Chapter 9.4, Problem 56E

Sometimes experiments involving success or failure responses are run in a paired or before/after manner. Suppose that before a major policy speech by a political candidate, n individuals are selected and asked whether (S) or not (F) they favor the candidate. Then after the speech the same n people are asked the same question. The responses can be entered in a table as follows:

where x₁ + x₂ + x₃ + x₄ = n. Let p₁, p₂, p₃, and p₄ denote the four cell probabilities, so that p₁ = P(S before and S after), and so on. We wish to test the hypothesis that the true proportion of supporters (S) after the speech has not increased against the alternative that it has increased.

a. State the two hypotheses of interest in terms of p₁, p₂, p₃, and p₄.

b. Construct an estimator for the after/ before difference in success probabilities.

c. When n is large, it can be shown that the rv (X_i − X_j)/n has approximately a normal distribution with variance given by [p_i + p_j − (p_i − p_j)²]/n. Use this to construct a test statistic with approximately a standard normal distribution when H₀ is true (the result is called McNemar’s test).

d. If x₁ = 350, x₂ = 150, x₃ = 200, and x₄ = 300, what do you conclude?