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: sales_t= ₀ + ₁t + ₂pop_t + ₃adv_t + ₄adv_t-1 + _t

where sales_t= sales in year t,

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

           adv_t = advertising expenditures in year t, and

           adv_t-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.

Name: The owner of a restaurant in Bloomington, Indianna, has recorded sales
Uploaded: 2024-03-20T07:07:12.000Z
Channel: Bartleby
Description: 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

Question

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: sales_t= ₀ + ₁t + ₂pop_t + ₃adv_t + ₄adv_t-1 + _t

where sales_t= sales in year t,

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

adv_t = advertising expenditures in year t, and

adv_t-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.

Accepted Answer

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