1) The data in dietstudy.csv come from a carefully controlled study of the long-term effectiveness of three diet plans. Most studies of long-term diet effectiveness are flawed by dropout (subjects leaving the study) and poor compliance (subjects not sticking with their assigned diet). These data came from a study where the main meal of the day was provided by the employer, so there was (relatively) little dropout and good compliance.

Here's a summary of the study:

322 overweight employees were randomly assigned to one of three diet plans: low fat, low carb, and mediterranean. Details of the diets are not important. 272 people completed the study.The response is the kg lost per subject 24 months after the start of the diet.

a) (2 pts) Test the null hypothesis that the mean weight loss is the same in the three diets. Report your test statistic, the p-value, and a short conclusion.

Your answer:

b) (2 pts) The low-fat and Mediterranean diets both have 1800 kcal/day for men and 1500 kcal/day for women. Use a linear contrast to evaluate whether these two diets have the same mean weight loss. Report the estimate and a 95% confidence interval.

Your answer:

c) (2 pts) The low-carb diet is somewhat different from the low-fat and Mediterranean diets. Estimate the difference in mean weight loss between the low-carb diet and the average of the low-fat and Med. diets. Report the estimate, the T statistic, and p-value.

Your answer:

d) (2 pts) Compared to other long-term diet studies, this study had a very low dropout rate. Even so, 50 of the initial 322 subjects did not complete the study. Why might dropout be a concern?(Hint: think about the Osborne chapter and how some of the concepts discussed there apply to this study).

Your answer:

2) The data in Trex.csv come from a study of two Oxygen isotopes in bones of a single dinosaur skeleton. The ratio of the two isotopes depends on the temperature of the bone when the Calcium Phosphate was laid down.So, if T rex was homeothermic (warm blooded), the temperature should be relatively similar throughout the body and the isotope ratio should be relatively similar. If T rex was poikilothermic (cold blooded), the isotopic ratio should vary more between the bones. The data are multiple measurements on small pieces of 12 bones from a single T rex skeleton.

a)(2 pts)Test the null hypothesis that all 12 bones have the same mean isotopic composition. Report your test statistic and p-value, and write a one-sentence conclusion.

Your answer:

b) (1 pt) Imagine that the investigators were specifically interested in the difference between bone 4 (dorsal vertebrae 1) and bone 11 (mid-caudal). Test whether this difference = 0. Consider this a pre-planned comparison. Report the appropriate p-value.

Your answer:

c) (1 pt) Actually, the contrast between bones 4 and 11 isn't a pre-planned question, so you should use a multiple comparisons adjustment.What multiple comparisons procedure is appropriate when you look at all pairwise differences of means?

Your answer:

d) (1 pt) Use the appropriate multiple comparisons procedure.What is the adjusted p-value for the comparison of bone 4 and 11?

Your answer:

e) (1 pt) Plot residuals vs predicted values for the 12 bones. Your answer is the plot.

Your answer:

f)(2 pts)Use the residual vs predicted value plot to assess the assumptions of equal variance and normality. Are these assumptions reasonable? Briefly explain your answer.

Your answer:

g) (2 pt) Would it be appropriate to log transform the oxygen isotope ratio before doing the analysis?Briefly explain why or why not.You do not need to repeat the analysis.

Your answer:

h)(2 pts)Deliberately mis-analyze the data by treating bone number as a continuous variable (not a categorical variable). Look at the ANOVA table for the "continuous bone" analysis. There is one number in this table that indicates you have done the wrong analysis. What is that number and why is it wrong?