A series of numbers is given in brackets for each problem; these are the point values assigned to the different parts of the problem. Thus, 1. [8/4/8] means that problem 1 was assigned 20 points: 8 for part (a), 4 for part (b) and 8 for part (c).

This is not meant to be a full solution set, but rather an indication of some of the important aspects of the problems, or an indication of the direction for a solution.

1. [8/4/8]a) Several answers. The most appropriate might be the chi-squared test for independence and logistic regression, although one could also compare mean Karnofsky scores between groups, or conduct a linear trend test. Measures of association do not directly test the hypothesis, although they may still be useful descriptors. b) Tests which take the ordering of the Karnofsky scores into account will be more powerful, and also may lead to interpretible parameters. c) Under the hypothesis of 1:3 randomization, the expected number randomized to Placebo is 53/4 = 13.25 and to MGDF 3*(53/4) = 39.75. Using the chi-squared goodness-of-fit test, we obtain chi-sq = (13-13.25)^2/13.25 + (40-39.75)^/39.75 = 0.005 with one degree of freedom. The p-value is nearly one; the actual allocations are about as close to the 1:3 ratio as they could be.

2. [4/4/4/4/4]a) 179/648 = 0.296 b) 72/119 = 0.605 110/306 = 0.359 44/223 = 0.197 c) 226/648 = 0.349 d) Independence of males' education level and their opinions about the role of women. e) The chi-squared statistic of 57.09 on 2 df (calculated by Stata) indicates a strong dependence between the variables. The answer to (b) above indicates that more inclusive views on women's roles are associated with higher levels of education among men.

3. [6/6/6] (+5 bonus)Note: A general description of the different sampling schemes does not related to Table 1-m, and consequently did not get full credit. a) Men were sampled for a fixed period of time; this resulted in completed surveys from 648 men, who were then cross-clasified by education and opinion. b) Men are sampled until 119 people with grade-school education, 306 with high-school education, and 223 with some college are obtained. (These numbers are specified in advance by the survey director.) They are then classified by opinion within each education level. c) Men are sampled until exactly 648 men are selected (this was the number which the survey budget would support). They were then cross-clasified by education and opinion. d) Men are sampled until 119 people with grade-school education, 306 with high-school education, and 223 with some college are obtained. (These numbers are specified in advance by the survey director.) These men are placed in a room with 226 cards that say "Agree" and 422 cards that say "Disagree". The group is instructed that each man must take one card, and that so far as possible, the cards should reflect the individuals' opinions about the question. The men are allowed to talk among themselves and to trade cards. The number of men from each education level who selected each type of card is recorded in the table.

4. [6/6/9]a) The male-female odds ratio for white victims for a "firearm" attack is 1.08, which represents only a small association between sex and type of attack. For those who prefer statistical guidance, the chi-squared statistic is 0.85 on 1 df (p=0.4). The relationship, if any, is very weak. b) The male-female odds ratio for black victims for a "firearm" attack is 1.21, which represents a modest association between sex and type of attack. The chi-squared statistic is 6.72 on 1 df (p=0.01). This suggests that there is a real relationship (with men being more likely than women to be assaulted using firearms/explosives), even though it is not a large effect. c) After adjusting for racial differences, female victims are less likely than male victims to be assualted with firearms/explosives (coeff = -0.13). After adjusting for any differences associated with the sex of the victim, black victims are less likely than white victims to be assaulted with firearms/explosives (coeff = -0.27). There is strong evidence that both of these effects are real (small p-values). A numerical interpretation would be that relative to black women, assualts on males have odds increased by 15% and assaults on whites have odds increased by an (additional) 31% for a firearm attack. [Note exp(0.273)=1.31 and exp(0.13)=1.15]

5. [8/7/6]a) 2 x [ -147.674 - ( -336.886 ) ] = 2 x 189.2 = 378.42 df = 2 (1 df each for A, B). b) 147.674 - 145.212 = 2.46 2 x 2.46 = 4.92. This is a 1 df comparison. The p-value is somewhat less than 0.05, which suggests that C is helpful in any model which contains A and B. c) If the regression coefficients in the logistic model are 6, 2, and 1.5, then then odds factors corresponding to the conditions are 403 for being in a single-parent household, 7.4 for overcrowded households, and 4.5 for foreign birth. The amount by which the odds are increased for any individual child is the product of the odds factors which pertain to the child's classification.