Question

To investigate the impact of advertising medias (say youtube) on sales, we construct the fol- lowing simple linear regression model Y; = Bo + B12; + &i with std N(0,0%) where Y is the sales and x is advertising budget in thousands of dollars. The summary table is given below:

Answer 1

(a). The regression line is $\hat{sales}=8.439112+0.047537*youtube$ . The sample size of the data set is 200.

(since the df for error is 198 and so the total df=error df+regression df=199 ie n=199+1=200)

(b). Multiple r-squared=0.6119. This indicates that the linear model could explain 61.19% of the total variation in sales. The Pearson correlation coefficient is the square root of it and is r=0.7822

(c). The hypothesis for F-test is:

There is no linear relationship between sales and you tube

Но

There is a linear relationship between sales and youtube.

H1

The Observed F-statistic=312.2289, DF of numerator=1, DF of denominator=198, p-value=2e-16.

(since we know that $t^2_{n}=F(1,n)$ ).

(d). The decision rule: Reject the null hypothesis. Hence, we conclude that the linear model fits the given data at 5% level of significance.

(e). The value of $SSx=2111986.737$

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We know that $t^2_{n}=F(1,n)$ . $F=17.67^2=312.2289$

Mean square error=residual se²=3.91²=15.2881

Regression SS=15.2881*312.2289=4773.3886

Error SS=198*15.2881=3027.0438

Total SS=Regression+Error=7800.4324

We know that $r=\frac{SS_{xy}}{\sqrt{SS_{xx}*SS_{yy}}}$

  $\frac{SS_{xy}}{\sqrt{SS_{xx}}}=69.0840$

Also since $\hat{\beta }_{1}=\frac{SS_{xy}}{SS_{xx}}$ we have $\frac{SS_{xy}}{SS_{xx}}=0.047537\Rightarrow SS_{xy}=0.047537*SS_{xx}$

Substituting the value of SSxy, in $\frac{SS_{xy}}{\sqrt{SS_{xx}}}=69.0840$ , we get $SS_{xx}=2111986.737$