1. (7 pts) Suppose you are a manufacturer of ball bearings, and you want to know if two production runs of ball bearings have significantly different size. You have three different assistants measure a random sample of 30 ball bearings from each run. The first assistant comes from the coatings division of the ball bearing plant, so she is interested in the surface area of the ball bearings, because the surface area determines how much anti-oxidant will be needed to coat the ball bearings. The second assistant comes from the warehouse division of the ball bearing plant, so she is interested in the radius of the ball bearings, because the radius determines how closely the bearings can be packed and stored. The third assistant comes from the shipping division of the ball bearing plant, so she is interested in the volume of the ball bearings, because the volume determines the weight and hence the shipping cost. All three assistants measure the same sample of ball bearings. Run the linked SPSS syntax file, and answer the following questions. (The latest version of SPSS seems to have trouble running the syntax all at once, unlike previous versions, so you may need to select small subsections and run them in sequence.) This program spews a lot of output. To save paper, include in your homework only the syntax or graphs that are really necessary. Please be sure that you keep things in sequence, clearly marked.

• Look at the syntax lines that generate the sample. Is the area measurement sampled from a normal population, or is it skewed left or right? Is the radius measurement sampled from a normal population, or is it skewed left or right? Is the volume measurement sampled from a normal population, or is it skewed left or right? Write your answers clearly next to the corresponding syntax lines.
• For the first ball bearing, verify that area, radius and volume have the correct numerical relationship. That is, starting with the value of the radius, verify that surface area is 4*pi*radius**2, and that volume is (4/3)*pi*radius**3. Write you answer next to the "Case Summaries" table.
• When the areas, radii, and volumes are ranked, are the ranks the same for the three measures or different? Annotate the "Case Summaries" table with your answer.
• In the t-tests, are the t values identical for the four different measures (i.e., area, radius, volume, and rank)? Do the four different measures lead to the same conclusion regarding the null hypothesis? Write your answer next to the t-test table.
• From the histograms and the descriptive statistics, do any of these measures look severely non-normal or non-homogeneous? Say why or why not, referring to the SD, skewness, and kurtosis values. Write your answer next to the histograms.
• What is the significance level of the Mann-Whitney test? How does that compare with the significance level of the t-test on ranks? Write your answer next to the Mann-Whitney output table.

2. (6 pts) Response time distributions are often heavily skewed to the right. This is intuitively plausible, because response times can only be so fast, but there is no limit to how slow they can be. The linked SPSS syntax file generates data skewed to the right. Run the file and do the following. Please keep your answer clear and organized.

• What does the t-test on the raw data tell you? Annotate the output with your answer.
• Make histograms of the data from each group separately, and determine the skewness and kurtosis of each group. (Use the commands in the SPSS file of the previous exercise as a guide, and check the SPSS textbook, too!) Are the data strongly non-normal? Annotate the "Descriptives" output with your answer.
• Transform the data to normal distributions. (Unlike data sampled from the real world, here you can look at the SPSS syntax file that generated the sample to figure out the exact transformation that will normalize the data.) Run a t-test on the transformed data. Is the conclusion of the t-test that same as for the raw data? Annotate the output table with your answer.
• Transform the data to ranks. Run a t-test. What is the conclusion of the test? Annotate the output table with your answer.
• Do a Mann-Whitney test. What is the conclusion? Annotate the output table with your answer.

3. (4 pts) Consider these data:
Group A: 0.0, 0.7, 0.7, 1.1, 1.1, 1.4
Group B: 1.1, 1.4, 1.4, 1.6, 1.6, 1.8
Which of the following transformations makes the group data most symmetric and homogeneous? exp(x), ln(x), x², sqrt(x). Provide numerical evidence for your claim. (Feel free to use SPSS for this problem.)

Consider now these data:
Group C: 0.0, 0.0, 0.7, 0.7, 1.6, 1.8
Group D: 0.0, 0.7, 1.6, 1.6, 1.8, 1.8
Which, if any, of the following transformations makes the group data symmetric and homogeneous? exp(x), ln(x), x², sqrt(x). (Feel free to use SPSS for this problem.) If no adequate transformation of the data can be found, what type of test could be conducted?

4. (8 pts) Suppose you are measuring individuals on some measure of polarity, and individuals tend to cluster toward -1 or +1. You have two groups of people, and you want to know whether the mean polarities of the groups differ. This is a difficult situation because the distributions of the scores are strongly bimodal; i.e., non-normal. The linked SPSS file specifies scores from two groups. Do the following:
(A) Run a t-test on the means of the two groups. Are the assumptions of the t-test satisfied? According to the t-test, what is the probability of getting a difference of means from the null hypothesis this large or larger?
(B) Use the SPSS program to generate a resampling distribution of the difference of means. According to the resampling distribution, are the means of the two groups significantly different? What is the probability of getting a difference of means from the null hypothesis this large or larger? Remember, it's a two-tailed test.