# STAT/ME 424 HW 5 solution

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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 0 to 100).
i
j 1 2 3 4 5 6 7 8 9 10 11 12
1 Low 7.6 8.2 6.8 5.8 6.9 6.6 6.3 7.7 6.0
2 Moderate 6.7 8.1 9.4 8.6 7.8 7.7 8.9 7.9 8.3 8.7 7.1 8.4
3 High 8.5 9.7 10.1 7.8 9.6 9.5

1. Write down a model appropriate for the data.

2. Let µj denote the true mean productivity improvement for the jth level of
expenditure. Obtain the values for the estimates ˆµj
, j = 1, 2, 3.

3. Obtain the residuals and plot them against the ˆµj

4. Make a normal quantile plot of the residuals. Does the normality assumption
appear to be reasonable here?

5. The economist wishes to investigate whether location of the firm’s home office
is related to productivity improvement. The home office locations are as follows
(U=U.S.; E=Europe):
i
j 1 2 3 4 5 6 7 8 9 10 11 12
1 Low U E E E E U U U U
2 Moderate E E E E U U U U U E E E
3 High E U E U U E

Make side-by-side boxplots of residuals by location of home office. Does it
appear that the ANOVA model could be improved by adding location of home
office as a second factor? Explain.

6. Obtain the ANOVA table without using location of home office as a second
factor.

7. Test whether or not the mean productivity improvement differs according to
the level of research and development expenditures. Use a significance level of
α = 0.05. State your conclusion.

8. What is the significance probability (p-value) of the preceding test?

9. What appears to be the nature of the relationship between research and development expenditures and productivity improvement?

10. Estimate the mean productivity improvement for firms with high research and
development expenditure levels with a 95% confidence interval.

11. Obtain a 95% confidence interval for µ2 − µ1. Interpret your interval estimate.

12. Obtain confidence intervals for all pairwise comparisons of the treatment means;
use the Tukey procedure and a 90% simultaneous confidence level. State your
findings.

13. Is the Tukey procedure employed in the preceding question the most efficient
one that could be used here? Explain.

14. Obtain a 95% confidence interval for (µ1 + µ2)/2 − µ3, the difference in mean
productivity improvement between firms with low or moderate research and
development expenditures and firms with high expenditures. Interpret your
interval estimate.

15. The sample sizes for the three treatment levels are proportional to the population sizes. The economist wishes to estimate the mean productivity gain last
year for all firms in the population. Find a 95% confidence interval for it.

16. Using the Scheff´e method, obtain 90% simultaneous confidence intervals for
these 4 contrasts:
µ3 − µ2, µ2 − µ1,
µ3 − µ1, (µ1 + µ2)/2 − µ3.
What do you conclude?

17. Would the Bonferroni method be more powerful than the Scheff´e method for
the previous question? Explain.