Question

Television Advertising ($1000s) Weekly Gross Revenue (1000s) 96 Newspaper Advertising ($1000s) 1 1.5 Moon 2.5 3.3 3.5 2 2.3 4.2 2.5 Predicted Revenue 5.3 3.6

Answer 1

I used R software to solve this question.

R codes and output:

revenue=c(96,90,96,92,96,95,95,94)
tv=c(5,2,5,2.5,4,3.5,2.5,3)
newsp=c(1.5,2,1.5,2.5,3.3,2.3,4.2,2.5)
fit=lm(revenue~tv+newsp)
summary(fit)

Call:
lm(formula = revenue ~ tv + newsp)

Residuals:
1 2 3 4 5 6 7 8
-0.09138 -0.51307 -0.09138 -0.31135 -0.66392 0.87884 0.15741 0.63485

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 83.3199 1.3723 60.717 2.29e-08 ***
tv 2.1076 0.2395 8.800 0.000314 ***
newsp 1.4890 0.3015 4.938 0.004330 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6353 on 5 degrees of freedom
Multiple R-squared: 0.9398, Adjusted R-squared: 0.9157
F-statistic: 39 on 2 and 5 DF, p-value: 0.0008905

Que.a

F test statistics = 39

P - value = 0.0008905

Conclusion: Since p - value is less than 0.01, hence we reject null hypothesis and conclude that either $\beta_1$ or $\beta_2$ are not equal to zero.

Que.b

t test statistics =  8.800

p value = 0.000314

Conclusion : Since p value is less than 0.05, we reject null hypothesis and conclude that $\beta_1$ is significantly different than zero.

We need not to drop x1

Que.c

t test statistics =  4.938

p value = 0.004330

Conclusion : Since p value is less than 0.05, we reject null hypothesis and conclude that $\beta_2$ is significantly different than zero.

We need not to drop x2

Que.d

Revenue = 83.3199 + 2.1076 TV + 1.4890 Newspaper

= 83.3199 + 2.1076 * 5.3 + 1.4890 *3.6

= 99.85058

Expected revenue is $100 (in $ 1000)