1. HW 24 Key

2. 25:39 Emerald Diamonds. This data table of 144 diamonds includes the price (in dollars), the weight (in carats), and the clarity grade of diamonds. The diamonds have clarity grade either VS1 or VVS1. VVS1 diamonds are nearly flawless; VS1 diamonds have more visible (but still small) flaws.

3. 25:39 a a.Would it be appropriate to use a two-sample t-test to compare the average prices of VS1 and VVS1 diamonds, or is this relationship confounded by the weights of the diamonds?

4. 25:39 a a.The two categories have similar average weight. Weight is unlikely to confound this analysis.

5. 25:39 b b. Perform the two-sample t-test to compare the prices of the two grades of diamonds. Summarize this analysis as if there are no lurking variables. Do you get the sort of difference that would be expected from the definitions of the categories?

6. 25:39 b b. There is a difference ($112) between the price means for VS1 vs VVS1 diamonds. This is expected as VVS1 is a higher quality.

7. 25:39 c c. Compare the prices of the two types of diamonds using an analysis of covariance. Summarize the comparison of prices based on this analysis. Use a dummy variable coded as 1 for VVS1 diamonds and 0 otherwise. (Assume for the moment that the model meets the conditions for the MRM.)

8. 25:39 c c.

9. 25:39 c c. Because the interaction is not statistically significant, remove it and refit the model. Given comparable weight, diamonds with clarity VVS1 average $127 more than those of VS1 clarity.

10. 25:39 d d. Compare the results from parts b and c. What can you conclude about the cost of diamonds of these two grades? You should take into account the precision of the estimates and your answer to part a. From the analysis of covariance, we see that when controlling for weight, grade of diamond still has a statistically significant affect on price. This matches with the two-sample t-test, which shows a difference between the price means of the two grades. So while weight is a confounding variable as mentioned in part a, it does not explain all of the variation in price.

11. 25:39 d d. The CI for the regression is narrower because the model removes the variation due to weight. There’s no confounding because the weights are comparable.

12. 25:39 e e. What problem bedevils the multiple regression used for the analysis of covariance that is not present in the two-sample t-test? Variances increase with price. Equal variance of the regression plot (big time funnel shape to the right “<”); it is heteroskedastic.

13. 25:41 Download

14. 25:41 a a.Appropriate to compare or would it be confounded? File sizes are paired and are not a confounding variable.

15. 25:41 b b. Two-sample t test to compare software performance of the two vendors.

16. 25:41 b b. A two sample t test finds a statistically significant difference in the performance of the software. Software labeled “MS” on average transfers files about 5.5 fewer seconds.

17. 25:41 c c. Compare download times using analysis of covariance.

18. 25:41 c c. The interaction is statistically significant: the two types of software have different transfer rates. Transfers using MS progressively take less time than NP. The small difference in the intercept occurs because of the interaction.

19. 25:41 d d. Compare b and c. Why do they agree/differ? The two sample comparison finds an average difference of 5.5 seconds (1 to 10 seconds), with MS faster. The analysis of covariance also identifies MS as faster, but shows that the gap is wider as the size increases.

20. 25:41 e e. Does the analysis of covariance meet similar variances condition? No. Hints of a problem are evident in a color coded plot of residuals on fitted values. Boxplots of residuals show different variances.