Sales (thousands of dollars) Year Population Advertising (thousands of dollars) Previous advertising (thousands of dollars) $15713 1 102558 $20 $30 $12937 2 101792 $15 $20 $12872 3 104347 $25 $15 $16227 4 106180 $30 $25 $15388 5 106562 $15 $30 $13180 6 105209 $25 $15 $17199 7 109185 $35 $25 $20674 8 109976 $40 $35 $20350 9 110659 $20 $40 $14444 10 111844 $25 $20         Consider the model: salest = 0 + 1t + 2popt + 3advt + 4advt-1 + t where salest = sales in year t,            popt = size of the population residing within 10 kilometres of the restaurant,            advt = advertising expenditures in year t, and            advt-1 = advertising expenditures in the previous year.   Based on the sample data, a regression analysis was performed and the results are shown below.   SUMMARY OUTPUT                           Regression Statistics           Multiple R 0,994630788           R Square 0,989290405           Adjusted R Square 0,980722728           Standard Error 393,1191142           Observations 10                         ANOVA               df SS MS F Significance F   Regression 4 71378769,21 17844692,3 115,4677605 4,12255E-05   Residual 5 772713,1899 154542,638       Total 9 72151482,4                         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 24799,62737 17008,82164 1,458045001 0,204627778 -18922,94058 68522,19531 Year 318,9594603 178,6691756 1,785195791 0,134293268 -140,3242772 778,2431977 Population -0,20186378 0,172483261 -1,17033838 0,294606217 -0,645246117 0,241518557 Advertising 134,947818 19,58889716 6,888995176 0,000986635 84,59295482 185,3026813 Previous_Advertising 295,5352939 19,53921967 15,12523524 2,2891E-05 245,3081307 345,762457     You are required to answer the following questions: Estimate the sample regression equation.  What proportion of variation in annual sales is explained by variation in year, population size, advertising expenditure for current year and advertising expenditure for previous year?  Construct a 99% confidence interval for From this confidence interval, can you conclude that previous year advertising expenditure influence current year sales?  Test at 5% level of significance the hypothesis that year, population size, advertising expenditure for current year and advertising expenditure for previous year jointly influence annual sales.

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The owner of a restaurant in Bloomington, Indianna, has recorded sales data for the past 10 years. He has also recorded data on potentially relevant variables. The following table gives data on sales and other potentially relevant variables for that particular restaurant.

 

 

Sales (thousands of dollars)

Year

Population

Advertising (thousands of dollars)

Previous advertising (thousands of dollars)

$15713

1

102558

$20

$30

$12937

2

101792

$15

$20

$12872

3

104347

$25

$15

$16227

4

106180

$30

$25

$15388

5

106562

$15

$30

$13180

6

105209

$25

$15

$17199

7

109185

$35

$25

$20674

8

109976

$40

$35

$20350

9

110659

$20

$40

$14444

10

111844

$25

$20

 

 

 

 

Consider the model: salest = 0 + 1t + 2popt + 3advt + 4advt-1 + t

where salest = sales in year t,

           popt = size of the population residing within 10 kilometres of the restaurant,

           advt = advertising expenditures in year t, and

           advt-1 = advertising expenditures in the previous year.

 

Based on the sample data, a regression analysis was performed and the results are shown below.

 

SUMMARY OUTPUT

           
             

Regression Statistics

         

Multiple R

0,994630788

         

R Square

0,989290405

         

Adjusted R Square

0,980722728

         

Standard Error

393,1191142

         

Observations

10

         
             

ANOVA

           

 

df

SS

MS

F

Significance F

 

Regression

4

71378769,21

17844692,3

115,4677605

4,12255E-05

 

Residual

5

772713,1899

154542,638

     

Total

9

72151482,4

 

 

 

 
             

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

24799,62737

17008,82164

1,458045001

0,204627778

-18922,94058

68522,19531

Year

318,9594603

178,6691756

1,785195791

0,134293268

-140,3242772

778,2431977

Population

-0,20186378

0,172483261

-1,17033838

0,294606217

-0,645246117

0,241518557

Advertising

134,947818

19,58889716

6,888995176

0,000986635

84,59295482

185,3026813

Previous_Advertising

295,5352939

19,53921967

15,12523524

2,2891E-05

245,3081307

345,762457

 

 

You are required to answer the following questions:

  1. Estimate the sample regression equation. 
  2. What proportion of variation in annual sales is explained by variation in year, population size, advertising expenditure for current year and advertising expenditure for previous year? 
  3. Construct a 99% confidence interval for From this confidence interval, can you conclude that previous year advertising expenditure influence current year sales? 
  4. Test at 5% level of significance the hypothesis that year, population size, advertising expenditure for current year and advertising expenditure for previous year jointly influence annual sales. 
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