject Ho; significant (critical F = 10.13)

27. Below you are given information on a woman's age and her annual expenditure on purchase of books.

Age

Annual Expenditure ($)

18

210

22

180

21

220

28

280

a.

Develop the least squares regression equation.

b.

Compute the coefficient of determination.

c.

Use a t test to determine whether the slope is significantly different from zero. Let a = 0.05.

d.

At 95% confidence, perform an F test and determine whether or not the model is significant.

ANS: 

a.

y= 54.834 + 7.536X

b.

R2 = 0.568

c.

t = 1.621; p-value (actual p-value using Excel = 0.2464) > a = .05; do not reject Ho; not significant (critical t = 4.303)

d.

F = 2.628; p-value (actual p-value using Excel = 0.2464) > a = .05; do not reject Ho; not significant (critical F = 18.51)

28. The following sample data contains the number of years of college and the current annual salary for a random sample of heavy equipment salespeople.

Years of College

Annual Income (In Thousands)

2

20

2

23

3

25

4

26

3

28

1

29

4

27

3

30

4

33

4

35

a.

Which variable is the dependent variable? Which is the independent variable?

b.

Determine the least squares estimated regression line.

c.

Predict the annual income of a salesperson with one year of college.

d.

Test if the relationship between years of college and income is statistically significant at the .05 level of significance.

e.

Calculate the coefficient of determination.

f.

Calculate the sample correlation coefficient between income and years of college. Interpret the value you obtain.

ANS: 

a.

Y (dependent variable) is annual income and X (independent variable) is years of college

b.

 = 21.6 + 2X

c.

$23,600

d.

The relationship is not statistically significant since t = 1.51; p-value (actual p-value using Excel = 0.1696) > a = .05 (critical t = 2.306)

e.

0.222

f.

0.471; there is a positive correlation between years of college and annual income

29. The following data shows the yearly income (in $1,000) and age of a sample of seven individuals.

Income (in $1,000)

Age

20

18

24

20

24

23

25

34

26

24

27

27

34

27

a.

Develop the least squares regression equation.

b.

Estimate the yearly income of a 30-year-old individual.

c.

Compute the coefficient of determination.

d.

Use a t test to determine whether the slope is significantly different from zero. Let a = 0.05.

e.

At 95% confidence, perform an F test and determine whether or not the model is significant.

ANS: 

a.

y= 16.204 + 0.3848X

b.

$27,748

c.

0.2266

d.

t = 1.21; p-value (actual p-value using Excel = 0.2803) > a = .05; not significant (critical t = 2.571)

e.

F = 1.46; p-value (actual p-value using Excel = 0.2803) > a = .05; not significant (critical F = 6.61)

30. The following data show the results of an aptitude test (Y) and the grade point average of 10 students.

Aptitude Test Score (Y)

GPA (X)

26

1.8

31

2.3

28

2.6

30

2.4

34

2.8

38

3.0

41

3.4

44

3.2

40

3.6

43

3.8

a.

Develop a least squares estimated regression line.

b.

Compute the coefficient of determination and comment on the strength of the regression relationship.

c.

Is the slope significant? Use a t test and let a = 0.05.

d.

At 95% confidence, test to determine if the model is significant (i.e., perform an F test).

ANS: 

a.

 = 8.171 + 9.4564X

b.

0.83; there is a fairly strong relationship

c.

t = 6.25; p-value (actual p-value using Excel = 0.0002) < a =.05; it is significant (critical t = 2.306)

d.

F = 39.07; p-value (actual p-value using Excel = 0.0002) < a =.05; it is significant (critical F = 5.32)

31. Shown below is a portion of the computer output for a regression analysis relating sales (Y in millions of dollars) and advertising expenditure (X in thousands of dollars).

Predictor

Coefficient

Standard Error

Constant

4.00

0.800

X

0.12

0.045

Analysis of Variance

 

 

SOURCE

DF

SS

Regression

  1

1,400

Error

18

3,600

a.

What has been the sample size for the above?

b.

Perform a t test and determine whether or not advertising and sales are related. Let a = 0.05.

c.

Compute the coefficient of determination.

d.

Interpret the meaning of the value of the coefficient of determination that you found in Part c. Be very specific.

e.

Use the estimated regression equation and predict sales for an advertising expenditure of $4,000. Give your answer in dollars.

ANS: 

a.

20

b.

t = 2.66; p-value is between 0.01 and 0.02; they are related (critical t = 2.101)

c.

R2 = 0.28

d.

28% of variation in sales is explained by variation in advertising expenditure.

e.

$4,480,000

32. A company has recorded data on the daily demand for its product (Y in thousands of units) and the unit price (X in hundreds of dollars). A sample of 15 days demand and associated prices resulted in the following data.

SX = 75

S (Y- )(X- ) = -59

SY = 135

S (X- )2 = 94

S (Y- )2 = 100

SSE = 62.9681

a.

Using the above information, develop the least-squares estimated regression line and write the equation.

b.

Compute the coefficient of determination.

c.

Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let a = 0.05.

d.

Would the demand ever reach zero? If yes, at what price would the demand be zero?

ANS: 

a.

y= 12.138 - 0.6277X

b.

R2 = 0.3703

c.

F = 7.65; p-value is between .01 and .025; reject Ho and conclude that demand and unit price are related (critical F = 4.67)

d.

Yes, at $1,934

33. A regression and correlation analysis resulted in the following information regarding an independent variable (x) and a dependent variable (y).

SX = 42

S (Y - )(X - ) = 37

SY = 63

S (X - )2 = 84

n = 7

S (Y - )2 = 28

a.

Develop the least squares estimated regression equation.

b.

At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero.

c.

Perform an F test to determine whether or not the model is significant. Let a = 0.05.

d.

Compute the coefficient of determination.

ANS: 

a.

 = 6.3571 + 0.4405x

b.

p-value < .05; reject Ho

c.

p-value < .05; reject Ho

d.

0.582