8.5 Section 8.1.1 gave the large-sample z or chi-squared McNemar test for comparing dependent proportions. The exact P-value, needed for small samples, uses the binomial distribution. For Table 8.1, consider Ha: π1+ > π+1, or equivalently, Ha: π12 > π21.

a. The exact P-value is the binomial probability of at least 132 successes out of 239 trials, when the parameter is 0.50. Explain why. (Software reports P-value = 0.060.)

b. For these data, how is the mid P-value (Section 1.4.5) defined in terms of binomial probabilities? (This P-value = 0.053.)

c. Explain why Ha: π1+ = π+1 has ordinary P-value = 0.120 and mid P-value = 0.106. (The large-sample McNemar test has P-value that is an approximation for the binomial mid P-value. It is also 0.106 for these data.)