## Information Pertaining to the Final Exam

The final exam is an open book(s) / open notes exam --- you can use whatever books and other printed or written material that you want to bring with you. I recommend that you also bring a basic scientific calculator. You will not be allowed to use a computer. The final exam is on Wednesday, Dec. 14, and goes from 7:30 PM until 10:15 PM. You are expected to take the final exam during this designated time period. (The exam may not be ready in time for anyone to take it early, and I don't expect that it will be easy for you to convince me for you to take it late.)

The exam will be worth 50 points, since it accounts for 50% of your final grade. There will be 10 questions of the True/False or Multiple Choice type. These will be worth 2 points each, or 20 points in all. Also there will be seven problems where you are expected to show your work and adequately justify your answers. These will be worth 6 points each, and I'll only count 5 of the 7, making this portion of the exam worth 30 points in all.

### Extra Office Hours

I'll be available for extra office hours on Sunday (12/9) from 3:00 to 4:00 PM and on Wednesday (12/12) from 5:00 to 6:00 PM.

### Study Guide

Eight of the ten True/False and multiple choice questions will deal with tests for the general two-sample problem (so Chapters 6 through 9). The other two questions in this portion of the exam will be based on Ch. 5 and Ch. 10 material. To study for this portion of the exam, I recommend focusing on the class notes I supplied as opposed to the text book.

Here are some specific things to study.
1. Given observations of iid continuous random variables, which of the one-sample and paired-samples tests are always valid for tests about distribution medians, tests about distribution means, and tests for a treatment effect? If we know that the underlying distribution is symmetric, how does that affect the validity? Given observations of independent continuous random variables, what is needed for the two-sample tests we studied to be valid for the general two-sample problem? If we can assume that the two distributions are identical or else one is stochastically larger than the other, do we have validity for tests about means? What about medians? What can be said about these issues for k-sample tests?
2. Without worrying about how to obtain the ARE values given on the top of p. 5-31 of the class notes, know how to extend the information given there by using those values to obtain the ARE of the t test w.r.t. the signed-rank test if the underlying distribution is logisitic, and the ARE of the signed-rank test w.r.t. the normal scores test if the underlying distribution is normal. (The key idea is that one can use ratios of two AREs to obtain another ARE.)
3. Understand how to take an ARE value and use it to figure out what sample size is needed for a test to have power characteristics approximately similar to another test with a given sample size for the other test. As an example, do Exercise 23 just using that the ARE of the sign test w.r.t. the z test in the normal setting is 2/π.
4. For a test of the general two-sample problem, know what tests are considered to be omnibus tests, and know what is meant by the term omnibus test. What tests are primarily sensitive to location differences and have poor power if the medians are nearly equal and the primary difference in the distributions is in the dispersion of the probability mass about the central values of the distributions? What tests are primarily sensitive to dispersion differences and have poor power if the dispersion characteristics are similar and the primary difference in the distributions is in the location of the bulk of the probability mass? What happens with these tests if the distributions differ greatly in both dispersion of probability mass and location of probability mass? (Some pertinent information is given on pages 9-2 and 9-3 of the course notes.)
5. Know how to apply the formulas given on p. 7-5 of the class notes to find the null mean and null variance of a linear rank statistic given the scores. (Two examples are given on p. 7-5, and the null variance of a percentile modified rank test statistic is worked out on p. 8-12.)
6. Understand what makes two linear rank tests equivalent or asymptotically equivalent. (An example of two equivalent tests are the Mann-Whitney test and the Wilcoxon rank-sum test. (Although the text doesn't have the Mann-Whitney test in the chapters about linear rank tests, I argued in class (see bottom part of p. 7-5 of the class notes) that it was one.) See the first new paragraph on p. 8-7 for the relationship between the scores used with the two tests, and comments about how the null distributions differ. Note that if you take any set of scores and add a constant to each one to obtain a new test, the new test will be equivalent to the test you got the scores from. Also, you can multiply scores by a constant and obtain an equivalent test. (As another example of equivalent tests, consider the rank-sum test and the percentile modified rank test for location differences if N is odd and k is (N - 1)/2.) An example of asymptotically equivalent tests are the Terry normal scores test (for two samples) and the similar test using van der Waerden scores (which are just approximations of the Terry test scores).

Five of the seven problems where you are to show your work are based on Ch. 10. One of them will be based on Ch. 4, and the other one will be based on either Ch. 11 or Ch. 12. You can try all of the problems, but only five of the scores will contribute towards your final score. I'll count your best 4 of 5 scores for the Ch. 10 problems, and I'll count your better score from the other two problems.

Here are some specific things to study. If you have a good understanding of the things indicated below, I don't think you should have any problem with the majority of the problems on the final exam. (Notes: (i) The exam won't have problems similar to all of the things listed below, but the exam's problems will be strongly based on a subset of these things. (ii) The Exercises referred to below refer to Exercises on the homework web page. The problem that will be based on either Ch. 11 or Ch. 12 will be a straightforward application of one of the major tests (from the text) that I discussed in class.
1. Understand the n = 2 example on this web page. (Note: This was something I created in 2008 when I noticed problems with the K-S test in a previous version of StatXact. The important thing for you to focus on is to be able to obtain the exact null sampling distribution of T in the n = 2 case for the simple discrete uniform distribution setting considered near the middleof the web page.)
2. Problem 10.1 on p. 378 of G&C. The main task involves generating the null distribution of Q. First determine all possible tables, and obtain the null probability for each one of them. If you decide to reject if you obtain one of the least likely tables, what is the size of the test? (You'll find that more than one table is tied with being the least likely. If we reject if one of the least-likely tables results from the observed data, the size of the test will be the sum of thes least-likely probabilities (the smallest probability for a table times the number of tables with this probability).) From all of the possible tables, obtain the value of the test statistic. (More than one possible table will result in the same value of the test statistic.) You should get a really simple null distribution for Q (since Q doesn't take on many different values). If you decide to reject if the largest possible value of Q is observed, what is size of the test? (You'll get the same size as you did using the other viewpoint, since the tables that are the least likely under the null hypothesis are exactly those that yield the largest possible value of the test statistic. I guess the point of the part of the problem other than obtaining the null distribution of Q is to note that the tables which are the least likely under the null hypothesis are the ones that yield the largest value for the test statistic.)
3. Problem 10.2 on p. 379 of G&C. Also determine the exact null distribution if the smallest two of the four data values are the same value (so that the midranks used are 1.5, 1.5, 3, and 4). Similarly, for the same sample sizes, determine the null distribution of the test statistic of the k-sample percentile modified rank test based on the scores -1, 0, 0, and 1. (To get practice working with midranks, obtain the exact null distribution of the Wilcoxon rank-sum statistic given the samples (with two tied values) on p. 8-4 of the course notes.)
4. Think about Problem 10.8 on p. 379 of G&C.
5. Consider Problem 10.10 on pp. 379-380 of G&C. Try to work out (2).
6. Obtain the three approximate p-values given for data sets 8 and 9 on p. 10-23 of the class notes.
7. Obtain the three exact p-values given for data sets 7 and 19 on p. 10-25 of the class notes.
8. Be able to do Exercise 35. (If there is something like this on the final exam, it'll involve an easier to deal with data set.)
9. Be able to do Exercise 36. (If there is something like this on the final exam, it'll involve an easier to deal with data set.)
10. Since, once you lump all of the treatment samples together, there is little difference between the C-D test and the two-sample control median test, practice working with the exact null sampling distribution of the C-D test by doing part (i) of Exercise 26.