Refer to Questionnaire color Problem 16.8. It has been suggested to the investigator that size of parking lot might be a useful concomitant variable. The number of spaces (Xij ) in each parking lot utilized in the study follow.
a. Obtain the residuals for covariance model (22.3).
b. For each treatment, plot the residuals against the fitted values. Also prepare a normal probability plot of the residuals and calculate the coefficient of correlation between the ordered residuals and their expected values under normality. What do you conclude from your analysis?
c. State the generalized regression model to be employed for testing whether or not the treatment regression lines have the same slope. Conduct this test using a = .005. State the alternatives, decision rule. and conclusion. What is the P-value of the test?
d. could you conduct a formal test here as to whether the regression functions are linear?
Explain.
Problem 16.8
Refer to Productivity improvement Problems 16.7 and 22.7. Assume that convince model (22.3) is appropriate.
a. Prepare a symbolic scatter plot of the data. Does it appear that there are effects of the level of research and development expenditures on mean productivity improvement? Discuss?
b. State the regression model equivalent to covariance model (22.3) for this case. use 1, -1 .indicator variables. Also state the reduced regression model for testing for treatment effects.
c. Fit the full and reduced regression models and test for treatment effects; Use a = .05 the alternatives, decision rule, and conclusion. What is the P-value of the test?'
d. Is MSE( F) for the covariance model substantially smaller than MSE for the analysis of variance model in Problem 16.7d? Does this affect the conclusion reached about treatment effects? Does it affect the P-value?
e. Estimate the mean productivity improvement for firms with moderate research and development
expenditures that had a prior productivity improvement of X = 9.0; use a 95 percent confidence interval.
f. Make all pairwise comparisons between the treatment effects; use either the Bonferroni or the Scheffe procedure with a 90 percent family confidence coefficient, whichever is more efficient. State your findings.
Problems 22.7
Refer to Productivity improvement Problem 16.7. The economist also has information on annual productivity improvement in the prior year and wishes to use this information as a concomitant variable. The data on the prior year's productivity improvement (Xij) follow.
a. Obtain the residuals for covariance model (22.3).
b. For each treatment, plot the residuals against the fitted values. Also prepare a normal probability plot of the residuals and calculate the coefficient of correlation between the ordered residuals and their expected values under normality. What do you conclude from your analysis?
c. State the generalized regression model to be employed for testing whether or not the treatment regression lines have the same slope. Conduct this test using α = .01. State the alternatives, decision rule, and conclusion. What is the P-value of the test?
d. Could you conduct a formal test here as to whether the regression functions are linear? If so, how many degrees of freedom are there for the denominator mean square in the test statistic?
Problem 16.7
Productivity improvement. An economist compiled data on productivity improvements last year for a sample of firms producing electronic computing equipment. The firms were classified according to the level of their average expenditures for research and development in the past three years (low, moderate, high). The results of the study follow (productivity improvement is measured on a scale from a to 100). Assume that ANOVA model (16.2) is appropriate.
a. Prepare aligned dot plots of the data. Do the factor level means appear to differ? Does the variability of the observations within each factor level appear to be approximately the same for all factor levels?
b. Obtain the fitted values.
c. Obtain the residuals. Do they sum to zero in accord with (16.21)?
d. Obtain the analysis of variance table.
e. Test whether or not the mean productivity improvement differs according to the level of research and development expenditures. Control the α risk at .05. State the alternatives, decision rule, and conclusion.
f. What is the P-value of the test in part (e)? How does it support the conclusion reached in part (e)? g. What appears to be the nature of the relationship between research and development expenditures and productivity improvement?