The Wilcoxon test differs from parametric procedures comparing means in the sense that

(a) It applies directly to ordinal as well as interval response variables.

(b) It is unnecessary to assume that the population distribution is normal.

(c) Random sampling is not assumed.

7.65.* A test consists of 100 true–false questions. Joe did not study, so on each question he randomly guesses the correct response.

(a) Find the probability that he scores at least 70, thus passing the exam. (Hint: Use either the binomial distribution or the sampling distribution for the proportion of correct responses.)

(b) Jane studied a little and has a 0.60 chance of a correct response for each question. Find the probability that her score is nonetheless lower than Joe’s. (Hint: Usethe sampling distribution of the difference of sample proportions.)

(c) How do the answers to

(a) and

(b) depend on the number of questions? Explain.

7.66.*Let yi1 denote the observation for subject i at time 1, yi2 the observation for subject i at time 2, and yi = yi2−yi1.

(a) Letting ¯y1, ¯y2, and ¯yd denote the means of these observations, show that ¯yd = ¯y2 − ¯y1.

(b) Is the median difference (i.e., the median of the yi values)

equal to the difference between the medians of the yi1 and yi2 values? Show this is true, or give a counterexample to show that it is false.