Question

Question: Promotion of supermarket vegetables. A supermarket chain is interested in exploring the relationship between the sales of its store-brand canned vegetables (y), the amount spent on promotion of the vegetables in local newspapers(x₁) , and the amount of shelf space allocated to the brand (x₂ ) . One of the chain’s supermarkets was randomly selected, and over a 20-week period, x₁ and x₂ were varied, as reported in the table.

Week	Sales, y	Advertising expenses,	Shelf space,	Interaction term,
1	2010	201	75	15075
2	1850	205	50	10250
3	2400	355	75	26625
4	1575	208	30	6240
5	3550	590	75	44250
6	2015	397	50	19850
7	3908	820	75	61500
8	1870	400	30	12000
9	4877	997	75	74775
10	2190	515	30	15450
11	5005	996	75	74700
12	2500	625	50	31250
13	3005	860	50	43000
14	3480	1012	50	50600
15	5500	1135	75	85125
16	1995	635	30	19050
17	2390	837	30	25110
18	4390	1200	50	60000
19	2785	990	30	29700
20	2989	1205	30	36150

1. Fit the following model to the data:$\mathbf{y}\mathbf{}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$
2. Conduct an F-test to investigate the overall usefulness of this model. Use$\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$ .
3. Test for the presence of interaction between advertising expenditures and shelf space. Use$\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$ .
4. Explain what it means to say that advertising expenditures and shelf space interact.
5. Explain how you could be misled by using a first-order model instead of an interaction model to explain how advertising expenditures and shelf space influence sales.
6. Based on the type of data collected, comment on the assumption of independent errors.

Short answer

Answer

1. The model equation can be written as $\hat{y}=1333.1782-0.1512x_1-2.6253x_2+0.0519x_1{x}_2$
2. At 95% significance level,${\beta }_1\ne {\beta }_2\ne {\beta }_3\ne 0$. Hence the model is not a good fit for the data.
3. At 95% significance level,${\beta }_3\ne 0$. Hence it can be concluded with enough evidence that $x_{1}and{x}_2$ interact in the model.
4. When it is said that advertising expense(x₁) and shelf space allotted to the brand (x₂) interact in the model, it means that the effect of shelf space allotted to the brand on the sales done by the brand depends on the advertising expenses incurred by the brand.
5. By using only first-order model instead of the interaction model the researcher might get biased results as the researcher is ignoring the effect of two or more independent variables combined have on the dependent variable which might be larger than the effects of the independent variables alone.
6. Assumptions of independent error terms –
7. $$\begin{gathered} Corr\left(x_i,\epsilon \right)=0 \\ Cov\text{(}{e}_{u,}{e}_v\text{)}=0 \\ \epsilon ~N(0,{\sigma }^2) \end{gathered}$$

Step-by-step solution

Given Information

It is given that y is the sales of its store brand canned vegetables whereas the amount spent on promotion of the vegetables in local newspapers is(x₁),and the amount of shelf space allocated to the brand is (x₂).To conduct this test one of the chain’s supermarkets was randomly selected for 20-week period.The model is fitted as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2+\epsilon$and one has to conduct F-Test and the presence of interaction between advertising expenditures and shelf space at .

Model for the data 

a.

To fit the model, excel function of data analysis is used, where the given values of y, $x_1,x_{2}and{x}_1,x_2$ are written in the table and the values are used for regression analysis.

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

The SSR is calculated by using $n\Sigma {\left(X_j-\overline{X_j}\right)}^2$, 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 bydividing the standard deviation by the sample size's square root.

The excel output is shown in the excel and the model summary is shown below,

So, the model equation can be written as $\hat{y}=1333.1782-0.1512x_1-2.6253x_2+0.0519x_1{x}_2$

Significance of the model

b.

.$$\begin{gathered} \text{Here}, F-\text{test} \text{statistic}=\frac{\frac{R^2}{k}}{\frac{\left(1-R^2\right)}{\left[n-\left(k+1\right)\right]}} \\ =\frac{568826.1953}{20-4} \\ =35551.63 \end{gathered}$$

Value of $F_{0.05,19,19}$ is 2.114

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

For$\alpha =0.05$ , since$F>F_{0.05,19,19}$ , there is a sufficient evidence to reject H₀ at 95% confidence level.

Therefore, ${\mathbf{\beta }}_{\mathbf{1}}\ne {\mathbf{\beta }}_{\mathbf{2}}\ne {\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{0}$

Hence, the model is not a good fit for the data.

Significance of  β3

c.

$$\begin{gathered} H_0\text{:} {\beta }_3=0 \\ {H}_a\text{:} {\beta }_3\ne 0 \end{gathered}$$

$$\begin{gathered} \text{Here}, t-\text{test} \text{statistic}=\frac{{\hat{\beta }}_3}{s{\hat{\beta }}_3} \\ =\frac{0.0519}{40.00686} \\ =7.5655 \end{gathered}$$

Value of $t_{0.025,19}$ is 2.093

H₀ is rejected if$>t_{0.025,19}$ .

For $\alpha =0.05$, since $t>t_{0.025,19}$, there is a sufficient evidence to reject H₀ at 95% confidence level.

Thereafter,${\mathbf{\beta }}_{\mathbf{3}}\ne \mathbf{0}$ .Hence it can be concluded with enough evidence that x₁ and x₂ interact in the model.

Interaction term explanation

d.

When it is said that advertising expense (x₁) and shelf space allotted to the brand interact in the model, it means that the effect of shelf space allotted to the brand on the sales done by the brand depends on the advertising expenses incurred by the brand(x₂). The effect of one independent variable is depended on that of the other independent variable(s).

Step 6:Significance of interaction model

e.

By using only first-order model instead of the interaction model the researcher might get biased results as the researcher is ignoring the effect of two or more independent variables combined have on the dependent variable which might be larger than the effects of the independent variables alone.

Comment on error term

f.

Assumptions of independent error terms –

$$\begin{gathered} Corr\left(x_i,\epsilon \right)=0 \\ Cov\text{(}{e}_{u,}{e}_v\text{)}=0 \\ \epsilon ~N(0,{\sigma }^2) \end{gathered}$$