Sample Questions for Final Exam

(Modified from 2001 Final)

All questions can be answered with no more than a scientific calculator.

All the questions on this exam relate to a series of studies on the reproductive biology of the chartreuse-footed booby. The males in this species come in two forms, white-winged and black-winged. Females mate with one male and then lay eggs.

(Yes, I seem to write a lot of questions about birds with different colored wings. Apparently my imagination is stuck in a rut.)

1. A complete survey of your population finds that 80% of males are white-winged and 20% are black-winged. You want to know if females favor one form over the other when mating.

   1. The first three females you observe all mate with black-winged males. What is the probability of getting this combination (three blacks in a row) by chance?
      
      P = 0.2³ = 0.008
      
      
      
   2. Ultimately, observations of 100 matings among random females found that 70 were with white-winged males and 30 were with black-winged males. Carry out an appropriate test of the null hypothesis that females mate with males at random.
      
      Goodness-of-fit chi-square analysis gives chi-square = 5.641; the critical value is 3.841, so females appear to prefer mating with black-winged males.
      
      
      
2. The next question you ask is whether the number of eggs laid by a female depends on the wing color of her mate. You randomly select 15 females that mated with black-winged males and 15 that mated with white-wings. The summary data was as follows:
   

   Black-wing maters: Mean = 6.6, SD = 1.4
   White-wing maters: Mean = 5.8, SD = 1.1
   

   Is there a significant difference between these two groups?
   

   A two-sample t-test can be carried out using the supplied values. The resulting t-stat is 1.740, less than the t-crit of 2.048, so we reject the hypothesis that clutch size is affected by mate choice.

   
   
3. You now wonder whether there may be competition among females for mates. With the idea that body size might contribute to a female's competitive ability, you look to see whether there is a difference in body size between females that mate with black-winged versus white-winged males. You also decide to include females that do not mate at all, with the idea that these females would be the least competitive. You randomly select eight females of each type and weigh them, giving you a mass for each animal. What method of analysis would you use on these data?
   

   Although officially body size is the independent variable and mate color the dependent, you are still simply asking whether three groups differ in the mean value of a continuous variable, so ANOVA is the way to go. If there were only two groups, you could use logistic regression to predict what mate a female of a particular body size would most likely get, but note that this is not quite what "you" are asking above.
    

4. Let's assume your results from the above experiments suggest that: a) mate choice might be important in determining the number of eggs a female lays, and b) that female size is related to mate choice. You would like to know whether it might be the difference in female size between groups that is driving the differences in clutch size. Being busy yourself, you hand off the data to Chim-Chim, the lab chimp, who brings back the results below in exchange for a banana. Alas, Chim-Chim cannot speak, and you must interpret the tables yourself. What sort of analysis did Chim-Chim do, and what to the results indicate?
   

   Source	SS	DF	MS	F	P
   Mate Color	1.929	1	1.929	0.752	0.392
   Mass	42.068	1	42.068	16.403	0.001
   Mate Color x Mass	3.762	1	3.762	1.467	0.234
   Error	92.326	36	2.565

   
   

   Source	SS	DF	MS	F	P
   Mate Color	5.458	1	5.458	2.102	0.156
   Mass	44.661	1	44.661	17.197	0.001
   Error	96.089	37	2.597

   
   You are trying to determine whether clutch size (a numeric variable) differs based on mate color (catergorical) and female size (numeric). Given this combination of data types, ANCOVA is the appropriate form of analysis. Even if you weren't sure about this, the tables above should help. The general setup indicates some sort of ANOVA-type analysis. Remember that the variables listed under "Source" are the independent variables in the analysis - the presence of a numeric variable here is a sure sign that this is Analysis of Covariance, since a numeric variable will always be a covariate rather than a factor. Once you know this is ANCOVA, the first table is obviously the test for homogeneity of slopes. Since the interaction term is non-significant, the slopes can be considered the same for all groups, and the main ANCOVA can be done. The second table shows that the size of females (Mass) had a significant effect on clutch size, but mate color did not.

   
   
5. A potential weakness of all your work to date is that it is observational, so it is difficult to know for certain whether the patterns you are finding in your studies represent true cause-and-effect relationships. Assuming you could capture the birds, and keep and breed them under controlled conditions, what sort of experiment(s) might you do to see whether
    
1. females prefer white-winged or black-winged males
2. female body size affects their ability to get mates of a particular color
3. male wing color influences clutch size
4. female body size influences clutch size.
   

Briefly discuss the experimental setups and the analyses you would use on the resulting data. Consider that you may need to control some variables while examining others.

Very briefly (and other answers are possible):

1. I would present single females with the choice of two males - one black and one white. Ideally, the males would be separated so that it was female choice, rather than competition between the males, that decided the final mating. Multiple trials on different females would allow a comparison of the observed number of times each color was chosen with the expected frequency if matings were random (50:50). Chi-square goodness-of-fit would be suitable for this.
    
2. Assuming I had established that there actually was a preference in mating choice, I might run trials in which I weighed two females and put both in with two males, one of each color. I would then record which female mated with which male. One way to analyze the results would be to simply classify each female as "bigger" or "smaller" and then look at whether bigger females mated with one color male more than expected by chance. Again this could be done by chi-square, with the expectation that bigger females should mate with each color 50% of the time if there was no effect of size on mating. I might also use a paired t-test, where male color was the grouping variable, and female size the dependent variable. If size has an effect on mate choice, one group should be significantly bigger than the other.
    
3. Create random pairs of males and females and count the number of eggs produced. For the analysis, I would compare this number for black vs. white males, and also include female size as a covariate, since I have reason to believe this effects clutch size as well. Analysis of covariance would allow me to see if color affects clutch size while controlling for female size. Alternatively, I could try to use females that were all about the same size.
    
4. The first experiment in "c" above would be one approach. But to make my experiment more powerful, I might just use one color male, and then analyze the results by regression to see if bosy size predicts clutch size. I could also try to improve my chances of finding any effect by focusing on the bigger and smaller females to improve the "signal-to-noise" ratio."