Question

Question: Impact of advertising on market share. The audience for a product’s advertising can be divided into four segments according to the degree of exposure received as a result of the advertising. These segments are groups of consumers who receive very high (VH), high (H), medium (M), or low (L) exposure to the advertising. A company is interested in exploring whether its advertising effort affects its product’s market share. Accordingly, the company identifies 24 sample groups of consumers who have been exposed to its advertising, six groups at each exposure level. Then, the company determines its product’s market share within each group.

Market Share within Group	Exposure Level
10.1	L
10.3	L
10.0	L
10.3	L
10.2	L
10.5	L
10.6	M
11.0	M
11.2	M
10.9	M
10.8	M
11.0	M
12.2	H
12.1	H
11.8	H
12.6	H
11.9	H
12.9	H
10.7	VH
10.8	VH
11.0	VH
10.5	VH
10.8	VH
10.6	VH

1. Write a regression model that expresses the company’s market share as a function of advertising exposure level. Define all terms in your model and list any assumptions you make about them.
2. Did you include interaction terms in your model? Why or why not?
3. The data in the table (previous page) were obtained by the company. Fit the model in part a to the data.
4. Is there evidence to suggest that the firm’s expected market share differs for different levels of advertising exposure? Test using a=0.5.

Short answer

Answers

1. The regression equation can be written as $E\left(y\right)=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3$.
2. 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.
3. The regression equation for a qualitative variable with 4 levels can be written as$E\left(y\right)=10.233+0.5x_1+2.01667x_2+0.6833x_3$ .
4. At 95% significance level${\beta }_1={\beta }_2={\beta }_3=0$, meaning that company’s expected market share differs for different level of advertising exposure.

Step-by-step solution

Given Information 

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.

Regression model

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$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$ .

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.

Interaction terms in the model 

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.

Regression model

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$n\Sigma \left(x_j-x_j....\right)$,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: $\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{n\left(\sum x_2\right)-\left(\sum x_2\right)}$ whereas the coefficient of intercept is calculated by $\frac{\left(\sum y\right)\left(\sum x_2\right)-\left(\sum x\right)\left(\sum xy\right)}{n\left(\sum x_2\right)-\left(\sum x_2\right)}$.

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

Hypothesis testing

d.

$H_0:{\beta }_1={\beta }_2={\beta }_3=0$

At least one of the parameters ${\beta }_1,{\beta }_2,{\beta }_3$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 ,$\alpha =0.05$since $F<F_{0.05,28,28}$not sufficient evidence to reject H₀ at 95% confidence interval.

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