91Ó°ÊÓ

The data for groups of consumers who receive different exposure to the advertising is provided. Consumers are divided into six different group of exposure levels.

a.

The regression model to express the company’s market share as a function of advertising exposure can be written as a regression model in qualitative variables where (k-1).

Here, advertising exposure has 4 levels; Very high, high, medium, and low therefore, 3 variables are introduced.

Mathematically, the regression equation can be written as$E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_3$ .

Where, when advertising exposure is very high, 0 otherwise, x₂,=1 when advertising exposure is high, 0 otherwise, x₃,=1 when advertising exposure is medium, 0 otherwise.

b.

The model given in the previous part did not include interaction terms since the researchers just want to know the effect of level of advertising exposure on the company’s market share. The variables introduced are qualitative variables not having any interactions amongst themselves.

c.

From the excel output, the summary of the data is given below. The regression equation becomes.$E\left(y\right)=10.233+0.5x_1+2.01667x_2+0.6833x_3$

The model can be fit using the excel function of data analysis. From the table, y, $x_1,x_{2}and{x}_3$ values can be taken and the data analysis function under the data tab in excel can be used to fit the regression model.

The data analysis function can be used to do a regression where values of dependent and independent variables need to be selected from the excel table and the excel runs and does the regression on its own.

For the anova table we need to calculate the mean if the independent variable and then calculate the SSR, SSE, and SST, after that one need to calculate the degrees of freedom and the mean squares and the F.

The SSR is calculated by using$n\mathrm{Î\pounds }\left(x_j-x_j....\right)$,and the SSE is calculated by squaring the each term and adding them all. The SST is the sum of SSR and SSE. The MS regression is calculated by dividing SST by degrees of regression and similarly the MS residual is calculated by dividing SSE by degrees of residual and F is calculated by dividing MS regression by MS residual.

The coefficients of x is calculated by using this formula: $\frac{n\left(∑xy\right)-\left(∑x\right)\left(∑y\right)}{n\left(∑x_2\right)-\left(∑x_2\right)}$ whereas the coefficient of intercept is calculated by $\frac{\left(∑y\right)\left(∑x_2\right)-\left(∑x\right)\left(∑xy\right)}{n\left(∑x_2\right)-\left(∑x_2\right)}$.

The standard error is calculated by dividing the standard deviation by the sample

d.

$H_0:{\mathrm{β}}_1={\mathrm{β}}_2={\mathrm{β}}_3=0$

At least one of the parameters ${\mathrm{β}}_1,{\mathrm{β}}_2,{\mathrm{β}}_3$is non zero.

$$\begin{gathered} \text{Here}, F-\text{test} \text{statistic}=\frac{R^2/k}{\left(1-R^2\right)/\left[n-\left(k+1\right)\right]} \\ =\frac{1.41}{24-4} \\ =0.0705 \end{gathered}$$

.

Value of is 2.0055

H₀ is rejected if $F-statistic>F_{0.05,28,28}$.

For ,$\mathrm{Î\pm }=0.05$since $F<F_{0.05,28,28}$not sufficient evidence to reject H₀ at 95% confidence interval.

Therefore,${\mathbf{β}}_{\mathbf{1}}\mathbf{=}{\mathbf{β}}_{\mathbf{2}}\mathbf{=}{\mathbf{β}}_{\mathbf{3}}\mathbf{=}\mathbf{0}$meaning that company’s expected market share differs for different level of advertising exposure.