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The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year.

Car

Age (years)

Selling Price ($000)

Car

Age (years)

Selling Price ($000)

1

9

8.1

7

8

7.6

2

7

6.0

8

11

8.0

3

11

3.6

9

10

8.0

4

12

4.0

10

12

6.0

5

8

5.0

11

6

8.6

6

7

10.0

12

6

8.0

a.

If we want to estimate selling price on the basis of the age of the car, which variable is the dependent variable and which is the independent variable?

   is the independent variable and  is the dependent variable.

b-1.

Determine the correlation coefficient. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

X

Y

( )2

( )2

( )( )

9.0  

8.1  

1.192  

0.007  

1.420  

0.099  

7.0  

6.0  

-0.908  

3.674  

0.825  

1.741  

11.0  

3.6  

2.083  

4.340  

10.945  

-6.892  

12.0  

4.0  

3.083  

9.507  

8.458  

-8.967  

8.0  

5.0  

-0.917  

-1.908  

3.642  

1.749  

7.0  

10.0  

-1.917  

3.092  

9.558  

-5.926  

8.0  

7.6  

-0.917  

0.692  

0.840  

-0.634  

11.0  

8.0  

2.083  

1.092  

4.340  

2.274  

10.0  

8.0  

1.083  

1.092  

1.174  

1.192  

12.0  

6.0  

3.083  

-0.908  

9.507  

0.825  

6.0  

8.6  

-2.917  

1.692  

8.507  

2.862  

-4.934  

6.0  

8.0  

-2.917  

1.092  

8.507  

1.192  

-3.184  

107.000  

82.900  

=

=

sx

=

sy

=

r

=

b-2.

Determine the coefficient of determination. (Round your answer to 3 decimal places.)

c.

Interpret the correlation coefficient. Does it surprise you that the correlation coefficient is negative? (Round your answer to nearest whole number.)

   correlation between age of car and selling price. So,  % of the variation in the selling price is explained by the variation in the age of the car.

The Student Government Association at Middle Carolina University wanted to demonstrate the relationship between the number of beers a student drinks and his or her blood alcohol content (BAC). A random sample of 18 students participated in a study in which each participating student was randomly assigned a number of 12-ounce cans of beer to drink. Thirty minutes after they consumed their assigned number of beers, a member of the local sheriff’s office measured their blood alcohol content. The sample information is reported below.

Student

Beers

BAC

Student

Beers

BAC

1

6

0.10

10

3

0.07

2

7

0.09

11

3

0.05

3

7

0.09

12

7

0.08

4

4

0.10

13

1

0.04

5

5

0.10

14

4

0.07

6

3

0.07

15

2

0.06

7

3

0.10

16

7

0.12

8

6

0.12

17

2

0.05

9

6

0.09

18

1

0.02

Use a statistical software package to answer the following questions.

Click here for the Excel Data File

a-1.

Choose a scatter diagram that best fits the data.

b.

Fill in the blanks below. (Round your answers to 3 decimal places.)

  sx

  sy

c.

Determine the coefficient of correlation and coefficient of determination. (Round your answers to 3 decimal places.)

  Coefficient of correlation

  Coefficient of determination

c-1.

 State the decision rule for .01 significance level: H0: ρ ≤ 0; H1: ρ > 0. (Round your answer to 3 decimal places.)

  Reject H0 if t >

c-2.

 Compute the value of the test statistic. (Round your answer to 2 decimal places.)

  Value of the test statistic

c-3.

 What is the p-value? (Hint: use Megastat) (Round p-value to 4 decimal places.)

  p-value

c-4.

 At the .01 significance level, is it reasonable to conclude that there is a positive relationship in the population between the number of beers consumed and the BAC?

   H0 . There is between beers consumed and BAC.

The following sample observations were randomly selected.

Click here for the Excel Data File

X:

 5

 3

6

 3

 4

 4

6

8

Y:

13

15

7

12

13

11

9

5

a.

Determine the regression equation. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

X

Y

( )2

( )2

( )( )

5  

13   

2.375  

5.641  

3  

15   

−1.875  

3.516  

−8.203    

6  

7   

13.141  

−4.078    

3  

12   

−1.875  

1.375  

4  

13   

−0.875  

0.766  

−2.078    

4  

11   

0.375  

0.141  

6  

9   

1.125  

−1.625  

8  

5   

31.641  

−17.578    

=

=

  sx

=

sy

=

r

=

b

=

a

=

  Y' = + X

b.

Determine the value of when X is 7. (Round your answer to 3 decimal places.)

The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year.

Car

Age (years)

Selling Price ($000)

Car

Age (years)

Selling Price ($000)

1

9

8.1

7

8

7.6

2

7

6.0

8

11

8.0

3

11

3.6

9

10

8.0

4

12

4.0

10

12

6.0

5

8

5.0

11

6

8.6

6

7

10.0

12

6

8.0

Click here for the Excel Data File

The regression equation is , the sample size is 12, and the standard error of the slope is 0.23. Use the .05 significance level. Can we conclude that the slope of the regression line is less than zero?

   H0 and conclude the slope is zero.

The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year.

Car

Age (years)

Selling Price ($000)

1  

9  

8.1  

2  

7  

6.0  

3  

11  

3.6  

4  

12  

4.0  

5  

8  

5.0  

6  

7  

10.0  

7  

8  

7.6  

8  

11  

8.0  

9  

10  

8.0  

10  

12  

6.0  

11  

6  

8.6  

12  

6  

8.0  

a.

Determine the standard error of estimate. (Round your answer to 3 decimal places.)

  Standard error of estimate

b.

Determine the coefficient of determination. (Round your answer to 3 decimal places.)

c.

Interpret the coefficient of determination. (Round your answer to the nearest whole number.)

   percent of the variation in the selling price is explained by the variation in the age of the car.

Thompson Photo Works purchased several new, highly sophisticated processing machines. The production department needed some guidance with respect to qualifications needed by an operator. Is age a factor? Is the length of service as an operator (in years) important? In order to explore further the factors needed to estimate performance on the new processing machines, four variables were listed:

X1 = Length of time an employee was in the industry

X2 = Mechanical aptitude test score

X3 = Prior on-the-job rating

X4 = Age

Performance on the new machine is designated y.

     Thirty employees were selected at random. Data were collected for each, and their performances on the new machines were recorded. A few results are:

Name

Performance on New Machine, Y

Length of Time in Industry, X1

Mechanical Aptitude Score, X2

Prior On-the-Job Performance, X3

Age, X4

   Mike Miraglia

112

12

312

121

52

   Sue Trythall

113

2

380

123

27

The equation is:

= 11.6 + 0.4X1 + 0.286X2 + 0.112X3 + 0.002X4

a.

What is this equation called?

b.

How many dependent and independent variables are there?

dependent, independent

c.

What is the number 0.286 called?

d.

As age increases by one year, how much does estimated performance on the new machine increase? (Round your answer to 3 decimal places.)

e.

Carl Knox applied for a job at Photo Works. He has been in the business for 6 years and scored 280 on the mechanical aptitude test. Carl’s prior on-the-job performance rating is 97, and he is 35 years old. Estimate Carl’s performance on the new machine. (Round your answer to 3 decimal places.)

Consider the ANOVA table that follows.

  Analysis of Variance

  Source

DF

SS

MS

F

  Regression

5

3710.00

742.00

12.89

  Residual Error

46

2647.38

57.55

  Total

51

6357.38

a-1.

Determine the standard error of estimate. (Round your answer to 2 decimal places.)

  Standard error of estimate

a-2.

About 95% of the residuals will be between what two values? (Round your answers to 2 decimal places.)

  95% of the residuals will be between and .

b-1.

Determine the coefficient of multiple determination. (Round your answer to 3 decimal places.)

  Coefficient of multiple determination value is .

b-2.

Determine the percentage variation for the independent variables. (Round your answer to 1 decimal place. Omit the "%" sign in your response.)

  The independent variables explain  % of the variation.

c.

Determine the coefficient of multiple determination, adjusted for the degrees of freedom. (Round your answer to 3 decimal places.)

  Coefficient of multiple determination

The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars.

  Predictor

Coeff

SE Coeff

t

p-value

  Constant

7.987

2.967

2.690

0.010

  X1

0.122

0.031

3.920

0.000

  X2

–1.120

0.053

–2.270

0.028

  X3

–0.063

0.039

–1.610

0.114

  X4

0.523

0.142

3.690

0.001

  X5

–0.065

0.040

–1.620

0.112

  Analysis of Variance

  Source

DF

SS

MS

F

p-value

  Regression

5

3710.00

742.00

12.89

0.000

  Residual Error

46

2647.38

57.55

  Total

51

6357.38

X1 is the number of architects employed by the company.

X2 is the number of engineers employed by the company.

X3 is the number of years involved with health care projects.

X4 is the number of states in which the firm operates.

X5 is the percent of the firm’s work that is health care–related.

a.

Write out the regression equation. (Round your answers to 3 decimal places. Negative answers should be indicated by a minus sign.)

  Ŷ = + X1 + X2 + X3 + X4 + X5.

b.

How large is the sample? How many independent variables are there?

  Sample n

  Independent variables k

c-1.

State the decision rule for .05 significance level: H0: β1 = β2 = β3 =β4 =β5 =0; H1: Not all β's are 0. (Round your answer to 2 decimal places.)

  Reject H0 if F >

c-2.

Compute the value of the F statistic. (Round your answer to 2 decimal places.)

  Computed value of F is

c-3.

Can we conclude that the set of regression coefficients could be different from 0? Use the .05 significance level.

   H0. of the regression coefficients are zero.

For X1

For X2

For X3

For X4

For X5

H0: β1 = 0

H0: β2 = 0

H0: β3 = 0

H0: β4 = 0

H0: β5 = 0

H1: β1 ≠ 0

H1: β2 ≠ 0

H1: β3 ≠ 0

H1: β4 ≠ 0

H1: β5 ≠ 0

d-1.

State the decision rule for .05 significance level. (Round your answers to 3 decimal places.)

  Reject H0 if t < or t > .

d-2.

Compute the value of the test statistic. (Round your answers to 2 decimal places. Negative answers should be indicated by a minus sign.)

t − value

  X1

  X2

  X3

  X4

  X5

d-3.

Which variable would you consider eliminating?

  Consider eliminating variables .

b-1.

Compute the coefficient of correlation. (Round your answer to 3 decimal places. Negative amount should be indicated by a minus sign.)

r  =  

b-2.

Compute the coefficient of determination. (Round your answer to 3 decimal places.)

r2 =

c.

Give a description of the degree of association between the variables.

There is  association between the variables.

d.

Determine the regression equation. Use assets as the independent variable. (Round your answers to 4 decimal places. Negative amounts should be indicated by a minus sign.)

b =  

a =  

e.

For a fund with $400.0 million in sales, determine the five-year rate of return (in percent). (Round your answer to 4 decimal places.)

=  The equation should be used with caution. Assets do not account for much of the variation in the rate of return.

Mr. James McWhinney, president of Daniel-James Financial Services, believes there is a relationship between the number of client contacts and the dollar amount of sales. To document this assertion, Mr. McWhinney gathered the following sample information. The X column indicates the number of client contacts last month, and the Y column shows the value of sales ($ thousands) last month for each client sampled.

Number of Contacts, X

Sales ($ thousands), Y

Number of Contacts, X

Sales ($ thousands), Y

14  

24  

23  

30  

12  

14  

48  

90  

20  

28  

50  

85  

16  

30  

55  

120  

46  

80  

50  

110  

Click here for the Excel Data File

a.

Determine the regression equation. (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations. Round final answers to 2 decimal places.)

X

Y

( ) 2

( )2

( )( )

14  

376.36  

1376.41  

719.74  

12  

14  

−21.4  

−47.1  

20  

−13.4  

179.56  

443.54  

16  

30  

−31.1  

967.21  

46  

12.6  

357.21  

23  

−10.4  

967.21  

48  

90  

28.9  

213.16  

421.94  

50  

85  

23.9  

275.56  

396.74  

55  

466.56  

3469.21  

1,272.24  

50  

110.0  

16.6  

48.9  

=

=

  sx

=

sy

=

r

=

   b =

   a  =

  Y' = + X

b.

Determine the estimated sales if 40 contacts are made. (Do not round intermediate calculations. Round final answers to 2 decimal places.)

On the first statistics exam, the coefficient of determination between the hours studied and the grade earned was 80%. The standard error of estimate was 10. There were 20 students in the class. Develop an ANOVA table for the regression analysis of hours studied as a predictor of the grade earned on the first statistics exam.

  Source

DF

SS

MS

  Regression

  Error

  Total