Question

A regression analysis relating a company’s sales, their advertising expenditure, price, and time resulted in the following.

Regression Statistics
Multiple R	0.8800
R Square	0.7744
Adjusted R Square	0.7560
Standard Error	232.29
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	3	53184931.86	17728310.62	328.56	0.0000
Residual	21	1133108.30	53957.54
Total	24	54318040.16
	Coefficients	Standard Error	t Stat	P-value
Intercept	927.23	1229.86	0.75	0.4593
Advertising (X1)	1.02	3.09	0.33	0.7450
Price (X2)	15.61	5.62	2.78	0.0112
Time (X3)	170.53	28.18	6.05	0.0000

a.	At 95% confidence, determine whether or not the regression model is significant. Fully explain how you arrived at your conclusion (give numerical reasoning) and what your answer indicates.
b.	At 95% confidence determine which variables are significant and which are not. Explain how you arrived at your conclusion (Give numerical reasoning).
c.	Fully explain the meaning of R-square, which is given in this model. Be very specific and give numerical explanation.

Answer 1

a. At 95% confidence, determine whether or not the regression model is significant. Fully explain how you arrived at your conclusion (give numerical reasoning) and what your answer indicates.

Answer: The regression model is statistically significant at the 5% significance level. From the given ANOVA output, we clearly see the p-value = 0.000 is less than the given significance level of 0.05 (5%), we, therefore, reject the null hypothesis and conclude that the regression model is significant at 5% level of significance.

b. At 95% confidence determine which variables are significant and which are not. Explain how you arrived at your conclusion (Give numerical reasoning).

Answer: The variables Price (X2) and Time (X3) are significant at 5% significance level, while the variable advertising (X1) is insignificant at 5% significance level. Looking at the table of coefficients, we see the p-values of variables Price (X2) and Time (X3) are less than 0.05 significance level, therefore, we reject the null hypotheses and conclude that these two variables are statistically significant at 5% level of significance and the p-value of variable Advertising (X1) is greater than 0.05 significance level, therefore, we fail to reject the null hypothesis and conclude that Advertising (X1) is not statistically significant at 0.05 significance level.

c. Fully explain the meaning of R-square, which is given in this model. Be very specific and give a numerical explanation.

Answer: R-square is a statistical measure that describes the percentage of variation in the dependent variable that is explained by the linear regression model. From the given outputs, we see the value of R-square = 0.7744 or 77.44%. Which means 77.44% of the variation in the company's sales is explained by the given regression model.