The managing partner of an advertising agency

Question 1: The managing partner of an advertising agency believes that his company's sales are related to the industry sales. He uses Microsoft Excel's Data Analysis tool to analyze the last 4 years of quarterly data (i.e., n = 60) with the following results:

Regression Statistics

Multiple R 0.946

R Square 0.895

Adjusted R Square 0.893

Standard Error SYX 0.9995

Observations 60

ANOVA

df SS MS F Sig.F

Regression 1 493.988 493.988 494.466 0.000

Error 58 57.944 0.999

Total 59 551.932

Predictor Coef StdError t Stat P-value

Intercept -7.7962 1.1549 -6.75 0.001

Industry 7.8742 0.3541 22.2 0.000

a) What is the value of the quantity that the least squares regression line minimizes? Explain your answer.

b) What is the prediction of Y for a quarter in which X = 100? Show how you obtain your answer.

c) What is the value for the coefficient of determination?

d) What does the coefficient of determination tell you?

1. Question 2: Your HR Director presents you the following data on your employees from a regression output

Dependent variable: Wage and Salary Income (INCWS)

Independent variable: a disability affecting work indicator

--------------------------------------------------------------------------------------------------

INCWS | Coefficient Std. Err. t P>|t| Beta

-------------+------------------------------------------------------------------------------------

Work Disability | -9303.723 4475.500 -2.079 0.038 -0.070

constant | 37931.813 1416.073 26.787 0.000

--------------------------------------------------------------------------------------------------

Write the regression equation for the above output.

What information does the constant give you?

What information does the slope of the coefficient give you?

Write the null and alternative hypotheses to test whether or not disabled people earn less than non-disabled people.

At the 5% significance level, do disabled people earn less than non-disabled people?

At the 1% significance level, do disabled people earn less than non-disabled people?

Question 3: Given below is the Excel output from regressing GPA on ACT scores using a data set of 75 randomly chosen students from a Big-Ten university, with GPA on a 5-point grading scale.

Regressing GPA on ACT

Regression Statistics

Multiple R

0.1027

R Square

0.0105

Adjusted R Square

-0.0030

Standard Error

9.8909

Observations

75

ANOVA

df

SS

MS

F

Significance F

Regression

1

76.124

76.12

0.778

0.3806

Residual

73

7141.58

97.83

Total

74

7217.71

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

4.0133

3.878

1.035

0.3041

-3.715

11.742

ACT

0.0943

0.107

0.882

0.3806

-0. 119

0.307

a) The interpretation of the coefficient of determination in this regression is

A) 1.05% of the total variation of ACT scores can be explained by GPA.

B) ACT scores account for 1.05% of the total fluctuation in GPA.

C) GPA accounts for 1.05% of the variability of ACT scores.

D) None of the above.

b) What is the value of the measured test statistic to test whether there is any linear relationship between GPA and ACT?

c) What is the predicted average value of GPA when ACT = 20? Show how you obtain your answer.

d) What are the decision and conclusion on testing whether there is any linear relationship at 5% level of significance between GPA and ACT scores? Explain your answer.

Question 4: Analysts at a major automobile company collected data on a variety of variables for a sample of 30 different cars and small trucks. Included among those data were the EPA highway mileage rating and the horsepower of each vehicle. The analysts were interested in the relationship between horsepower (x) and highway mileage (y). Given below is the Excel output from regressing starting miles per gallon (MPG) on number of horsepower for a sample of 30 students.

Note: Some of the numbers in the output are purposely erased.

Regression Statistics

Multiple R

0.5493

R Square

0.3016

Adjusted R Square

0.2766

Standard Error

3.5532

Observations

30

ANOVA

df

SS

MS

F

Significance F

Regression

1

152.655

12.091

12.091

Residual

12.625

Total

30

506.167

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

31.1658

1.9332

16.12

0.0000

27.2058

35.1258

Horsepower

-0.0286

0.0082

-3.4772

0.0017

-0.0454

-0.0117

a) What is the estimated average change in MPG as a result of an extra unit of horsepower?

b) What is the value of the measured t-test statistic and its p-value to test whether average MPG depends linearly on Horsepower?

c) What is the error sum of squares (SSE) of the above regression? Show how you obtain your answer.

f) The 99% confidence interval for the average change in MPG as a result of increased horsepower is:

A) wider than [-0.0454, -0.0117].

B) narrower than [-0.0454, -0.0117].

C) wider than [27.2058, 35.1258].

D) narrower than [27.2058, 35.1258].

Explain your reasoning.

Question 5: A shipping company believes that the variation in the cost of a customer’s shipment can be explained by differences in the weight of the package being shipped. To investigate whether this relationship is useful, a random sample of 20 customer shipments was selected, and the weight (in lb.) and the cost (in dollars, rounded) for each shipment were recorded. The following results were obtained:

Weight (lbs.)

Cost (Dollars)

8

11

6

8

5

11

7

11

12

17

9

11

17

27

13

16

8

9

18

25

17

21

17

24

10

16

20

24

9

21

5

10

13

21

6

16

6

11

12

20

a) Construct a scatter plot for these data. What, if any, relationship appears to exist between the two variables?

b) Compute the linear regression model based on the sample data. Interpret the slope and regression coefficients.

c) Test the significance of the overall regression model using a significance level of 0.05.

d) What percentage of the total variation in shipping cost can be explained by the regression model you developed in part b?