This week’s assignment requires that you demonstrate the relationship between a dependent variable and multiple independent variables.  You will be required to use the least squares approach as well a

TIM-7100

DATA FILE 7

ASSIGNMENT 8

1. Suppose you fit the multiple regression model

y = β0 + β1x1 + β2x2 + ϵ

to n = 30 data points and obtain the following result:

The estimated standard errors of and are 1.86 and .29, respectively.

1. Test the null hypothesis H0: β2 = 0 against the alternative hypothesis Ha: β2 ≠0. Use α = .05.

2. Test the null hypothesis H0: β3 = 0 against the alternative hypothesis Ha: β3 ≠0. Use α = .05.

3. The null hypothesis H0: β2 = 0 is not rejected. In contrast, the null hypothesis H0: β3 = 0 is rejected. Explain how this can happen even though >

1. Use SPSS to fit a second-order model to the following data:

1. 1. Find SSE and s2.

   2. Do the data provide sufficient evidence to indicate that the second-order term provides information for the prediction of y? [Hint: Test H0: β2 = 0].

   3. State the least squares prediction equation.

1. Running a manufacturing operation efficiently requires knowledge of the time it takes employees to manufacture the product, otherwise the cost of making the product cannot be determined. Estimates of production time are frequently obtained using time studies. The data in the table below came from a recent time study of a sample of 15 employees performing a particular task on an automobile assembly line.

1. 1. Run the multiple linear regression model in SPSS. State the least squares prediction equation.

   2. Test the null hypothesis H0: β2 = 0 against the alternative Ha: β2 ≠0. Use α = .01. Does the quadratic term make an important contribution to the model?

   3. Your conclusion in part b should have been to drop the quadratic term from the model. Do so and fit the “reduced model” y = β0 + β1x + ϵ to the data.

   4. Define β1 in the context of this exercise. Find a 90% confidence interval for β1 in the reduced model of part c.

1. Suppose you fit the model y = β0 + β1x1 + β2x2 + β3x1x2 + β4x12+ β5x22ϵ to n = 30 data points and get SSE = .46 and R2 = .87.

   1. Do the values of SSE and R2 suggest that the model provides a good fit to the data? Explain.

   2. Is the model of any use in predicting y? Test the null hypothesis that E(y) = β0; that is, test

H0: β1 = β2 = β3 = β4 = β5 = 0

against the alternative hypothesis

Ha: At least one of the parameters β1, β2, … β5 is nonzero.

Use α = .05.

1. A company that services copy machines is interested in developing a regression model that will assist in personnel planning. It needs a model that describes the relationship between the time spent on a preventive maintenance service call to a customer, y, and two independent variables: the number of copy machines to be serviced, x1, and the service person’s experience in preventive maintenance, x2.

1. 1. Fit the model y = β0 + β1x1 + β2x2 + ϵ to the data.

   2. Investigate whether the model is useful. Test α = .10.

   3. Find R2 for the fitted model. Interpret your results.

   4. Fit the model y = β0 + β1x1 + β2x2 + β3x1x2 + ϵ to the data.

   5. Find R2 for the model in part d.

   6. Explain why you should not rely solely on a comparison of the two R2 values for drawing conclusions about which model is more useful for predicting y.

   7. Do the data provide sufficient evidence to indicate that the interaction term, x1x2 contributes information for the prediction of y? [Hint: Test H0: β3 = 0] Which model is more useful for predicting y?

   8. Can you be certain that the model you selected in part g is the best model to use in predicting maintenance time? Explain.