P553 Statistics, Prof. Kruschke, Homework 12

## Homework 12: Review Due Thursday 18 December 2008 at the final exam (12:30pm).

Do these exercises by hand, except the t-tests, for which you may use SPSS if you like. On the other hand, you'll need to do t-tests by hand on the final exam, so it's good to get the practice now!

Because you will not be getting any corrective feedback on this homework before the exam, you are welcome and invited to discuss it thoroughly on the web forum. If you like, you can even post candidate answers and get verification (or not!) from other people.

1. (6 pts)
(A) We flip a coin 5 times and find that it comes up heads 3 times. Test the hypothesis that the coin is fair. What assumptions have you made?
(B) Now the coin is flipped 500 times and comes up heads 300 times. Test whether the coin is fair.
(C) If we have an alternative hypothesis that the probability of heads is .6, what are the powers of the tests (for N=5 and N=500)?

2. (6 pts) Consumers of cake mixes were observed in rural and urban areas, and the frequencies with which three brands were purchased are shown below:

```                Duncan Hines      Pillsbury       Betty Crocker
urban          250             200             550
rural          100             150             300
```
Test the hypothesis that the distributions of cake mix purchases for the urban and rural consumers come from the same underlying distribution. What assumptions have you made?

3. (7 pts) A group of 500 (otherwise normal) adults viewed a recent popular film which explored the psychopathologies of serial killers (in graphic detail). After the film, half the audience was randomly assigned to a debriefing seminar, and the other half went to a control group situation (the restrooms). Measurements of blood pressure were taken immediately after the film and after the post-film treatment. Here are data from 4 subjects in each group:

```     seminar                    control
before after               before after
145    136                 117   105
128    124                 128   108
110    109                 110   104
117    115                 145   115
```
(A) Test the hypothesis that the reduction is the same for both groups (using a t-test).
(B) Are the difference scores approximately normally distributed? (I'm looking for the answer "no".) What ways have we learned to deal with non-normal distributions?
(C) Apply a square root transformation to the data and test whether the reduction is the same for both groups. (Careful: The transformation is applied to the data that go into the t-test; i.e., to the difference scores.)
(D) Transform the scores to ranks and conduct a t-test on the rank scores.
(E) If you were to conduct a randomization test (treating the data as the entire population and merely permuting the assignment of groups to scores as done in the textbook), how many differences of means would you have to compute to determine the full sampling distribution? (In other words, how many distinct ways are there of reassigning scores to groups?)

4. (6 pts)
(A) Test the null hypothesis that these four groups are equally probable:

```            group   A   B   C   D
observed freq   8   6   4   2
```
Have the conventional rules for good approximation by the continuous chi-square been satisfied?

(B) Now test the null hypothesis for these results:

```            group   A   B   C   D
observed freq  80  60  40  20
```
Which case (N=20 or N=200) gave the chi-square value with higher significance (i.e., smaller probability that the null hypothesis would yield a value that large or larger)?