7.65. A test consists of 100 true-false questions. Joe did not study, so on each question, he randomly *7.66.

guesses the correct response.

(a) Find the probability that he scores at least 70, thus passing the exam. (Hint: Use 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: Use the sampling distribution of the difference of sample proportions.)

(c) How do the answers to

(a) and

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

Let yi\ denote the observation for subject / at time 1, yi2 the observation for subject i at time 2, and yi = ya - yn-

(a) Letting y^, y^ and y^ denote the means of these observations, show that ^

_ yi-

(b) Is the median difference (i.e., the median of the y, values) equal to the difference between the medians of the y,| and ya values? Show that this is true, or give a counterexample to show that it is false.