Book Homework (does not require R)
Note: This may be hand written or typed. Answers should be clearly marked. Please put your name in the upper right corner.
1. Recall problem 1 of homework 2 : A rehabilitation center researcher was interested in examining the relationship between physical fitness prior to surgery and the time (in days) for successful completion of physical therapy. The fitness levels were measured as “Below Average” ( group I), “Average” (group II), or “Above Average” (group III). Summary statistics follow:
(a) Using the sample mean values and the value of MSE, calculate the value of φ, the non-central parameter used for calculating power.
(b) Estimate the power of a test where α = 0.01 (use the tables provided online).
(c) Estimate the effect size, ∆/σ, using the sample values and MSE.
(d) If we wanted to have a power of 0.95 using an α of 0.10, what sample size would we need in each group (at least)? Use the tables provided online.
2. Continue with problem 1.
(a) Write down the appropriate ANOVA model if we are interested in how each group differs from the overall mean, and specify all assumptions this model makes.
(b) Estimate all values of γi (gamma).
(c) Interpret the estimated value of γ2, where i = 2 corresponds to group II, in terms of the problem.
(d) Show that the estimated values of γi sum to zero.
(e) If the ANOVA model assumptions hold, what is the distribution of γˆi? Be as specific as you can.
3. Continue with problem 1.
(a) Find the 95% confidence interval for µI , and interpret it in terms of the problem.
(b) Find the 99% confidence interval for µI − µIII , and interpret it in terms of the problem.
(c) The rehabilitation center suggests that average time to recovery for completion of physical therapy for an “Below Average” patient is at least 30 days. Does your confidence interval from (a) agree with that statement? Explain.
(d) What is the largest difference in average recovery time you expect an ”above average” patient to have compare to a “below average” patient? Explain your answer.
4. Recall problem 3 of homework 2: A consumer organization studied the effect of age of automobile owner on cash offer for a used car (in hundreds of dollars). The age groups were “Young” (Y ), “Middle”‘ (M), and “Old” (O). Summary statistics follow:
(a) Interpret the power of the test in terms of the problem.
(b) Using the sample mean values and the value of MSE, calculate the value of φ, the non-central parameter used for calculating power.
(c) Estimate the power of a test where α = 0.01 (use the tables provided online).
(d) Explain why the value calculated in (c) is so high.
5. Continue with problem 4.
(a) Write down the appropriate ANOVA model if we are interested in the group mean of each group, and specify all assumptions this model makes.
(b) Estimate the difference in population means for the M group vs. the Y group.
(c) Estimate the standard deviation for the difference in sample means between group M and group Y .
(d) Assuming the assumptions for ANOVA hold, estimate the probability that an error (ij ) is larger than 2.5.
6. Continue with problem 4.
(a) Find the 95% confidence interval for µM − µY +µO 2 .
(b) Interpret the confidence interval from (a) in terms of the problem.
(c) Find the 95% confidence interval for µO − µY .
(d) Interpret your interval from (c) in terms of the problem.
(e) Using the results of this problem, which means do you believe are significantly different and why?
7. For the following problems, assume that µ1 = µ2 = 13, µ3 = µ4 = 18, σ = 3.5, and ni = 6 for all i.
(a) Calculate the power of the test for equal means when α = 0.05.
(b) Calculate the power of the test for equal means when α = 0.01.
(c) What trend is suggested by (a), (b)?
(d) What sample size would we need for the power to be approximately 0.80? Assume α = 0.05.
(e) What sample size would we need for the power to be approximately 0.90? Assume α = 0.05. (f) What trend is suggested by (d), (e)? 8. Answer the following questions with TRUE or FALSE. It is good practice to explain your answers. (a) As α increases, the power of a test tends to decrease. (b) If a confidence interval for some µi − µi‘ does not contain zero at (1 − α)100% confidence, this suggests we would reject the null hypothesis of single factor ANOVA. (c) As the sample size per group increases, the F teststatistic for single factor ANOVA tends to decrease. (d) For a (1−α)100% confidence interval for µi , it is appropriate to say the probability that µi is in the interval is 1 − α.
R Homework (requires some use of R)
Note: You do not have to use R Markdown to turn in the homework, but the homework must be turned in in a reasonable format. The answers to the questions should be in the body of the homework, and the code used to obtain those answers should be in an appendix. There should be no code in the body of the homework. You can accomplish this in R, Word, LaTex, Google Docs, etc.
I. Recall problem I from homework 2: Online you will find the file “Cancer.csv”. The csv file has the following columns: Column 1. Survival: The survival time of the patient in days Column 2. Organ: The organ where cancer was present - Stomach, Bronchus, Colon, Ovary, Breast Data Source : From the article ”Supplemental Ascorbate in the Supportive Treatment of Cancer: Reevaluation of Prolongation of Survival Times in Terminal Human Cancer” by Ewan Cameron and Linus Pauling, Proceedings of the National Academy of Sciences of the United States of America, Vol. 75, No. 9 (Sep., 1978), pp. 4538-4542. (a) Estimate the power of the test using th
(a) Estimate the power of the test using the sample values, and assuming α = 0.05.
(b) If we wanted the power to be 0.99 using α = 0.05, what is the smallest sample size we would need for each group?
(c) If we wanted the power to be 0.99 using α = 0.01, what is the smallest sample size we would need for each group?
(d) If the calculated value of FS is very large, do we expect the power of a test to be large, or small? Explain your answer.
II. Continue with problem I
(a) Find the 95% confidence interval for µStomach − µBronchus.
(b) Does this interval suggest a significant difference in the means? Explain your answer.
(c) Find the 95% confidence interval for µColon − µStomach+µBronchus 2
(d) Do your results suggest a significant difference between the survival time for Colon, Stomach, and Bronchus cancers? Explain your answer.
III. Online you will find the file “Green.csv”. The csv file has the following columns:
Column 1. weight: The dry weight of the plant Column 2. group: What group the plant was in, with levels ctrol (Control), trt1 (Treatment 1), and trt2 (Treatment 2) Data Source: Dobson, A. J. (1983) An Introduction to Statistical Modelling. London: Chapman and Ha
(a) Find the p-value for the test of equal means, and state your conclusion in terms of the problem if α = 0.10.
(b) Find the estimated values of γi , the factor effects.
(c) Interpret the power in terms of the problem (do not calculate it, yet!).
(d) Interpret the value of 1 −α in terms of the problem.
IV. Continue with problem III
(a) Estimate the power, using the sample values. You may assume α = 0.10.
(b) If we wanted the power to be 0.99 using α = 0.05, what is the smallest sample size we would need for each group?
(c) If we wanted to increase the power of the test, what is one method we could use to do that?
(d) Find a 95% confidence interval to compare the average of means of the control and treatment 1 group to the average of the treatment 2 group.
(e) Interpret the interval from (d) in terms of the problem.
