Question

A large sports supplier has many stores located world wide. A regression model is to be constructed to predict the annual revenue of a particular store based upon the population of the city or town where the store is located, the annual expenditure on promotion for the store and the distance of the store to the center of the city.

Data has been collected on 30 randomly selected stores: show data

a)Find the multiple regression equation using all three explanatory variables. Assume that X₁ is population, X₂ is annual promotional expenditure and X₃ is distance to city center. Give your answers to 3 decimal places.

y^ =  + population + promo. expenditure + dist. to city

b)At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis is, is not rejected.

For parts c) and d), using the data, separately calculate the correlations between the response variable and each of the three explanatory variables.

c)The explanatory variable that is most correlated with annual revenue is:

population
promotional expenditure
distance to city

d)The explanatory variable that is least correlated with annual revenue is:

population
promotional expenditure
distance to city

e)The value of R² for this model, to 2 decimal places, is equal to

f)The value of s_e for this model, to 3 decimal places, is equal to

g)Construct a new multiple regression model by removing the variable distance to city center. Give your answers to 3 decimal places.

The new regression model equation is:

y^ =  + population + promo. expenditure

h)In the new model compared to the previous one, the value of R² (to 2 decimal places) is:

increased
decreased
unchanged

i)In the new model compared to the previous one, the value of s_e (to 3 decimal places) is:

increased
decreased
unchanged

Answer 1

ANSWER:

Given that,

We have excel output for multiple regression as :

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.991553
R Square	0.983177
Adjusted R Square	0.981236
Standard Error	30.27639
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	3	1392884	464294.5	506.507	3.59E-23
Residual	26	23833.15	916.6597
Total	29	1416717
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-11.5673	22.9015	-0.50509	0.617751	-58.642	35.5074
Population	0.805069	0.023907	33.675	5.64E-23	0.755927	0.85421
Promotional exp	0.494499	0.140416	3.521672	0.001605	0.20587	0.783128
Dist to city	-0.33873	1.148964	-0.29481	0.770481	-2.70046	2.023005

a)

We have coefficients for x1,x2,x3 as

b1 = 0.805, b2 = 0.494, b3 = -0.339

intercept = a =- 11.567

Hence the regression is

b)

The calculated value of F = 506.507

p value = 3.59 E-23 = 0.0000

Here p value < ( 0.05)

Hence we reject null hypothesis.

At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis is rejected.

Using excel formula ' =correl( data 1 , data 2 ) " we get correlations.

So correlation between y and x1 is r1 = 0.99

correlation between y and x2 is r2 = 0.43

correlation between y and x3 = r3 = -0.24

c)

The explanatory variable that is most correlated with annual revenue is: Population

d)

The explanatory variable that is least correlated with annual revenue is: distance to city.

e)

R² = 0.98

f)

Se = Standard error = 30.276

g)

The new regression model is given by

Regression Statistics
Multiple R	0.991525
R Square	0.983121
Adjusted R Square	0.981871
Standard Error	29.76004
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1392804	696402	786.3084	1.17E-24
Residual	27	23912.82	885.6601
Total	29	1416717
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-15.8435	17.42022	-0.90949	0.37114	-51.5868	19.89984
Population	0.807002	0.022598	35.7107	2.89E-24	0.760634	0.85337
Promotional exp	0.489327	0.13694	3.5733	0.001352	0.20835	0.770304

Here a = -15.844, b1 = 0.807, b2 = 0.489

= -15.844 + 0.807 x1 + 0.489 x2

h)

Here R² = 0.98

In the new model compared to the previous one, the value of R2 (to 2 decimal places) is: Unchanged

i)

Se = 29.760

In the new model compared to the previous one, the value of se (to 3 decimal places) is: decreased