Cincinnati Paint Company sells quality brands of paints through hardware stores throughout the United States. The company maintains a large sales force whose job it is to call on existing customers as well as look for new business. The national sales manager is investigating the relationship between the number of sales calls made and the miles driven by the sales representative. Also, do the sales representatives who drive the most miles and make the most calls necessarily earn the most in sales commissions? To investigate, the vice president of sales selected a sample of 25 sales representatives and determined:

%u2022 The amount earned in commissions last month (Y).

%u2022 The number of miles driven last month (X2)

%u2022 The number of sales calls made last month (X1)

($000) Calls Driven Commissions

($000) Calls Driven

22 139 2,371 38 146 3,290

13 132 2,226 44 144 3,103

33 144 2,731 29 147 2,122

38 142 3,351 38 144 2,791

23 142 2,289 37 149 3,209

47 142 3,449 14 131 2,287

29 138 3,114 34 144 2,848

38 139 3,342 25 132 2,690

41 144 2,842 27 132 2,933

32 134 2,625 25 127 2,671

20 135 2,121 43 154 2,988

13 137 2,219 34 147 2,829

47 146 3,463

The objective is to test if the number of miles driven and the number of calls made affects the amount earned in sales commissions in one month by a sales representative.

Y: The amount earned in commissions last month

X₁: The number of sales calls made last month

X₂: The number of miles driven last month

Develop a regression equation including an interaction term. Is there a significant interaction between the number of sales calls and the miles driven?

The linear regression model with more than one explanatory variable is:

E(Y)= α + β₁X₁ + β₂X₂ +...+ βkXk where k= number of explanatory variables.

For this sample of 25 sales representatives you have two explanatory variables so your model will be:

E(Y)= α + β₁X₁ + β₂X₂

To estimate the regression plane you have to estimate the intercept (α) and both slopes (β₁ and ₂)

α⇒ a= -101.31

β₁⇒ b₁= 0.63

β₂⇒ b₂= 15.69

I've run the data using a statics software to estimate the parameters.

The estimated regression plane is:

Yi= -101.31 + 0.63X₁ + 15.69X₂

Where

$ -101.31 is the estimated average commission's earnings of the representatives when the miles were driven and the number of sales calls is zero.

0.63calls/$ is the modification of the estimated average commission's earnings of the representatives when the number of calls increases one unit and the miles are driven remain constant.

15.69miles/$ is the modification of the estimated average commission's earnings of the representatives when the miles driven increases one unit and the number of sales calls remains constant.

The coefficient of determination for the multiple linear regression is:

R²= 0.84

This means that 84% of the variability of the commission's earnings of the sales representatives are explained by the miles driven and the number of sales calls made per month, for the estimated model Yi= -101.31 + 0.63X₁ + 15.69X₂.

The coefficient of determination takes values from 0% to 100%, the closer it is to 100% the stronger is the relationship between the variables. So judging by the value obtained, the regression seems strong, however, this coefficient is a sample measurement and isn't enough to conclude there is a significant interaction between the variables. To reach that kind of conclusion you need to make a hypothesis test.

The statistical hypotheses for the multiple linear regression are:

H₀: β₁ = β₂ = 0

H₁: One of the βi ≠ 0 ∀i= 1, 2

α: 0.05

The way to test these hypotheses is by applying an ANOVA for the variables. The test statistic is the Snedecor's F and it is always one-tailed to the right, meaning that you will reject the null hypothesis to high values of F.

[tex]F= \frac{MSReg}{MSError} ~F_{DFreg;DFerror}[/tex]

[tex]F= \frac{1033.96}{18.54}= 55.78[/tex]

The p-value of the test is <0.0001

When you compare it to the significance level, the decision is to reject the null hypothesis. Then there is significant interaction between the variables.

I hope it helps!