| Stat 251: Statistical Methods I | Last updated 09/19/2008 |
1) Complete the following exercises from your text: pages 91-97, #35, 43, 44, 50. Show all your work and justify your answers completely.
These exercises are reproduced below.
#35) Suppose that a group of 20 subjects contains 10 men and 10 women. Suppose that these subjects are to be randomly assigned to two treatment groups.
(a) Create a graph of the hypergeometric probability distribution of X, the number of women randomly assigned to group A. (Hint: Use Minitab.)
(b) Use the hypergeometric distribution to determine the probability that 5 of each gender end up in each group. Is this outcome more likely than not?
(c) Use the hypergeometric distribution to determine the probability that all 10 of one gender end up in the same group.
(d) If the experiment does end up with all men in one group and all women in the other, would you have reason to doubt that randomization was applied correctly? Explain.
Now suppose that a group of 50 men and 50 women are to be randomly assigned to two treatment groups A and B.
(e) Without doing any calculations, produce a rough sketch of the probability distribution of the number of women randomly assigned to group A.
(f) Would the probability of an exactly even gender split be larger, smaller, or the same as in (b)? Explain, without providing any calculations.
(g) Would the probability of finding all men in one group and all women in the other be larger, smaller, or the same as in (c)? Explain, without providing any calculations.
#43)
In a now famous study of gender discrimination, Rosen and Jerdee (1974)
reported on a study of 48 male bank supervisors (attending a management institute
at the University of North Carolina) who were each sent the same personnel file
and asked to judge whether the person should be promoted to a branch manager
position. The files were identical except that half of them were
randomly assigned to be the file of a female and the other half indicated that the file
was that of a male. The researchers suspected that a higher proportion of
"males" would be recommended for promotion. Of the 24 "male" files, 21
were recommended for promotion. Of the 24 "female" files, 14 were
recommended for promotion. (a) Is this an experiment or an observational study? Explain. (b) In this study, the subjects believed that they were all participating in
an identical exercise dealing with personnel problems in the banking industry.
Explain why it is important for the subject to be "blind" to the fact that there
were other versions and that the researchers were focusing on the gender of the
applicant. (c) Calculate the p-value from Fisher's exact test for this study.
(Show your calculations!) (d) Write a paragraph describing the results of your analysis to the
researchers. In addition to stating your conclusion, explain the reasoning
process that leads to this conclusion. What does this p-value tell you? (e) Does the design of this study allow you to draw a cause-and-effect
conclusion regarding gender and likelihood of promotion? Justify your
answer.
#44) Recall the bank manager gender discrimination study (from the previous exercise). The results that you already examined were based on the subjects being told that the nature of the manager’s job was “routine.” The researchers also examined how the subjects would respond to the applicant’s gender when the nature of the managerial position was described as “complex.” They expected that the greater tendency to promote the male would be even stronger for the more complex job. In this case, 5 of 25 women were recommended for promotion and 11 of 20 men were recommended for promotion.
(a) Do these data provide convincing evidence that men were more likely to be recommended for promotion than women for a complex job? Justify your answer with appropriate calculations (show your work and/or output) and explanations. Clearly explain in your own words what the calculated probability represents.
(b) How do these results compare to those of the “routine” job? Does the tendency to promote men seem even stronger for the complex job? Explain.
#50) An instructor suspected that male students are more likely to wear a baseball cap in class than female students. While proctoring an exam, he noted that 5 of 18 male students were wearing a cap, and none of the 6 female students were. Suppose for now that you were to randomly distribute the five caps to be worn among these 24 students, and let the random variable X be the number of males who would be given caps.
Explain why X has a hypergeometric distribution, and state the values of its parameters.(a)
(b) Determine and interpret the expected value of X, that is, the expected number of caps distributed to men.
(c) List the possible values of X and their probabilities. Also provide a graph of this probability distribution. (Hint: You are strongly encouraged to use Minitab.)
(d) Do the observed data provide strong evidence that male students tend to wear caps in class more than female students do? Report the p-value of Fisher's exact test, and explain the reasoning behind your answer.
(e) Explain and show how this p-value could be calculated in two other ways. (Hint: Refer to the discussion on pages 67-68, or the class notes from Day 8).