# CHAPTER 15 MULTIPLE REGRESSION

**CHAPTER FIFTEEN**

# MULTIPLE REGRESSION

## MULTIPLE CHOICE QUESTIONS

In the following multiple choice questions, circle the correct answer.

- For a multiple regression model, SSR = 600 and SSE = 200. The multiple coefficient of determination is
- 0.333
- 0.275
- 0.300
- 0.75

- In a multiple regression analysis involving 15 independent variables and 200 observations, SST = 800 and SSE = 240. The coefficient of determination is
- 0.300
- 0.192
- 0.500
- 0.700

- A regression model involved 5 independent variables and 136 observations. The critical value of t for testing the significance of each of the independent variable’s coefficients will have
- 121 degrees of freedom
- 135 degrees of freedom
- 120 degrees of freedom
- 4 degrees of freedom

- In order to test for the significance of a regression model involving 3 independent variables and 47 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are
- 47 and 3
- 3 and 47
- 2 and 43
- 3 and 43

- In regression analysis, an outlier is an observation whose
- mean is larger than the standard deviation
- residual is zero
- mean is zero
- residual is much larger than the rest of the residual values

- A variable that cannot be measured in terms of how much or how many but instead is assigned values to represent categories is called
- an interaction
- a constant variable
- a category variable
- a qualitative variable

- A variable that takes on the values of 0 or 1 and is used to incorporate the effect of qualitative variables in a regression model is called
- an interaction
- a constant variable
- a dummy variable
- None of these alternatives is correct.

- In a multiple regression model, the error term e is assumed to be a random variable with a mean of
- zero
- -1
- 1
- any value

- In regression analysis, the response variable is the
- independent variable
- dependent variable
- slope of the regression function
- intercept

- The multiple coefficient of determination is
- MSR/MST
- MSR/MSE
- SSR/SST
- SSE/SSR

- A multiple regression model has the form

= 7 + 2 x_{1} + 9 x_{2}

As x_{1} increases by 1 unit (holding x_{2} constant), y is expected to

- increase by 9 units
- decrease by 9 units
- increase by 2 units
- decrease by 2 units

- A multiple regression model has
- only one independent variable
- more than one dependent variable
- more than one independent variable
- at least 2 dependent variables

- A measure of goodness of fit for the estimated regression equation is the
- multiple coefficient of determination
- mean square due to error
- mean square due to regression
- sample size

- The numerical value of the coefficient of determination
- is always larger than the coefficient of correlation
- is always smaller than the coefficient of correlation
- is negative if the coefficient of determination is negative
- can be larger or smaller than the coefficient of correlation

- The correct relationship between SST, SSR, and SSE is given by
- SSR = SST + SSE
- SSR = SST – SSE
- SSE = SSR – SST
- None of these alternatives is correct.

**Exhibit 15-1**

In a regression model involving 44 observations, the following estimated regression equation was obtained.

= 29 + 18X_{1} +43X_{2} + 87X_{3}

For this model SSR = 600 and SSE = 400.

- Refer to Exhibit 15-1. The coefficient of determination for the above model is
- 0.667
- 0.600
- 0.336
- o.400

** **

- Refer to Exhibit 15-1. MSR for this model is
- 200
- 10
- 1,000
- 43

- Refer to Exhibit 15-1. The computed F statistics for testing the significance of the above model is
- 1.500
- 20.00
- 0.600
- 0.6667

- In a multiple regression analysis SSR = 1,000 and SSE = 200. The F statistic for this model is
- 5.0
- 1,200
- 800
- Not enough information is provided to answer this question.

- The ratio of MSE/MSR yields
- SST
- the F statistic
- SSR
- None of these alternatives is correct.

- In a multiple regression model, the variance of the error term e is assumed to be
- the same for all values of the dependent variable
- zero
- the same for all values of the independent variable
- -1

- The adjusted multiple coefficient of determination is adjusted for
- the number of dependent variables
- the number of independent variables
- the number of equations
- detrimental situations

- In multiple regression analysis, the correlation among the independent variables is termed
- homoscedasticity
- linearity
- multicollinearity
- adjusted coefficient of determination

- In a multiple regression model, the values of the error term ,e, are assumed to be
- zero
- dependent on each other
- independent of each other
- always negative

- In multiple regression analysis,
- there can be any number of dependent variables but only one independent variable
- there must be only one independent variable
- the coefficient of determination must be larger than 1
- there can be several independent variables, but only one dependent variable

- In a multiple regression model, the error term e is assumed to
- have a mean of 1
- have a variance of zero
- have a standard deviation of 1
- be normally distributed

- In a multiple regression analysis involving 12 independent variables and 166 observations, SSR = 878 and SSE = 122. The coefficient of determination is
- 0.1389
- 0.1220
- 0.878
- 0.7317

- A regression analysis involved 8 independent variables and 99 observations. The critical value of t for testing the significance of each of the independent variable’s coefficients will have
- 98 degrees of freedom
- 97 degrees of freedom
- 90 degrees of freedom
- 7 degrees of freedom

- In order to test for the significance of a regression model involving 14 independent variables and 255 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are
- 14 and 255
- 255 and 14
- 13 and 240
- 14 and 240

**Exhibit 15-2**

A regression model between sales (Y in $1,000), unit price (X_{1} in dollars) and television advertisement (X_{2} in dollars) resulted in the following function:

= 7 – 3X_{1} + 5X_{2}

For this model SSR = 3500, SSE = 1500, and the sample size is 18.

- Refer to Exhibit 15-2. The coefficient of the unit price indicates that if the unit price is
- increased by $1 (holding advertising constant), sales are expected to increase by $3
- decreased by $1 (holding advertising constant), sales are expected to decrease by $3
- increased by $1 (holding advertising constant), sales are expected to increase by $4,000
- increased by $1 (holding advertising constant), sales are expected to decrease by $3,000

- Refer to Exhibit 15-2. The coefficient of X
_{2}indicates that if television advertising is increased by $1 (holding the unit price constant), sales are expected to - increase by $5
- increase by $12,000
- increase by $5,000
- decrease by $2,000

- Refer to Exhibit 15-2. To test for the significance of the model, the test statistic F is
- 2.33
- 0.70
- 17.5
- 1.75

- Refer to Exhibit 15-2. To test for the significance of the model, the
*p*-value is - less than 0.01
- between 0.01 and 0.025
- between 0.025 and 0.05
- between 0.05 and 0.10

- Refer to Exhibit 15-2. The multiple coefficient of correlation for this problem is
- 0.70
- 0.8367
- 0.49
- 0.2289

**Exhibit 15-3**

In a regression model involving 30 observations, the following estimated regression equation was obtained:

= 17 + 4X_{1} – 3X_{2} + 8X_{3} + 8X_{4}

For this model SSR = 700 and SSE = 100.

- Refer to Exhibit 15-3. The coefficient of determination for the above model is approximately
- -0.875
- 0.875
- 0.125
- 0.144

- Refer to Exhibit 15-3. The computed F statistic for testing the significance of the above model is
- 43.75
- 0.875
- 50.19
- 7.00

- Refer to Exhibit 15-3. The critical F value at 95% confidence is
- 2.53
- 2.69
- 2.76
- 2.99

- Refer to Exhibit 15-3. The conclusion is that the
- model is not significant
- model is significant
- slope of X
_{1}is significant - slope of X
_{2}is significant

**Exhibit 15-4**

- Y = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2}+ e

- E(Y) = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2}+ e

- = b
_{o}+ b_{1}X_{1}+ b_{2}X_{2}

- E(Y) = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2}

- Refer to Exhibit 15-4. Which equation describes the multiple regression model?

a

b

c

d

- Refer to Exhibit 15-4. Which equation gives the estimated regression line?

a

b

c

d

- Refer to Exhibit 15-4. Which equation describes the multiple regression equation?

a

b

c

d

**Exhibit 15-5**

Below you are given a partial Minitab output based on a sample of 25 observations.

**Coefficient** **Standard Error**

Constant 145.321 48.682

X_{1} 25.625 9.150

X_{2} ‑5.720 3.575

X_{3} 0.823 0.183

- Refer to Exhibit 15-5. The estimated regression equation is
- Y = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2}+ b_{3}X_{3}+ e - E(Y) = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2}+ b_{3}X_{3} - = 145.321 + 25.625X
_{1}– 5.720X_{2}+ 0.823X_{3} - = 48.682 + 9.15X
_{1}+ 3.575X_{2}+ 0.183X_{3}

- Refer to Exhibit 15-5. The interpretation of the coefficient on X
_{1}is that - a one unit change in X
_{1}will lead to a 25.625 unit change in Y - a one unit change in X
_{1}will lead to a 25.625 unit increase in Y when all other variables are held constant - a one unit change in X
_{1}will lead to a 25.625 unit increase in X_{2}when all other variables are held constant - It is impossible to interpret the coefficient.

- Refer to Exhibit 15-5. We want to test whether the parameter b
_{1}is significant. The test statistic equals - 0.357
- 2.8
- 14
- 1.96

- Refer to Exhibit 15-5. The t value obtained from the table to test an individual parameter at the 5% level is
- 2.06
- 2.069
- 2.074
- 2.080

- Refer to Exhibit 15-5. Carry out the test of significance for the parameter b
_{1}at the 5% level. The null hypothesis should be - rejected
- not rejected
- revised
- None of these alternatives is correct.

**Exhibit 15-6**

Below you are given a partial computer output based on a sample of 16 observations.

**Coefficient** **Standard Error**

Constant 12.924 4.425

X_{1} -3.682 2.630

X_{2} 45.216 12.560

Analysis of Variance

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 4,853 2,426.5

Error 485.3

- Refer to Exhibit 15-6. The estimated regression equation is
- Y = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2}+ e - E(Y) = b
_{0}+ b_{1}X_{1}+ b_{2}X_{2} - = 12.924 ‑ 3.682 X
_{1}+ 45.216 X_{2} - = 4.425 + 2.63 X
_{1}+ 12.56 X_{2}

- Refer to Exhibit 15-6. The interpretation of the coefficient of X
_{1}is that - a one unit change in X
_{1}will lead to a 3.682 unit decrease in Y - a one unit increase in X
_{1}will lead to a 3.682 unit decrease in Y when all other variables are held constant - a one unit increase in X
_{1}will lead to a 3.682 unit decrease in X_{2}when all other variables are held constant - It is impossible to interpret the coefficient.

- Refer to Exhibit 15-6. We want to test whether the parameter b
_{1}is significant. The test statistic equals - -1.4
- 1.4
- 3.6
- 5

- Refer to Exhibit 15-6. The t value obtained from the table which is used to test an individual parameter at the 1% level is
- 2.65
- 2.921
- 2.977
- 3.012

- Refer to Exhibit 15-6. Carry out the test of significance for the parameter b
_{1}at the 1% level. The null hypothesis should be - rejected
- not rejected
- revised
- None of these alternatives is correct.

- Refer to Exhibit 15-6. The degrees of freedom for the sum of squares explained by the regression (SSR) are
- 2
- 3
- 13
- 15

- Refer to Exhibit 15-6. The sum of squares due to error (SSE) equals
- 37.33
- 485.3
- 4,853
- 6,308.9

- Refer to Exhibit 15-6. The test statistic used to determine if there is a relationship among the variables equals
- -1.4
- 0.2
- 0.77
- 5

- Refer to Exhibit 15-6. The F value obtained from the table used to test if there is a relationship among the variables at the 5% level equals
- 3.41
- 3.63
- 3.81
- 19.41

- Refer to Exhibit 15-6. Carry out the test to determine if there is a relationship among the variables at the 5% level. The null hypothesis should
- be rejected
- not be rejected
- revised
- None of these alternatives is correct.

- A regression model in which more than one independent variable is used to predict the dependent variable is called
- a simple linear regression model
- a multiple regression model
- an independent model
- None of these alternatives is correct.

- A term used to describe the case when the independent variables in a multiple regression model are correlated is
- regression
- correlation
- multicollinearity
- None of the above answers is correct.

- A variable that cannot be measured in numerical terms is called
- a nonmeasurable random variable
- a constant variable
- a dependent variable
- a qualitative variable

- A multiple regression model has the form

= 5 + 6X + 7W

As X increases by 1 unit (holding W constant), Y is expected to

- increase by 11 units
- decrease by 11 units
- increase by 6 units
- decrease by 6 units

** **

**Exhibit 15-7**

A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of squares.

SSR = 165

SSE = 60

- Refer to Exhibit 15-7. The coefficient of determination is
- 0.3636
- 0.7333
- 0.275
- 0.5

- Refer to Exhibit 15-7. If we want to test for the significance of the model at 95% confidence, the critical F value (from the table) is
- 3.06
- 3.48
- 3.34
- 3.11

- Refer to Exhibit 15-7. The test statistic from the information provided is
- 2.110
- 3.480
- 4.710
- 6.875

**Exhibit 15-8**

The following estimated regression model was developed relating yearly income (Y in $1,000s) of 30 individuals with their age (X_{1}) and their gender (X_{2}) (0 if male and 1 if female).

= 30 + 0.7X_{1} + 3X_{2}

Also provided are SST = 1,200 and SSE = 384.

- Refer to Exhibit 15-8. From the above function, it can be said that the expected yearly income of
- males is $3 more than females
- females is $3 more than males
- males is $3,000 more than females
- females is $3,000 more than males

- Refer to Exhibit 15-8. The yearly income of a 24-year-old female individual is
- $19.80
- $19,800
- $49.80
- $49,800

- Refer to Exhibit 15-8. The yearly income of a 24-year-old male individual is
- $13.80
- $13,800
- $46,800
- $49,800

- Refer to Exhibit 15-8. The multiple coefficient of determination is
- 0.32
- 0.42
- 0.68
- 0.50

- Refer to Exhibit 15-8. If we want to test for the significance of the model, the critical value of F at 95% confidence is
- 3.33
- 3.35
- 3.34
- 2.96

- Refer to Exhibit 15-8. The test statistic for testing the significance of the model is
- 0.73
- 1.47
- 28.69
- 5.22

- Refer to Exhibit 15-8. The model
- is significant
- is not significant
- would be significant is the sample size was larger than 30
- None of these alternatives is correct.

- Refer to Exhibit 15-8. The estimated income of a 30-year-old male is
- $51,000
- $5,100
- $510
- $51

- For a multiple regression model, SST = 200 and SSE = 50. The multiple coefficient of determination is
- 0.25
- 4.00
- 250
- 0.75

- In a multiple regression analysis involving 10 independent variables and 81 observations, SST = 120 and SSE = 42. The coefficient of determination is
- 0.81
- 0.11
- 0.35
- 0.65

- A regression model involved 18 independent variables and 200 observations. The critical value of t for testing the significance of each of the independent variable’s coefficients will have
- 18 degrees of freedom
- 200 degrees of freedom
- 199 degrees of freedom
- 181 degrees of freedom

- In order to test for the significance of a regression model involving 8 independent variables and 121 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are
- 8 and 121
- 7 and 120
- 8 and 112
- 7 and 112

- In a multiple regression analysis involving 5 independent variables and 30 observations, SSR = 360 and SSE = 40. The coefficient of determination is
- 0.80
- 0.90
- 0.25
- 0.15

- A regression analysis involved 6 independent variables and 27 observations. The critical value of t for testing the significance of each of the independent variable’s coefficients will have
- 27 degrees of freedom
- 26 degrees of freedom
- 21 degrees of freedom
- 20 degrees of freedom

78 In order to test for the significance of a regression model involving 4 independent variables and 36 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are

- 4 and 36
- 3 and 35
- 4 and 31
- 4 and 32

- In logistic regression,
- there can only be two independent variables
- there are two dependent variables
- the dependent variable only assumes two discrete values
- the dependent variable only assumes two continuous values

- In a situation where the dependent variable can assume only one of the two possible discrete values,
- we must use multiple regression
- there can only be two independent variables
- logistic regression should be applied
- all the independent variables must have values of either zero or one

**PROBLEMS**

- Multiple regression analysis was used to study how an individual’s income (Y in thousands of dollars) is influenced by age (X
_{1}in years), level of education (X_{2}ranging from 1 to 5), and the person’s gender (X_{3}where 0 =female and 1=male). The following is a partial result of a computer program that was used on a sample of 20 individuals.

**Coefficient** **Standard Error**

X_{1} 0.6251 0.094

X_{2} 0.9210 0.190

X_{3} -0.510 0.920

**Analysis of Variance**

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 84

Error 112

- Compute the coefficient of determination.
- Perform a t test and determine whether or not the coefficient of the variable “level of education” (i.e., X
_{2}) is significantly different from zero. Let a = 0.05. - At a = 0.05, perform an F test and determine whether or not the regression model is significant.
- As you note the coefficient of X
_{3}is -0.510. Fully interpret the meaning of this coefficient.

- A multiple regression analysis between yearly income (Y in $1,000s), college grade point average (X
_{1}), age of the individuals (X_{2}), and the gender of the individual (X_{3}; zero representing female and one representing male) was performed on a sample of 10 people, and the following results were obtained.

**Coefficient** **Standard Error**

Constant 4.0928 1.4400

X_{1} 10.0230 1.6512

X_{2} 0.1020 0.1225

X_{3} -4.4811 1.4400

**Analysis of Variance**

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 360.59

Error 23.91

- Write the regression equation for the above.
- Interpret the meaning of the coefficient of X
_{3}. - Compute the coefficient of determination.
- Is the coefficient of X
_{1}significant? Use a = 0.05. - Is the coefficient of X
_{2}significant? Use a = 0.05. - Is the coefficient of X
_{3}significant? Use a = 0.05. - Perform an F test and determine whether or not the model is significant.

- The following results were obtained from a multiple regression analysis.

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 384 48

Error 20

Total 704

- How many independent variables were involved in this model?
- How many observations were involved?
- Determine the F statistic.

- Shown below is a partial computer output from a regression analysis.

**Coefficient** **Standard Error**

Constant 10.00 2.00

X_{1} -2.00 1.50

X_{2} 6.00 2.00

X_{3} -4.00 1.00

**Analysis of Variance**

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 60

Error

Total 19 140

- Use the above results and write the regression equation.
- Compute the coefficient of determination and fully interpret its meaning.
- At a = 0.05, test to see if there is a relation between X
_{1}and Y. - At a = 0.05, test to see if there is a relation between X
_{3}and Y. - Is the regression model significant? Perform an F test and let a = 0.05.

- In order to determine whether or not the sales volume of a company (Y in millions of dollars) is related to advertising expenditures (X
_{1}in millions of dollars) and the number of salespeople (X_{2}), data were gathered for 10 years. Part of the regression results is shown below.

**Coefficient** **Standard Error**

Constant 7.0174 1.8972

X_{1} 8.6233 2.3968

X_{2} 0.0858 0.1845

**Analysis of Variance**

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 321.11

Error 63.39

- Use the above results and write the regression equation that can be used to predict sales.
- Estimate the sales volume for an advertising expenditure of 3.5 million dollars and 45 salespeople.
**Give your answer in dollars.** - At a = 0.05, test to determine if the fitted equation developed in Part a represents a significant relationship between the independent variables and the dependent variable.
- At a = 0.05, test to see if b
_{1}is significantly different from zero. - Determine the multiple coefficient of determination.
- Compute the adjusted coefficient of determination.

- In order to determine whether or not the number of automobiles sold per day (Y) is related to price (X
_{1}in $1,000), and the number of advertising spots (X_{2}), data were gathered for 7 days. Part of the regression results is shown below.

**Coefficient** **Standard Error**

Constant 0.8051

X_{1} 0.4977 0.4617

X_{2} 0.4733 0.0387

**Analysis of Variance**

**Source of Degrees Sum of Mean**

**Variation** **of Freedom** **Squares** **Square** **F**

Regression 40.700

Error 1.016

- Determine the least squares regression function relating Y to X
_{1}and X_{2}. - If the company charges $20,000 for each car and uses 10 advertising spots, how many cars would you expect them to sell in a day?
- At a = 0.05, test to determine if the fitted equation developed in Part a represents a significant relationship between the independent variables and the dependent variable.
- At 95% confidence, test to see if price is a significant variable.
- At 95% confidence, test to see if the number of advertising spots is a significant variable.
- Determine the multiple coefficient of determination.

- The following is part of the results of a regression analysis involving sales (Y in millions of dollars), advertising expenditures (X
_{1}in thousands of dollars), and number of salespeople (X_{2}) for a corporation. The regression was performed on a sample of 10 observations.

**Coefficient** **Standard Error**

Constant -11.340 20.412

X_{1} 0.798 0.332

X_{2} 0.141 0.278

- Write the regression equation.
- Interpret the coefficients of the estimated regression equation found in Part (a).
- At a =0.05, test for the significance of the coefficient of advertising.
- At a =0.05, test for the significance of the coefficient of number of salespeople.
- If the company uses $50,000 in advertisement and has 800 salespersons, what are the expected sales? Give your answer in dollars.

- The following is part of the results of a regression analysis involving sales (Y in millions of dollars), advertising expenditures (X
_{1}in thousands of dollars), and number of sales people (X_{2}) for a corporation:

**ANALYSIS OF VARIANCE**

**Source of Degrees of Sum of Mean**

**Variation** **Freedom** **Squares** **Square** **F**

Regression 2 822.088

Error 7 736.012

- At a = 0.05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable.
- Determine the multiple coefficient of determination.
- Determine the adjusted multiple coefficient of determination.
- What has been the sample size for this regression analysis?

- Below you are given a partial computer output based on a sample of 12 observations relating the number of personal computers sold by a computer shop per month (Y), unit price (X
_{1}in $1,000) and the number of advertising spots (X_{2}) used on a local television station.

**Coefficient** **Standard Error**

Constant 17.145 7.865

X_{1} -0.104 3.282

X_{2} 1.376 0.250

- Use the output shown above and write an equation that can be used to predict the monthly sales of computers.
- Interpret the coefficients of the estimated regression equation found in Part a.
- If the company charges $2,000 for each computer and uses 10 advertising spots, how many computers would you expect them to sell?
- At a = 0.05, test to determine if the price is a significant variable.
- At a = 0.05, test to determine if the number of advertising spots is a significant variable.

** **

- Below you are given a partial computer output based on a sample of 12 observations relating the number of personal computers sold by a computer shop per month (Y), unit price (X
_{1}in $1,000) and the number of advertising spots (X_{2}) they used on a local television station.

**ANALYSIS OF VARIANCE**

**Source of Sum of Degrees of Mean**

**Variation** **Squares** **Freedom** **Square** **F**

Regression 655.955 2

Error 9

__ __

Total 838.917

- At a = 0.05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable.
- Determine the multiple coefficient of determination.
- Determine the adjusted multiple coefficient of determination.

- Below you are given a partial computer output based on a sample of 30 days of the price of a company’s stock (Y in dollars), the Dow Jones industrial average (X
_{1}), and the stock price of the company’s major competitor (X_{2}in dollars).

**Coefficient** **Standard Error**

Constant 20.000 5.455

X_{1} 0.006 0.002

X_{2} -0.70 0.200

- Use the output shown above and write an equation that can be used to predict the price of the stock.
- If the Dow Jones Industrial Average is 10,000 and the price of the competitor is $50, what would you expect the price of the stock to be?
- At a = 0.05, test to determine if the Dow Jones average is a significant variable.
- At a = 0.05, test to determine if the stock price of the major competitor is a significant variable.

- Below you are given a partial computer output relating the price of a company’s stock (Y in dollars), the Dow Jones industrial average (X
_{1}), and the stock price of the company’s major competitor (X_{2}in dollars).

**ANALYSIS OF VARIANCE**

**Source of Degrees of Sum of Mean**

**Variation** **Freedom** **Squares** **Square** **F**

Regression

Error 20 40

__ __

Total 800

- What has been the sample size for this regression analysis?
- At a = 0.05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable.
- Determine the multiple coefficient of determination.

- A regression was performed on a sample of 16 observations. The estimated equation is = 23.5 – 14.28X
_{1}+ 6.72X_{2}+ 15.68X_{3}. The standard errors for the coefficients are S_{b1}= 4.2, S_{b2}= 5.6, and S_{b3}= 2.8. For this model, SST = 3809.6 and SSR = 3285.4.

- Compute the appropriate t ratios.
- Test for the significance of b
_{1}, b_{2}, and b_{3}at the 5% level of significance. - Do you think that any of the variables should be dropped from the model? Explain.
- Compute R
^{2}and R_{a}^{2}. Interpret R^{2}. - Test the significance of the relationship among the variables at the 5% level of significance.

- The following results were obtained from a multiple regression analysis of supermarket profitability. The dependent variable, Y, is the profit (in thousands of dollars) and the independent variables, X
_{1}and X_{2}, are the food sales and nonfood sales (also in thousands of dollars).

**Coefficient** **Standard Error**

Constant -15.062

X_{1} 0.0972 0.054

X_{2} 0.2484 0.092

Analysis of Variance

**Degrees of Sum of**

**Source** **Freedom** **Squares** **F**

Regression 2 562.363 11.23

Error 9 225.326

Coefficient of determination = 0.7139

- Write the estimated regression equation for the relationship between the variables.
- What can you say about the strength of this relationship?
- Carry out a test of whether Y is significantly related to the independent variables. Use a 0.05 level of significance.
- Carry out a test of whether X
_{1}and Y are significantly related. Use a .05 level of significance. - How many supermarkets are in the sample used here?

- A regression was performed on a sample of 20 observations. Two independent variables were included in the analysis, X and Z. The relationship between X and Z is Z = X
^{2}. The following estimated equation was obtained.

= 23.72 + 12.61X + 0.798Z

The standard errors for the coefficients are S_{b1} = 4.85 and S_{b2} = 0.21

For this model, SSR = 520.2 and SSE = 340.6

- Estimate the value of Y when X = 5.
- Compute the appropriate t ratios.
- Test for the significance of the coefficients at the 5% level. Which variable(s) is (are) significant?
- Compute the coefficient of determination and the adjusted coefficient of determination. Interpret the meaning of the coefficient of determination.
- Test the significance of the relationship among the variables at the 5% level of significance.

- A student used multiple regression analysis to study how family spending (Y) is influenced by income (X
_{1}), family size (X_{2}), and additions to savings (X_{3}). The variables Y, X_{1}, and X_{3}are measured in thousands of dollars. The following results were obtained.

**Degrees of Sum of**

**Source** **Freedom** **Squares** **F**

Regression 3 45.9634 64.28

Error 11 2.6218

**Predictor** **Coefficient** **Standard Error**

Constant 0.0136

X_{1} 0.7992 0.074

X_{2} 0.2280 0.190

X_{3} -0.5796 0.920

Coefficient of determination = 0.946

- Write out the estimated regression equation for the relationship between the variables.
- What can you say about the strength of this relationship?
- Carry out a test of whether Y is significantly related to the independent variables. Use a .05 level of significance.
- Carry out a test to see if X
_{3}and Y are significantly related. Use a .05 level of significance.

- A regression model involving 3 independent variables for a sample of 20 periods resulted in the following sum of squares.

**Sum of Squares**

Regression 90

Residual (Error) 100

- Compute the coefficient of determination and fully explain its meaning.
- At a = 0.05 level of significance, test to determine whether or not there is a significant relationship between the independent variables and the dependent variable.

** **

- A regression model involving 8 independent variables for a sample of 69 periods resulted in the following sum of squares.

SSE = 306

SST = 1800

- Compute the coefficient of determination.
- At a = 0.05, test to determine whether or not the model is significant.

- In a regression model involving 46 observations, the following estimated regression equation was obtained.

= 17 + 4X_{1} – 3X_{2} + 8X_{3} + 5X_{4} + 8X_{5}

For this model, SST = 3410 and SSE = 510.

- Compute the coefficient of determination.
- Perform an F test and determine whether or not the regression model is significant.

- A microcomputer manufacturer has developed a regression model relating his sales (Y in $10,000s) with three independent variables. The three independent variables are price per unit (Price in $100s), advertising (ADV in $1,000s) and the number of product lines (Lines). Part of the regression results is shown below.

**Coefficient** **Standard Error**

Intercept 1.0211 22.8752

Price -0.1524 0.1411

ADV 0.8849 0.2886

Lines -0.1463 1.5340

**Analysis of Variance**

**Source of Degrees Sum of**

**Variation** **of Freedom** **Squares**

Regression 2708.61

Error (Residuals) 14 2840.51

- Use the above results and write the regression equation that can be used to predict sales.
- If the manufacturer has 10 product lines, advertising of $40,000, and the price per unit is $3,000, what is your estimate of their sales?
**Give your answer in dollars**__.__ - Compute the coefficient of determination and fully interpret its meaning.
- At a = 0.05, test to see if there is a significant relationship between sales and unit price.
- At a = 0.05, test to see if there is a significant relationship between sales and the number of product lines.
- Is the regression model significant? (Perform an F test.)
- Fully interpret the meaning of the regression (coefficient of price) per unit that is, the slope for the price per unit.
- What has been the sample size for this analysis?

- The following is part of the results of a regression analysis involving sales (Y in millions of dollars), advertising expenditures (X
_{1}in thousands of dollars), and number of salespeople (X_{2}) for a corporation. The regression was performed on a sample of 10 observations.

**Coefficient** **Standard Error**

Intercept 40.00 7.00

X_{1} 8.00 2.50

X_{2} 6.00 3.00

- If the company uses $40,000 in advertisement and has 30 salespersons, what are the expected sales?
**Give your answer in dollars.** - At a = 0.05, test for the significance of the coefficient of advertising.
- At a = 0.05, test for the significance of the coefficient of the number of salespeople.

- Sherri Cola Company has developed a regression model relating its sales (Y in $10,000s) with four independent variables. The four independent variables are price per unit (PRICE, in dollars), competitor’s price (COMPRICE, in dollars), advertising (ADV, in $1,000s) and type of container used

(CONTAIN; 1 = Cans and 0 = Bottles). Part of the regression results is shown below. (Assume n = 25)

**Coefficient** **Standard Error**

Intercept 443.143

PRICE -57.170 20.426

COMPRICE 27.681 19.991

ADV 0.025 0.023

CONTAIN -95.353 91.027

- If the manufacturer uses can containers, his price is $1.25, advertising $200,000, and his competitor’s price is $1.50, what is your estimate of his sales?
**Give your answer in dollars.** - Test to see if there is a significant relationship between sales and unit price. Let a = 0.05.
- Test to see if there is a significant relationship between sales and advertising. Let a = 0.05.
- Is the type of container a significant variable?

Let a = 0.05.

- Test to see if there is a significant relationship between sales and competitor’s price. Let a = 0.05.

- The Brock Juice Company has developed a regression model relating sales (Y in $10,000s) with four independent variables. The four independent variables are price per unit (X
_{1}, in dollars), competitor’s price (X_{2}, in dollars), advertising (X_{3}, in $1,000s) and type of container used (X_{4}) (1 = Cans and 0 = Bottles). Part of the regression results are shown below:

**Analysis of Variance**

**Source of Degrees of Sum of**

**Variation** **Freedom** **Squares**

Regression 4 283,940.60

Error (Residuals) 18 621,735.14

- Compute the coefficient of determination and fully interpret its meaning.
- Is the regression model significant? Explain what your answer implies. Let a = 0.05.
- What has been the sample size for this analysis?

- The following regression model has been proposed to predict sales at a furniture store.

= 10 – 4X_{1} + 7X_{2} + 18X_{3}

where

X_{1} = competitor’s previous day’s sales (in $1,000s)

X_{2} = population within 1 mile (in 1000s)

X_{3} = 1 if any form of advertising was used, 0 if otherwise

= sales (in $1,000s)

- Fully interpret the meaning of the coefficient of X
_{3}. - Predict sales (in dollars) for a store with competitor’s previous day’s sale of $3,000, a population of 10,000 within 1 mile, and six radio advertisements.

- A sample of 30 houses that were sold in the last year was taken. The value of the house (Y) was estimated. The independent variables included in the analysis were the number of rooms (X
_{1}), the size of the lot (X_{2}), the number of bathrooms (X_{3}), and a dummy variable (X_{4}), which equals 1 if the house has a garage and equals 0 if the house does not have a garage. The following results were obtained:

**Coefficient** **Standard Error**

Constant 15,232.5 8,462.5

X_{1} 2,178.4 778.0

X_{2} 7.8 2.2

X_{3} 2,675.2 2,229.3

X_{4} 1,157.8 463.1

**Analysis of Variance**

Source of Degrees Sum of Mean

**Variation** **of Freedom** **Squares** **Squares**

Regression 204,242.88 51,060.72

Error (Residuals) 205,890.00 8,235.60

- Write out the estimated equation.
- Interpret the coefficient on the number of rooms (X
_{1}). - Interpret the coefficient on the dummy variable (X
_{4}). - What are the degrees of freedom for the sum of squares explained by the regression (SSR) and the sum of squares due to error (SSE)?
- Test whether or not there is a significant relationship between the value of a house and the independent variables. Use a .05 level of significance. Be sure to state the null and alternative hypotheses.
- Test the significance of b
_{1}at the 5% level. Be sure to state the null and alternative hypotheses. - Compute the coefficient of determination and interpret its meaning.
- Estimate the value of a house that has 9 rooms, a lot with an area of 7,500, 2 bathrooms, and a garage.

- A sample of 25 families was taken. The objective of the study was to estimate the factors that determine the monthly expenditure on food for families. The independent variables included in the analysis were the number of members in the family (X
_{1}), the number of meals eaten outside the home (X_{2}), and a dummy variable (X_{3}) that equals 1 if a family member is on a diet and equals 0 if there is no family member on a diet. The following results were obtained.

**Coefficient** **Standard Error**

Constant 450.08 53.6

X_{1} 49.92 9.6

X_{2} 10.12 2.2

X_{3} -.60 12.0

**Analysis of Variance**

**Source of Degrees of Sum of Mean**

**Variation** **Freedom** **Squares** **Squares**

Regression 3,078.39 1,026.13

Error 2,013.90 95.90

- Write out the estimated regression equation.
- Interpret all coefficients.
- Test for the significance of b
_{1}, b_{2}, and b_{3}at the 1% level of significance. - What are the degrees of freedom for the sum of squares explained by the regression (SSR) and the sum of squares due to error (SSE)?
- Test whether of not there is a significant relationship between the monthly expenditure on food and the independent variables. Use a .01 level of significance. Be sure to state the null and alternative hypotheses.
- Compute the coefficient of determination and explain its meaning.
- Estimate the monthly expenditure on food for a family that has 4 members, eats out 3 times, and does not have any member of the family on a diet.

- The following regression model has been proposed to predict sales at a fast food outlet.

= 18 – 2X_{1} + 7X_{2} + 15X_{3}

where

X_{1} = the number of competitors within 1 mile

X_{2} = the population within 1 mile (in 1,000s)

X_{3} = 1 if drive-up windows are present, 0 otherwise

= sales (in $1,000s)

- What is the interpretation of 15 (the coefficient of X
_{3}) in the regression equation? - Predict sales for a store with 2 competitors, a population of 10,000 within one mile, and one drive-up window (give the answer in dollars).
- Predict sales for the store with 2 competitors, a population of 10,000 within one mile, and no drive-up window (give the answer in dollars).

- The following regression model has been proposed to predict sales at a computer store.

= 50 – 3X_{1} + 20X_{2} + 10X_{3}

where

X_{1} = competitor’s previous day’s sales (in $1,000s)

X_{2} = population within 1 mile (in 1,000s)

= sales (in $1000s)

Predict sales (in dollars) for a store with the competitor’s previous day’s sale of $5,000, a population of 20,000 within 1 mile, and nine radio advertisements.

- The following regression model has been proposed to predict monthly sales at a shoe store.

= 40 – 3X_{1} + 12X_{2} + 10X_{3}

where

X_{1} = competitor’s previous month’s sales (in $1,000s)

X_{2} = Stores previous month’s sales (in $1,000s)

= sales (in $1000s)

- Predict sales (in dollars) for the shoe store if the competitor’s previous month’s sales were $9,000, the store’s previous month’s sales were $30,000, and no radio advertisements were run.
- Predict sales (in dollars) for the shoe store if the competitor’s previous month’s sales were $9,000, the store’s previous month’s sales were $30,000, and 10 radio advertisements were run.