Question

Multiple Linear Regression
20. The Excel file Concert Sales provides data on sales dollars and the number of radio, TV, and newspa-per ads promoting the concerts for a group of cities. Develop simple linear regression models for predict-ing sales as a function of the number of each type of ad. Compare these results to a multiple linear regres-sion model using both independent variables. State each model and explain R-Square, Significance F, and p-values

from Business Analytics Methods, Models, and Decisions
James R. Evans 3rd

A 0 0 25 25 30 30 30 35 35 25 Concert Sales Thousands of Thousands of Sales ($1000) Radio&TV ads Newspaper ads $1,119.00 40 $973.00 40 $875.00 25 $625.00 25 $910.00 $971.00 30 $931.00 35 35 $1,177.00 $882.00 40 $982.00 25 $1,628.00 45 $1,577.00 $1,044.00 50 50 $914.00 50 50 $1,329.00 $1,330.00 55 $1,405.00 60 30 $1,436.00 60 $1,521.00 65 $1,741.00 $1,866.00 70 $1,717.00 70 40 45 45 45 55 20 20 30 35 35 40 65 40

Answer 1

Simple regression output b/w sales and radio:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.696608416
R Square	0.485263285
Adjusted R Square	0.459526449
Standard Error	254.0524474
Observations	22
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1216939.671	1216939.671	18.85481532	0.000316108
Residual	20	1290852.92	64542.64601
Total	21	2507792.591
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	699.9571823	132.5217655	5.281828082	3.60821E-05	423.5216236	976.392741
Radio	12.1620442	2.80088602	4.342213182	0.000316108	6.319498341	18.00459006

So,

Sales = 699.9 + 12.16*Radio

Simple regression output b/w sales and newspaper:

Regression Statistics
Multiple R	0.264271891
R Square	0.069839632
Adjusted R Square	0.023331614
Standard Error	341.5149542
Observations	22
ANOVA
	df	SS	MS	F	Significance F
Regression	1	175143.3125	175143.3125	1.501668632	0.234650477
Residual	20	2332649.278	116632.4639
Total	21	2507792.591
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	877.2432432	293.0840043	2.993146096	0.007185881	265.8807233
News	10.20486486	8.327606655	1.225425898	0.234650477	-7.166218221

So,

Sales = 877.2 + 10.2 * newspaper

Now, multiple linear regres-sion model using both independent variables:

Regression Statistics
Multiple R	0.737809661
R Square	0.544363095
Adjusted R Square	0.496401316
Standard Error	245.2327416
Observations	22
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1365149.738	682574.8688	11.34993534	0.00057132
Residual	19	1142642.853	60139.09754
Total	21	2507792.591
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	385.3814221	237.7349328	1.621055087	0.121483601	-112.2035109
Radio	12.03232224	2.704912709	4.448321824	0.00027571	6.370874871
News	9.391870119	5.9826236	1.5698581	0.132952375	-3.129904984

So,

Sales = 385.38 + 9.39*newspaper + 12.03*Radio

R-squared is a goodness-of-fit measure for linear regression models. This statistic indicates the percentage of the variance in the dependent variable that the independent variables explain

Model	R square	F
multiple linear regression	0.54	11.34
Simple regression output b/w sales and newspaper	0.0698	1.5
Simple regression output b/w sales and Radio	0.48	18.8

R square is very low for 'Simple regression output b/w sales and newspaper'. Hence this model is not fit.

The F-test of overall significance indicates whether your linear regression model provides a better fit to the data than a model that contains no independent variables. So a large F value indicates that a linear model is more accurate. Again F value is very low for 'Simple regression output b/w sales and newspaper'. Hence this model is not fit.

The p-value for each term tests the null hypothesis that the coefficient is equal to zero (no effect). A low p-value (< 0.05) indicates strong relation between dependent and independent variable. ... Typically, you use the coefficient p-values to determine which terms to keep in the regression model.

The p-value (0.23) is high for 'Simple regression output b/w sales and newspaper'. This means that this linear model is not statistically fit.