To test the null hypothesis that the difference between two population proportions equals some constant \(\delta_{0}\), not necessarily 0 , we can use the statistic

which, for large samples, is a random variable having the standard normal distribution.

(a) With reference to Exercise 10.32, use this statistic to test at the 0.05 level of significance whether the true proportion of units requiring rework is now at least \(4 \%\) less than before the improvements were made.

(b) In a true-false test, a test item is considered to be good if it discriminates between well-prepared students and poorly prepared students. If 205 of 250 well-prepared students and 137 of 250 poorly prepared students answer a certain item correctly, test at the 0.01 level of significance whether for the given item the proportion of correct answers can be expected to be at least \(15 \%\) higher among well-prepared students than among poorly prepared students.

Data From Exercise 10.32

n1 Z = X X1 1 n ni n2 + ni X2 n2 - X2 1 - n2 X n2