Stat 5870 Homework #7: Due 22 Oct, by 11:59 pm.

Conceptual problems:
Chapter 5: 2, 4, 7, 9, 10 10, 11, 12
Chapter 6: 6, 8, 9, 10, 11

Computational problems: (turn in)
Note: For all problems this week, analyze the data as presented. No need to worry about whether or not to transform the observations.

1) You have fit two models to a set of data. The equal means model specifies that all groups have the same mean. The error SS for this model is 70907 with 31 degrees of freedom. The different means model specifies that each group has a different mean. The error SS for this model is 35088 with 24 df.
a) Fill in and complete the ANOVA table.
SourcedfSum of squaresMean SquareF
Between????????
Within????
C. Total????

b) Which model is the reduced model and which is the full model?
c) How many groups were used in this study?
d) How many observations were used in this study?
e) Is the error mean square (MSE) larger or smaller than 1500?

2) This problem evaluates 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 one of the best controlled studies of diet. 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. The response is the kg lost per subject 24 months after the start of the diet for the 272 people who completed the study. The data are in dietstudy.csv.

a) Test the null hypothesis that the mean weight loss is the same in the three diets. Report your test statistic, the numerator and denominator degrees of freedom, the p-value, and a short conclusion.
b) The low-fat and Mediterranean diets both have 1800 kcal/day for men and 1500 kcal/day for women. Use a contrast to test whether these two diets have the same mean weight loss. Report the estimate and p-value.
c) The foods in the low-carb diet are somewhat different from those in 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 and test whether that is zero. Report the estimate and p-value.
Note: JMP users will have to compute the estimate by hand, certainly for 2c and perhaps for 2b.
d) 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?
Note: We haven't talked about this. I want you to think about it. It may help to know that some folks fail to complete a study because they move out of the area or change employers; many folks decide on their own to stop participating, especially when they believe their assigned treatment is not working. What would happen to group averages when individuals with especially low outcomes disappear?

3) Lettuce yield. These data come from a randomized experiment on the response of lettuce to additions of nitrogen fertilizer.  The experiment was conducted on irrigated plots in Arizona. Plots were randomly assigned to one of five fertilizer levels: 0, 50, 100, 150, or 200 lbs N /acre. The response is the number of lettuce heads harvested from each plot. There were four replicates of each treatment. The data are in lettuce.csv


a) Use an F test to test the null hypothesis of no difference among treatments. Report the F statistic and p-value and write a 1 sentence conclusion.
b) Use a linear trend contrast to test the null hypothesis of no linear trend in yield. Report the coefficients you used for your contrast and the p-value.
c) Estimate the mean difference between no fertilizer (0 level) and any fertilizer (the average of the four non-zero levels). Report the coefficients you used for your contrast, the estimate, and p-value for the test of difference = 0.
d) Do these data provide evidence that adding N fertilizer increases lettuce yield? Briefly explain why or why not.