Treatment
Blocks		1	2	3	4
  1		8.0	3.5	5.3	5.2
  2		10.5	5.7	10.1	7.4
  3		10.8	8.0	8.4	6.9

A) Conduct the appropriate nonparametric test of the null hypothesis that the effects of the treatments are the equal. What is your conclusion?

B) Now conduct the appropriate nonparametric test if there was no blocking variable (just 3 people assigned to each treatment). What would th e conclusion be?

C) Why does it make sense that the test in A had a smaller p-value than the test in B?

D) Now imagine that the experimenter didn't realize the data was supposed to be in blocks and it got shuffled up. He decides that since the blocked design gives a smaller p-value that he'll make up some blocks and use those.

		Treatment
Blocks		1	2	3	4
  1		10.8    3.5     10.1    6.9
  2		10.5    5.7     5.3     5.2
  3		8.0     8.0     8.4     7.4

E) Based on the p-values for the three tests you've done, what conclusion can you reach about whether blocking is helpful or not?

2) Construct a data set of five points that has a Spearman's rho and a Kendall's tau of 1, but would have a correlation coefficient that is considerably smaller than 1. (A bonus point to the person with the smallest value).

2) Consider problem 4 on page 365.
a) State the appropriate null and alternate hypothesis.
b) Perform the Wilcoxon signed rank test at alpha=0.05 for your hypothesis in a, and state your conclusion in terms of the problem.
c) For each assumption, either check it, or say why it can't be checked.
d) Also perform the paired t-test and the quantile test (for p=0.5). Explain whether or not the relationship between the p-values for the three tests you've done seem to make sense based on what the AREs are and the shape of the distribution.
e) Construct a (95%) CI based on the Wilcoxon signed rank test. Give two reasons why you should be suprised that you could reject your null hypothesis in b, but that the CI includes 0, even though you told it to use alpha=0.05 for both.

3) A simulation is run to compare the power of one-way ANOVA and the KW test when the data is heavy tailed (t with df=7) and from a small sample size (n=10). Ten thousand data sets are simulated where the actual difference in the means is 1. In 4,149 cases both tests reject the null; in 289 just the t-test rejects; and in 436 cases just the KW test rejects. What do you estimate the power of each test to be, and are these values statistically significantly different.

Page 93 #7 (show all of your work)

Page 113 #2. For part c, find the power when p is really 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. (Hint: The binomial distribution will give you the probabilities you need. dbinom(x,size=n,prob=p) in R will give you P[X=x] for a binomial with parameters n and p.

1) If the jury was selected at random, is the number of women selected a binomial random variable, or a hypergeometric random variable?
2) If the defense attorney assumed this was a binomial experiment, what would they get for: the probability that the number of women selected would be 12, the expected number of women selected, and the standard deviation of the number of women selected.
3) If the defense attorney assumed this was a hypergeometric experiment, what would they get for: the probability that the number of women selected would be 12, the expected number of women selected, and the standard deviaition of the number of women selected.

You will probably want a calculator or computer program for calculating (e.g. excel or R) for this problem. Note that the function choose(n,k) in R will give you the binomial coefficients, e.g. 8!/(6!2!) is 8 choose 6 is choose(8,6) = 28. A ^ can be used to get powers, e.g. 2³ is 2^3 = 8. A * is used for multiplication so that 2 times 4 is 2*4 = 8.

• Pg. 12 #2, also, what if each letter is used only once?
• Pg. 21 #6
• Pg. 33 #2a
• Pg. 63 #4
• Consider flipping two fair coins. Let A={1st coin heads}, B={2nd coin heads}, C={both coins the same}. Show that each pair A-B, B-C, and A-C are independent, but that the three together are not mutually independent.

It is desired to see if the mean score for all such students on the analysis exam differ from the mean score on the algebra exam. State the appropriate hypotheses and assumptions for conducting a paired t-test. Use both SAS and R to produce the results of the hypothesis test (at alpha=0.05) and any plots needed to check the assumptions. Be sure to include the code used and any menu options. Briefly summarize your findings.