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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鈥檚 supermarkets was randomly selected for 20-week period.The model is fitted as $y={尾}_0+{尾}_1{x}_1+{尾}_2{x}_2+{尾}_3{x}_1{x}_2+蔚$and one has to conduct F-Test and the presence of interaction between advertising expenditures and shelf space at .

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 $n危{\left(X_j-\overline{X_j}\right)}^2$, 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: $\frac{n\left(鈭�xy\right)-\left(鈭�x\right)\left(鈭�y\right)}{n\left(鈭�x_2\right)-\left(鈭�x_2\right)}$, whereas the coefficient of intercept is calculated by$\frac{\left(鈭�y\right)\left(鈭�x_2\right)-\left(鈭�x\right)\left(鈭�xy\right)}{n\left(鈭�x_2\right)-\left(鈭�x_2\right)}$ .

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 $\hat{y}=1333.1782-0.1512x_1-2.6253x_2+0.0519x_1{x}_2$

b.

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

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

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

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

Therefore, ${\mathbf{尾}}_{\mathbf{1}}鈮�{\mathbf{尾}}_{\mathbf{2}}鈮�{\mathbf{尾}}_{\mathbf{3}}鈮�\mathbf{0}$

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

c.

$$\begin{gathered} H_0\text{:}鈥�{尾}_3=0 \\ {H}_a\text{:}鈥�{尾}_3鈮�0 \end{gathered}$$

$$\begin{gathered} \text{Here},鈥�t-\text{test}鈥�\text{statistic}=\frac{{\widehat{尾}}_3}{s{\widehat{尾}}_3} \\ =\frac{0.0519}{40.00686} \\ =7.5655 \end{gathered}$$

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

H₀ is rejected if$>t_{0.025,19}$ .

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

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

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).

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.

f.

Assumptions of independent error terms 鈥�

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