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e. At = 0.05, test to see if there is a relation between x3 and y.
5. In order to determine whether or not the sales volume of a company (y in millions of
dollars) is related to advertising expenditures (x1 in millions of dollars) and the number of
salespeople (x2), data were gathered for 10 years. Part of the Excel output is shown
below.
ANOVA
df
SS
MS
F
Regression
321.11
Residual
63.39
Coefficients
Standard Error
Intercept
7.0174
1.8972
x1
8.6233
2.3968
x2
0.0858
0.1845
a. Use the above results and write the regression equation that can be used to predict
sales.
b. Estimate the sales volume for an advertising expenditure of 3.5 million dollars and 45
salespeople. Give your answer in dollars.
c. At = 0.01, test to determine if the fitted equation developed in Part a represents a
significant relationship between the independent variables and the dependent
variable.
d. At = 0.05, test to see if
1 is significantly different from zero.
e. Determine the multiple coefficient of determination.
f. Compute the adjusted coefficient of determination.
6. In order to determine whether or not the number of automobiles sold per day (y) is
related to price (x1 in $1,000), and the number of advertising spots (x2), data were
gathered for 7 days. Part of the Excel output is shown below.
ANOVA
df
SS
MS
F
Regression
40.700
Residual
1.016
Coefficients
Standard Error
Intercept
0.8051
X1
0.4977
0.4617
X2
0.4733
0.0387
a. Determine the least squares regression function relating y to x1 and x2.
b. 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?
c. At = 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.
d. At = 0.05, test to see if
1 is significantly different from zero.
e. Determine the multiple coefficient of determination.
7. The following is part of the results of a regression analysis involving sales (y in millions
of dollars), advertising expenditures (x1 in thousands of dollars), and number of
salespeople (x2) for a corporation. The regression was performed on a sample of 10
observations.
Coefficient
Standard Error
Constant
-11.340
20.412
x1
0.798
0.332
x2
0.141
0.278
a. Write the regression equation.
b. Interpret the coefficients of the estimated regression equation found in Part (a).
c. At =0.05, test for the significance of the coefficient of advertising.
d. At =0.05, test for the significance of the coefficient of number of salespeople.
e. If the company uses $50,000 in advertisement and has 800 salespersons, what are the
expected sales? Give your answer in dollars.
8. The following is part of the results of a regression analysis involving sales (y in millions
of dollars), advertising expenditures (x1 in thousands of dollars), and number of sales
people (x2) for a corporation:
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F
Regression
2
822.088
Error
7
736.012
a. At = 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.
b. Determine the multiple coefficient of determination.
c. Determine the adjusted multiple coefficient of determination.
d. What has been the sample size for this regression analysis?
9. Below you are given a partial Excel output based on a sample of 12 observations relating
the number of personal computers sold by a computer shop per month (y), unit price (x1
in $1,000) and the number of advertising spots (x2) used on a local television station.
Coefficient
Standard Error
Intercept
17.145
7.865
x1
-0.104
3.282
x2
1.376
0.250
a. Use the output shown above and write an equation that can be used to predict the
monthly sales of computers.
b. Interpret the coefficients of the estimated regression equation found in Part a.
c. If the company charges $2,000 for each computer and uses 10 advertising spots, how
many computers would you expect them to sell?
d. At = 0.05, test to determine if the price is a significant variable.
e. At = 0.05, test to determine if the number of advertising spots is a significant
variable.
EMBS4 TB13 - 20
10. Below you are given a partial ANOVA table based on a sample of 12 observations
relating the number of personal computers sold by a computer shop per month (y), unit
price (x1 in $1,000) and the number of advertising spots (x2) they used on a local
television station.
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F
Regression
2
655.955
Error
9
Total
838.917
a. At = 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.
b. Determine the multiple coefficient of determination.
c. Determine the adjusted multiple coefficient of determination.
11. Below you are given a partial Excel output based on a sample of 30 days of the price of a
company's stock (y in dollars), the Dow Jones industrial average (x1), and the stock price
of the company's major competitor (x2 in dollars).
Coefficient
Standard Error
Intercept
20.000
5.455
x1
0.030
0.010
x2
-0.70
0.200
a. Use the output shown above and write an equation that can be used to predict the
price of the stock.
b. If the Dow Jones Industrial Average is 2650 and the price of the competitor is $45,
what would you expect the price of the stock to be?
c. At = 0.05, test to determine if the Dow Jones average is a significant variable.
d. At = 0.05, test to determine if the stock price of the major competitor is a
significant variable.
12. Below you are given a partial ANOVA table relating the price of a company's stock (y in
dollars), the Dow Jones industrial average (x1), and the stock price of the company's
major competitor (x2 in dollars).
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F
Regression
Error
20
40
Total
800
a. What has been the sample size for this regression analysis?
b. At = 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.
c. Determine the multiple coefficient of determination.
13. A regression was performed on a sample of 16 observations. The estimated equation is
ˆ
y
= 23.5 − 14.28x1 + 6.72x2 + 15.68x3. The standard errors for the coefficients are Sb1 =
4.2, Sb2 = 5.6, and Sb3 = 2.8. For this model, SST = 3809.6 and SSR = 3285.4.
a. Compute the appropriate t ratios.
b. Test for the significance of
1,
2, and
3 at the 5% level of significance.
c. Do you think that any of the variables should be dropped from the model? Explain.
d. Compute R2 and Ra2. Interpret R2.
e. Test the significance of the relationship among the variables at the 5% level of
significance.
14. 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, x1 and x2, are the food sales and nonfood sales (also in thousands
of dollars).
ANOVA
df
SS
MS
F
Regression
2
562.363
11.23
Error
9
225.326
Coefficients
Standard Error
Intercept
-15.0620
x1
0.0972
0.054
x2
0.2484
0.092
Coefficient of determination = 0.7139
a. Write the estimated regression equation for the relationship between the variables.
b. What can you say about the strength of this relationship?
c. Carry out a test of whether y is significantly related to the independent variables. Use
a .01 level of significance.
d. Carry out a test of whether x1 and y are significantly related. Use a .05 level of
significance.
e. How many supermarkets are in the sample used here?
15. 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 = x2. The
following estimated equation was obtained.
ˆ
y
= 23.72 + 12.61x + 0.798z
The standard errors for the coefficients are Sb1 = 4.85 and Sb2 = 0.21
For this model, SSR = 520.2 and SSE = 340.6
a. Estimate the value of y when x = 5.
b. Compute the appropriate t ratios.
c. Test for the significance of the coefficients at the 5% level. Which variable(s) is
(are) significant?
d. Compute the coefficient of determination and the adjusted coefficient of
determination. Interpret the meaning of the coefficient of determination.
e. Test the significance of the relationship among the variables at the 5% level of
significance.
16. A student used multiple regression analysis to study how family spending (y) is
influenced by income (x1), family size (x2), and additions to savings (x3). The variables y,
x1, and x3 are measured in thousands of dollars. The following results were obtained.
ANOVA
df
SS
MS
F
Regression
3
45.9634
64.28
Error
11
2.6218
Coefficients
Standard Error
Intercept
0.0136
x1
0.7992
0.074
x2
0.2280
0.190
x3
-0.5796
0.920
Coefficient of determination = 0.946
a. Write out the estimated regression equation for the relationship between the
variables.
b. What can you say about the strength of this relationship?
c. Carry out a test of whether y is significantly related to the independent variables. Use
a .05 level of significance.
d. Carry out a test to see if x3 and y are significantly related. Use a .05 level of
significance.
e. Why would a coefficient of determination very close to 1.0 be expected here?
17. 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
a. Compute the coefficient of determination and fully explain its meaning.
b. At = 0.05 level of significance, test to determine whether or not there is a
significant relationship between the independent variables and the dependent
variable.
18. A regression model involving 8 independent variables for a sample of 69 periods resulted
in the following sum of squares.
SSE = 306
SST = 1800
a. Compute the coefficient of determination.
b. At = 0.05, test to determine whether or not the model is significant.
19. In a regression model involving 46 observations, the following estimated regression
equation was obtained.
ˆ
y
= 17 + 4x1 – 3x2 + 8x3 + 5x4 + 8x5
For this model, SST = 3410 and SSE = 510.
a. Compute the coefficient of determination.
b. Perform an F test and determine whether or not the regression model is significant.
20. 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 (Adver in $1,000s) and the number of product lines
(Lines). Part of the regression results is shown below.
ANOVA
df
SS
MS
F
Regression
2708.61
Error
14
2840.51
Coefficients
Standard Error
Intercept
1.0211
22.8752
Price
-0.1524
0.1411
Adver
0.8849
0.2886
Lines
-0.1463
1.5340
a. Use the above results and write the regression equation that can be used to predict
sales.
b. 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.
c. Compute the coefficient of determination and fully interpret its meaning.
d. At = 0.05, test to see if there is a significant relationship between sales and unit
price.
e. At = 0.05, test to see if there is a significant relationship between sales and the
number of product lines.
f. Is the regression model significant? (Perform an F test.)
g. Fully interpret the meaning of the regression (coefficient of price) per unit that is, the
slope for the price per unit.
h. What has been the sample size for this analysis?
21. The following is part of the results of a regression analysis involving sales (y in millions
of dollars), advertising expenditures (x1 in thousands of dollars), and number of
salespeople (x2) for a corporation. The regression was performed on a sample of 10
observations.
Coefficient
Standard Error
Intercept
40.00
7.00
x1
8.00
2.50
x2
6.00
3.00
a. If the company uses $40,000 in advertisement and has 30 salespersons, what are the
expected sales? Give your answer in dollars.
b. At = 0.05, test for the significance of the coefficient of advertising.
c. At = 0.05, test for the significance of the coefficient of the number of salespeople.
22. The Natural Drink 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
a. 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.
b. Test to see if there is a significant relationship between sales and unit price. Let =
0.05.
c. Test to see if there is a significant relationship between sales and advertising. Let
= 0.05.
d. Is the type of container a significant variable?
Let = 0.05.
e. Test to see if there is a significant relationship between sales and competitor's price.
Let = 0.05.
23. The Very Fresh 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 (x1, in dollars), competitor's price (x2, in dollars), advertising (x3, in $1,000s) and
type of container used (x4 where 1 = Cans and 0 = Bottles). Part of the regression results
are shown below:
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F
Regression
4
283,940.60
Error
18
621,735.14
Total
a. Compute the coefficient of determination and fully interpret its meaning.
b. Is the regression model significant? Explain what your answer implies. Let = 0.05.
c. What has been the sample size for this analysis?
24. The following regression model has been proposed to predict sales at a furniture store.
ˆ
y
= 10 – 4x1 + 7x2 + 18x3
where
x1 = competitor's previous day's sales (in $1,000s)
x2 = population within 1 mile (in 1000s)
x3 = 1 if any form of advertising was used, 0 if otherwise
ˆ
y
= sales (in $1,000s)
a. Fully interpret the meaning of the coefficient of x3.
b. 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.
25. 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 (x1), the size of the lot (x2), the number of bathrooms (x3), and a dummy variable
(x4), 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:
ANOVA
df
SS
MS
F
Regression
204,242.88
51,060.72
Error
205,890.00
8,235.60
Coefficients
Standard Error
Intercept
15,232.5
8,462.5
x1
2,178.4
778.0
x2
7.8
2.2
x3
2,675.2
2,229.3
x4
1,157.8
463.1
a. Write out the estimated equation.
b. Interpret the coefficient on the number of rooms (x1).
c. Interpret the coefficient on the dummy variable (x4).
d. 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)?
e. 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.
f. Test the significance of
1 at the 5% level. Be sure to state the null and alternative
hypotheses.
g. Compute the coefficient of determination and interpret its meaning.
h. Estimate the value of a house that has 9 rooms, a lot with an area of 7,500, 2
bathrooms, and a garage.
26. 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 (x1), the number of
meals eaten outside the home (x2), and a dummy variable (x3) 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.
ANOVA
df
SS
MS
F
Regression
3,078.39
1,026.13
Error
2,013.90
95.90
Coefficients
Standard Error
Intercept
150.08
53.6
x1
49.92
9.6
x2
10.12
2.2
x3
-.60
12.0
a. Write out the estimated regression equation.
b. Interpret all coefficients.
c. Compute the appropriate t ratios.
d. Test for the significance of
1,
2, and
3 at the 1% level of significance.
e. 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)?
f. 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.
g. Compute the coefficient of determination and explain its meaning.
h. 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.
i. At 95% confidence determine which parameter is not statistically significant.
27. The following regression model has been proposed to predict sales at a fast food outlet.
ˆ
y
= 18 – 2x1 + 7x2 + 15x3
where
x1 = the number of competitors within 1 mile
x2 = the population within 1 mile (in 1,000s)
x3 = 1 if drive-up windows are present, 0 otherwise
ˆ
y
= sales (in $1,000s)
a. What is the interpretation of 15 (the coefficient of x3) in the regression equation?
b. 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).
c. 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).
28. The following regression model has been proposed to predict sales at a computer store.
ˆ
y
= 50 – 3x1 + 20x2 + 10x3
where
x1 = competitor's previous day's sales (in $1,000s)
x2 = population within 1 mile (in 1,000s)
0 if multimedia advertising was used
=
1 if only TV advertising was used
3
x
ˆ
y
= 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.
29. The following regression model has been proposed to predict monthly sales at a shoe
store.
ˆ
y
= 40 – 3x1 + 12x2 + 10x3
where
x1 = competitor's previous month's sales (in $1,000s)
x2 = Stores previous month's sales (in $1,000s)
if radio advertising was used
=
0 if otherwise
3
1
x
ˆ
y
= sales (in $1000s)
a. 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.
b. 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.
30. A company has recorded data on the weekly sales for its product (y), the unit price of the
competitor’s product (x1), and advertising expenditures (x2). The data resulting from a
random sample of 7 weeks follows. Use Excel’s Regression Tool to answer the
following questions.
Week
Price
Advertising
Sales
1
.33
5
20
2
.25
2
14
3
.44
7
22
4
.40
9
21
5
.35
4
16
6
.39
8
19
7
.29
9
15
a. What is the estimated regression equation?
b. Determine whether the model is significant overall. Use = 0.10.
c. Determine if price is significantly related to sales. Use = 0.10.
d. Determine if advertising is significantly related to sales. Use = 0.10.
e. Find and interpret the multiple coefficient of determination.
