Biological Data Analysis: Homework 2

Due Tuesday, Sept. 11

You must type this and all other homework assignments. Do not e-mail the assignment to me; turn it in early (at 322 Wolf) for a foreseeable absence, or turn it in late after an unexpected absence from class.

1. In human males, the left testis hangs lower than the right in 57 to 80% of adult men. You want to know whether the same kind of asymmetry occurs in chimpanzees (Pan troglodytes). You will observe some wild chimpanzees and record which testis hangs lower. (The testes of chimpanzees are so large that they couldn't walk if the testes hung at the same level, so you don't have to worry about ties; each chimp will be either "right" or "left.") You are going to analyze your data with an exact binomial test. Should you do a one-tailed test or a two-tailed test? What are the advantages and disadvantages of a one-tailed test for this experiment?

Advantage of one-tailed test: Gives you more power to detect a difference in chimps that's in the same direction as in humans (left hanging lower); in other words, if more chimps have the left testis hanging lower, it might be significant with a one-tailed test but insignificant with a two-tailed test.

Advantage of a two-tailed test: it could show a significant difference if the majority of chimps have the right testis hanging lower. This is how the data actually turned out; see this paper. In addition, if your results would be significant with a one-tailed test but not significant with a two-tailed test, people may be skeptical when you tell them that you intended to do a one-tailed test.

2. If you observe 20 chimpanzees, how many would have to have the left testis lower in order to have a signficant (P<0.05) result with a one-tailed test? Same question, but for a two-tailed test? Use the exact binomial test spreadsheet, and use trial and error to find the answers.

For a two-tailed test, P would be less than 0.05 with 15 left and 5 right (P=0.041), but not significant with 14 and 6. For a one-tailed test, P would be less than 0.05 with 15 left and 5 right (P=0.021), but not significant with 14 and 6. This shows that the improvement in power with a one-tailed test isn't that great; for 20 chimps, there is no result that would be signficant with a one-tailed test but not significant with a two-tailed test.

3. Using either a one-tailed or two-tailed test (whichever you think is more appropriate), how many chimps would have to have the left testis lower in order to have a signficant result if the sample size was 50, or 100, or 1000?

One tailed: 32, 59 and 527; two-tailed: 33, 61 and 532. This shows that larger sample sizes can detect smaller deviations from the null hypothesis. It also shows that some results would be significant with a one-tailed test but not significant with a two-tailed test.

4. You want to know how many male chimps you'll need to observe to have a 90% chance of getting a signficant result, if the true proportion is 60% left-testis-lower. Use the exact binomial power calculator for this. Then answer the same question if you want an 80% chance of getting a signficant result, or a 99% chance.

One-tailed test: 224, 169, 405; Two-tailed test: 274, 210, 468. This shows that it takes a bigger sample size to give you more power, and that a one-tailed test requires a smaller sample size for the same power.

5. Answer all parts of question 4, except for a true proportion of 75% left-testis-lower.

One-tailed test: 35, 28, 60; Two-tailed test: 42, 35, 72. This shows that much smaller sample sizes are needed if the effect size is larger.

Here are four more practice questions for the final exam. For each experiment, list the variables that are mentioned in the description, and say whether each is a nominal, measurement, or ranked variable. Don't list variables that are not mentioned in the description; for example, don't list "weight of mother" for the first experiment.

6. You want to know whether aspirin taken during pregnancy has an effect on the size of offspring. You ask 1072 new mothers whether they took aspirin during the first three months of their pregnancy. The average weight of newborn babies of mothers who took aspirin is 103 grams less than babies of mothers who didn't take aspirin.

aspirin or not: nominal
weight of baby: measurement

7. You want to know what affects the breakdown of fructose at high temperatures (due to caramelization and Maillard reactions) in apples. You bake 8 Winesap apples, 8 Rome Beauty apples, 8 Jonathan apples, and 8 Granny Smith apples for 90 minutes at 180 C, and you bake another set of 8 apples of each variety for 90 minutes at 200 C. You measure the amount of fructose (in milligrams of fructose per gram of baked apple) in each apple.

type of apple: nominal
temperature: nominal (because there are only two values, 180 C and 200 C)
amount of fructose: measurement

8. You want to know whether certain "home remedies" used for ant control really work. You find 40 houses that are infested with pavement ants (Tetramorium caespitum). In 10 of the houses, you place bay leaves along the baseboards; you sprinkle boric acid along the baseboards of 10 houses, sprinkle diatomaceous earth along the baseboards of 10 others, and leave the last 10 houses untreated. After two weeks, you place sticky traps in each house and count the number of ants caught in a 12-hour period.

treatment (bay leaves, boric acid, etc.): nominal
number of ants caught: measurement

9. In order to increase rotation speed during a figure skating jump, skaters must be strong enough to pull their arms in towards the center quickly after taking off. Strength tests have shown that skaters may not have the upper body strength necessary to overcome the centrifugal forces and pull their arms in, so you decide to put female figure skaters through a strength training program. Using a high-speed camera, you measure the rotation speed during the first spin of a triple Lutz-double toe loop combination of 9 top skaters, put them through a 12-week strength training program, then measure the rotation speed again.

before vs. after training program: nominal
rotation speed: measurement
name of skater: nominal

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