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. On Mythbusters, Jamie and Adam wanted to know whether it was true that toast always falls buttered-side down when you drop it on the floor. They dropped 48 pieces of toast from the roof of their building; 29 landed buttered-side up, and 19 landed buttered-side down. Do the exact binomial test on these data using the spreadsheet linked from the Handbook of Biological Statistics, under "how to do the test." Report the P-value. Then do the test using Richard Lowry's web page linked there, and report the P-value.
The P-value is 0.193 using either the web page or the spreadsheet.
2. Write a sentence or two explaining what the P-value you got in question 1 means. (If you got different P-values, use whichever one you have the most confidence in.)
If the null hypothesis is true that the toast lands butter-side-down 50% of the time, you'd get 19 or fewer buttered-side down, or 29 or more buttered-side down, 19.3% of the time.
3. Flip a coin 20 times, and record the number of heads and tails. Report your results, in order: for example, HHTHTTHTHHHHTTHTHHTT. Do the exact binomial test, using whichever method you prefer.
4. As I was typing this assignment, our cat Gus wanted me to pet him, so he patted me on the arm with his left paw 9 times and his right paw 2 times. Is this significantly different from a 1:1 ratio? Analyze the data using all four of the goodness-of-fit tests we've learned, report the P-values, and write a sentence interpreting any differences among the results.
Exact binomial test: P=0.065
Chi-squared test: P=0.035 (without Yates correction, which we didn't talk about in class); P=0.07 with Yates correction
G-test: P=0.028 without Yates correction, 0.063 with Yates
Randomization test: I got P=0.076 with 4000 replicates.
The P-values are all different from each other; the chi-squared and G-tests, in particular, give P<0.05 even though the correct P-value, as found with the exact test, is 0.065.
5. Falk and Ayala (1971) collected data on 1187 individuals, recording whether each one clasped their hands with the right thumb on top (R) or the left thumb on top (L). There were 535 R individuals and 652 L individuals. Is this significantly different from a 1:1 ratio of R and L individuals? Try to analyze the data using all four goodness-of-fit tests, report the P-values, and write a sentence interpreting any differences among the results.
Exact binomial test: my spreadsheet says "N is too big", Richard Lowry's web page gives dashes.
Chi-squared test: P=0.0007 (with or without Yates correction)
G-test: P=0.0007 (with or without Yates correction)
Randomization test: My spreadsheet won't handle a sample this big.
With big samples, the binomial and randomization tests may not be practical. (I could have done them in SAS, but I'm not making you learn SAS until after the first exam).
6. When I come home in the evening, Gus the cat is usually sleeping on the couch. Our couch has 4 cushions of equal size. Imagine I record the cushion that Gus is sleeping on for 14 days, and get the following: 7 times on the far left cushion, 3 times on the middle left, 0 times on the middle right, and 4 times on the far right. Is Gus sleeping on the cushions randomly? Analyze the data using the chi-square test, G test, and randomization test. For the randomization test, use at least 1000 replicates (and be sure to report how many replicates you used).
Chi-squared test: P=0.067 without Williams correction, 0.081 with it
G-test: P=0.020 without Williams correction, 0.026 with it)
Randomization test: P=0.219 for 10000 replicates
The chi-squared and G-tests give P-values that are way too low.
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 mice" for the first experiment.
7. 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 vs. no asprin: nominal. Baby weight: measurement.
8. 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.
Apple variety: nominal. Baking temperature: nominal. Fructose amount: measurment.
9. 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: nominal. Number of ants: measurement.
10. 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.
Rotation speed: measurement. Before vs. after training program: nominal. Name of skater: nominal (it's not obvious that this is a variable--don't worry if you didn't get it).
Falk, C.T., and F.J. Ayala. 1971. Genetic aspects of arm folding and hand-clasping. Japanese journal of human genetics 15: 241-247.
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This page was last revised September 16, 2009. Its URL is http://udel.edu/~mcdonald/stathw2.html