3. (30 points total, 6 points each) A testing laboratory studies and compares th

Question

3. (30 points total, 6 points each) A testing laboratory studies and compares the relationship between tire tread wear per 1000 miles (y) and average driving speed (%) for two competing tire types (A and B). a. Define an indicator variable x2 to represent the tire type. b. Give the regression model using x, and the indicator variable as predictors. Include an interaction term. c. Give the estimated regression equation for each tire type Predictor Coef T P Constant -1.497.28 .000 X 0.03 10.81.000 X2 .08 4.72 .000 X1X2 -.02 5.5 .000 d. Estimate the change in mean tire tread as driving speed increases by one mph for type A. Do the same for type B. These numbers should not be the same because there is interaction. e. Test for interaction between tire type and average driving speed at = .05. Give the hypotheses, test statistic, p-value, and conclusion.

Explanation / Answer

a.
Let the indicator variable x2 = 0 for tire type A and x2 = 1 tire type B

b.
The regression model using x1 and indicator variable as predictors is
y = a0 + a1 x1 + a2 x2 + a3 x1x2

c.
The estimated regression equation is,
y = -1.49 + 0.03 x1 + 1.08 x2 - 0.02 x1x2

For tire A, x2 = 0

y = -1.49 + 0.03 x1 + 1.08 * 0 - 0.02 x1 * 0

=> y = -1.49 + 0.03 x1

For tire B, x2 = 1

y = -1.49 + 0.03 x1 + 1.08 * 1 - 0.02 x1 * 1

=> y = (-1.49 + 1.08) + (0.03 - 0.02) x1

=> y = -0.41 + 0.01 x1

d.

For type A, the regression equation is,

y = -1.49 + 0.03 x1

The slope of the line is 0.03. So, the change in mean tire tread is 0.03 as driving speed increases by one mph for type A.

For type B, the regression equation is,

y = -0.41 + 0.01 x1

The slope of the line is 0.01. So, the change in mean tire tread is 0.01 as driving speed increases by one mph for type B.

Null Hypothesis H0: The coeffcient of interaction in the model is 0 and is not significant.

Alternative Hypothesis H1: The coeffcient of interaction in the model is not 0 and is significant.

The coefficient of interaction term is -0.02 and the test statistic is -5.5.

P-value for the coefficient of interaction term is 0.000. As, p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the coeffcient of interaction in the model is not 0 and is significant. So, there is significant interaction between tire type and average driving speed.

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