Enterprise Industries produces Fresh, a brand of liquid laundry detergent. In arder to manage its inventory more effectively and make revenue projections, the company would like to better predict demand for Fresh. To develop a prediction model, the company has gathered data concerning demand for Fresh over the last 30 sales periods (each sales period is defined to be a four week period). The demand data are presented in table below conceming y(demand for Fresh liquid laundry detergent), x (the price of Fresh), x (the average industry price of competitors' similar detergents), and x3 (Enterprise Industries' advertising expenditure for Fresh). To ultimately increase the demand for Fresh, Enterprise Industries' marketing department is comparing the effectiveness of three different advertising campaigns. These campaigns are denoted as campaigns A, B, and C Campaign A consists entirely of television commercials, campaign B consists of a balanced mixture of television and radio commercials, and campaign Č consists of a balanced mixture of television, radio, newspaper, and magazine ads. To conduct the study, Enterprise Industries has randomly selected one advertising campaign to be used in cach of the 30 sales periods in table below. Although logic would indicate that each of campaigns A, B, and Cshould be used in 10 of the 30 sales periods, Enterprise Industries has made previous commitments to the advertising media involved in the study. As a result, campaigns A, R and C were randomly assigned to, respectively. 9, 11, and 10 sales periods. Furthermore, advertising was done in only the first three weeks of each sales period, so that the carryover effect of the campaign used in a sales period to the next sales period would be minimized. Table lists the campaigns used in the sales periods. To compare the effectiveness of advertising campaigns A B, and C, we define two dummy variables. Specifically, we define the dummy variable ag to equal 1 if campaign Bis used in a sales period and O otherwise. Furthermore, we define the dummy variable De to equal 1f campalgn Cis used in a sales period and O otherwise. Table presents the JMP output of a regression analysis of the Fresh demand dats bu uriDg the modol
Enterprise Industries produces Fresh, a brand of liquid laundry detergent. In arder to manage its inventory more effectively and make revenue projections, the company would like to better predict demand for Fresh. To develop a prediction model, the company has gathered data concerning demand for Fresh over the last 30 sales periods (each sales period is defined to be a four week period). The demand data are presented in table below conceming y(demand for Fresh liquid laundry detergent), x (the price of Fresh), x (the average industry price of competitors' similar detergents), and x3 (Enterprise Industries' advertising expenditure for Fresh). To ultimately increase the demand for Fresh, Enterprise Industries' marketing department is comparing the effectiveness of three different advertising campaigns. These campaigns are denoted as campaigns A, B, and C Campaign A consists entirely of television commercials, campaign B consists of a balanced mixture of television and radio commercials, and campaign Č consists of a balanced mixture of television, radio, newspaper, and magazine ads. To conduct the study, Enterprise Industries has randomly selected one advertising campaign to be used in cach of the 30 sales periods in table below. Although logic would indicate that each of campaigns A, B, and Cshould be used in 10 of the 30 sales periods, Enterprise Industries has made previous commitments to the advertising media involved in the study. As a result, campaigns A, R and C were randomly assigned to, respectively. 9, 11, and 10 sales periods. Furthermore, advertising was done in only the first three weeks of each sales period, so that the carryover effect of the campaign used in a sales period to the next sales period would be minimized. Table lists the campaigns used in the sales periods. To compare the effectiveness of advertising campaigns A B, and C, we define two dummy variables. Specifically, we define the dummy variable ag to equal 1 if campaign Bis used in a sales period and O otherwise. Furthermore, we define the dummy variable De to equal 1f campalgn Cis used in a sales period and O otherwise. Table presents the JMP output of a regression analysis of the Fresh demand dats bu uriDg the modol
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 31EQ
Related questions
Question
excel:
| Price | IndPrice | PriceDif | AdvExp | Demand | AdCamp | DA | DB | DC |
| X1 | X2 | X4 | X3 | Y | ||||
| 3.85 | 3.80 | -0.05 | 5.50 | 7.38 | B | 0 | 1 | 0 |
| 3.75 | 4.00 | 0.25 | 6.75 | 8.51 | B | 0 | 1 | 0 |
| 3.70 | 4.30 | 0.60 | 7.25 | 9.52 | B | 0 | 1 | 0 |
| 3.70 | 3.70 | 0.00 | 5.50 | 7.50 | A | 1 | 0 | 0 |
| 3.60 | 3.85 | 0.25 | 7.00 | 9.33 | C | 0 | 0 | 1 |
| 3.60 | 3.80 | 0.20 | 6.50 | 8.28 | A | 1 | 0 | 0 |
| 3.60 | 3.75 | 0.15 | 6.75 | 8.75 | C | 0 | 0 | 1 |
| 3.80 | 3.85 | 0.05 | 5.25 | 7.87 | C | 0 | 0 | 1 |
| 3.80 | 3.65 | -0.15 | 5.25 | 7.10 | B | 0 | 1 | 0 |
| 3.85 | 4.00 | 0.15 | 6.00 | 8.00 | C | 0 | 0 | 1 |
| 3.90 | 4.10 | 0.20 | 6.50 | 7.89 | A | 1 | 0 | 0 |
| 3.90 | 4.00 | 0.10 | 6.25 | 8.15 | C | 0 | 0 | 1 |
| 3.70 | 4.10 | 0.40 | 7.00 | 9.10 | C | 0 | 0 | 1 |
| 3.75 | 4.20 | 0.45 | 6.90 | 8.86 | A | 1 | 0 | 0 |
| 3.75 | 4.10 | 0.35 | 6.80 | 8.90 | B | 0 | 1 | 0 |
| 3.80 | 4.10 | 0.30 | 6.80 | 8.87 | B | 0 | 1 | 0 |
| 3.70 | 4.20 | 0.50 | 7.10 | 9.26 | B | 0 | 1 | 0 |
| 3.80 | 4.30 | 0.50 | 7.00 | 9.00 | A | 1 | 0 | 0 |
| 3.70 | 4.10 | 0.40 | 6.80 | 8.75 | B | 0 | 1 | 0 |
| 3.80 | 3.75 | -0.05 | 6.50 | 7.95 | B | 0 | 1 | 0 |
| 3.80 | 3.75 | -0.05 | 6.25 | 7.65 | C | 0 | 0 | 1 |
| 3.75 | 3.65 | -0.10 | 6.00 | 7.27 | A | 1 | 0 | 0 |
| 3.70 | 3.90 | 0.20 | 6.50 | 8.00 | A | 1 | 0 | 0 |
| 3.55 | 3.65 | 0.10 | 7.00 | 8.50 | A | 1 | 0 | 0 |
| 3.60 | 4.10 | 0.50 | 6.80 | 8.75 | A | 1 | 0 | 0 |
| 3.65 | 4.25 | 0.60 | 6.80 | 9.21 | B | 0 | 1 | 0 |
| 3.70 | 3.65 | -0.05 | 6.50 | 8.27 | C | 0 | 0 | 1 |
| 3.75 | 3.75 | 0.00 | 5.75 | 7.67 | B | 0 | 1 | 0 |
| 3.80 | 3.85 | 0.05 | 5.80 | 7.93 | C | 0 | 0 | 1 |
| 3.70 | 4.25 | 0.55 | 6.80 | 9.26 | C | 0 | 0 | 1 |
![y = Bo + B1 x1 + B2 x2 + B3 3 + B4DB + B5DC + E
I Click here for the Excel Data File
(a) In this model the parameter B, represents the effect on mean demand of advertising campaign B compared to advertising
campaign A, and the parameter B, represents the effect on mean demand of advertising campaign C compared to advertising
campaign A. Use the regression output to find and report a point estimate of each of the above effects and to test the significance of
each of the above effects. Also, find and report a 95 percent confidence interval for each of the above effects. Interpret your results.
(Round your answers to 4 decimal places.)
The point estimate of the effect on the mean of campaign B compared to campaign A is b4 =
0.2695
0.1262.
The 95% confidence interval = [
The point estimate of the effect on the mean of campaign C compared to campaign A is b5 =
0.4128
0.4396
The 95% confidence interval = [
0.2944.
0.5847 ]-
Campaign
is probably most effective even though intervals overlap.
(b) The prediction results at the bottom of the output correspond to a future period when Fresh's price will be x, = 3.70, the average
price of similar detergents will be x, = 3.90, Fresh's advertising expenditure will be x = 6.50, and advertising campaign C will be
used. Show how ý = 8.61621 is calculated. Then find, report, and interpret a 95 percent confidence interval for mean demand and a 95
percent prediction interval for an individual demand when x, = 3.70, x, = 3.90, x = 6.50, and campaign Cis used. (Round your
answers to 5 decimal places.)
у-hat
Confidence interval
Prediction interval
(c) Consider the alternative model
y = Bo + B1 x + B2 x2 + B3 3 + B4DA + B5Dc + E
Here D, equals 1 if advertising campaign A is used and equals 0 otherwise. Describe the effect represented by the regression
parameter 85-
B5 = effect on mean of Campaign
compared to Campaign B.
А
B](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedd60f12-fe5d-409b-96fb-afd8188a6209%2Fb38c18ae-941e-42d4-8c7f-2be306bfe20b%2Fdxgqp6k_processed.png&w=3840&q=75)
Transcribed Image Text:y = Bo + B1 x1 + B2 x2 + B3 3 + B4DB + B5DC + E
I Click here for the Excel Data File
(a) In this model the parameter B, represents the effect on mean demand of advertising campaign B compared to advertising
campaign A, and the parameter B, represents the effect on mean demand of advertising campaign C compared to advertising
campaign A. Use the regression output to find and report a point estimate of each of the above effects and to test the significance of
each of the above effects. Also, find and report a 95 percent confidence interval for each of the above effects. Interpret your results.
(Round your answers to 4 decimal places.)
The point estimate of the effect on the mean of campaign B compared to campaign A is b4 =
0.2695
0.1262.
The 95% confidence interval = [
The point estimate of the effect on the mean of campaign C compared to campaign A is b5 =
0.4128
0.4396
The 95% confidence interval = [
0.2944.
0.5847 ]-
Campaign
is probably most effective even though intervals overlap.
(b) The prediction results at the bottom of the output correspond to a future period when Fresh's price will be x, = 3.70, the average
price of similar detergents will be x, = 3.90, Fresh's advertising expenditure will be x = 6.50, and advertising campaign C will be
used. Show how ý = 8.61621 is calculated. Then find, report, and interpret a 95 percent confidence interval for mean demand and a 95
percent prediction interval for an individual demand when x, = 3.70, x, = 3.90, x = 6.50, and campaign Cis used. (Round your
answers to 5 decimal places.)
у-hat
Confidence interval
Prediction interval
(c) Consider the alternative model
y = Bo + B1 x + B2 x2 + B3 3 + B4DA + B5Dc + E
Here D, equals 1 if advertising campaign A is used and equals 0 otherwise. Describe the effect represented by the regression
parameter 85-
B5 = effect on mean of Campaign
compared to Campaign B.
А
B
Transcribed Image Text:rewenue
Enterprise Industries produces Fresh, a brand of liquid laundry detergent. In order to manage its inventory more effectively and make
revenue projections, the company would like to better predict demand for Fresh. To develop a prediction model, the company has
gathered data concerning demand for Fresh over the last 30 sales periods (each sales period is defined to be a four -week period). The
demand data are presented in table below conceming y(demand for Fresh lquid laundry detergent), x1 (the price of Fresh), xz (the
average industry price of competitors' similar detergents), and x (Enterprise Industries' advertising expenditure for Fresh). To
ultimately increase the demand for Fresh, Enterprise Industries' marketing department is comparing the effectiveness of three different
advertising campaigns. These campaigns are denoted as campaigns A, B, and c Campaign Aconsists entirely of television
commercials, campaign B consists of a balanced mixture of television and radio commercials, and campaign C consists of a balanced
mixture of television, radio, newspaper, and magazine ads. To conduct the study, Enterprise Industries has randomly selected one
advertising campaign to be used in each of the 30 sales periads in table below. Although logic would indicate that each of campaigns
A, B, and Cshould be used in 10 of the 30 sales periods, Enterprise Industries has made previous commitments to the advertising
media involved in the study. As a result, campaigns A, B, and Cwere randomly assigned to, respectively, 9, 11, and 10 sales periods.
Furthermore, advertising was done in only the first three weeks of each sales period, so that the carryover effect of the campaign used
In a sales period to the next sales period would be minimized. Table lists the campaigns used in the sales periods.
To compare the effectiveness of advertising campaigns A B, and C, we define two dummy variables. Specifically, we define the
dummy variable D, to equal 1 if campaign Bis used in a sales period and O otherwise. Furthermore, we define the dummy variable De
to equal 1if campaign Cis used in a sales period and O otherwise. Table presents the JMP output of a regression analysis of the Fresh
demand data by using the model
Iliatorical Data Concerning Demand for Freah Detergent
Advertining
Espenditure
for Freah, x for Freh, y
Average
Industry
Priee, *2
Price for
Sales
Denand
Treah, x
Period
1
3.85
3.80
5.50
7.38
6. 75
3.75
4.00
6.75
a.51
3.70
4.30
7.25
9.52
3.70
3.70
5.50
7.50
3.60
1.85
7.00
2.33
3.60
3.80
6.50
a.28
3.60
3.80
3.75
6.75
a.75
3.85
5.25
7.87
3.80
1.65
5.25
7.10
10
3.85
4.00
6.00
a.00
1
1.90
4.10
6.50
7.89
12
1.90
4.00
6.25
1.15
1.70
13
3.70
4.10
7.00
9.10
14
3.75
4.20
6.90
a.86
15
1.75
4.10
6.80
a.90
16
3.80
4.10
6.80
1.07
a.87
1.70
17
3.70
4.20
7.10
9.26
3. 80
3. 70
4 30
18
3.80
4.30
7.00
2.00
19
3.70
4.10
6.80
a.75
1. 95
20
3.80
3.75
6.50
7.95
21
3.80
3.75
6.25
1.65
7.65
22
3.75
3.65
6.00
7.27
23
3.70
3.90
6.50
a.00
24
3.55
3.65
7.00
a.50
25
3.60
4.10
6.80
a.75
26
3.65
4.25
6.80
9.21
27
3.70
3.65
6.50
a.27
28
3.75
3.75
5.75
7.67
29
3.80
3.85
5.80
7.93
30
3.70
4.25
6.80
9.26
Advertining Carpaigna
Uaed
by Enter prine Industrie
Advertising
d
Sales
Period
Carpaign
11
14
30
Sumary of rit
Riquare
RSquare Adj
Rpot Hean Square Error
0.959727
0.951337
0.15028
Maan of Remponae
Obeervatione (or Sun Wgta)
30
Analyaia ot Variance
Sun of
Hean
F Ratio
Source
Dr
Squares
Square
5 12.916568
114, 362
Prob> 7
K.0001
Model
2.58331
Error
C. Total
24
0.542019
0.02258
29 13.4585BT
Tatimate Std Error t Ratio Probt| Lower 35 Upper 95
3.50 <D.0001
4.68 <D.0001
Turn
Intercept
1.715414
1.584933
5.4442724
11.114555
Price(XI)
IndPriee (X2)
AdvExp(X3
-2.76BD24
0.414437
-3.62338
-1.112669
1.6666921
0.191332
1.71
<D.001+
1.2718024
0.4927425
4.11 <D.0001.
0.326298
0.2694
a.06945
1.80
D.0007
0.1261574 0.4128347
DC
0.4395621
0.070335
4.25 <D.001
0.29439T9
0.547264
Lover 95
Predicted
Demand
Upper 95
Maan Denand
Upper 958
Indiv Denand
Lower 95
Mean Denand
Indiv Denand
11
B.616211576
8.5137990349
B.T18424117
8.289578079
1.9421450729
EClick here for the Excel Data File
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