Chapter 14.5, Problem 23E

The χ² test used in Exercise 14.22 is equivalent to the two-tailed Z test of Section 10.3, provided α is the same for the two tests. Show algebraically that the χ² test statistic X² is the square of the test statistic Z for the equivalent test.

14.22 A study to determine the effectiveness of a drug (serum) for the treatment of arthritis resulted in the comparison of two groups each consisting of 200 arthritic patients. One group was inoculated with the serum whereas the other received a placebo (an inoculation that appears to contain serum but actually is not active). After a period of time, each person in the study was asked whether his or her arthritic condition had improved. The results in the accompanying table were observed. Do these data present sufficient evidence to indicate that the proportions of arthritic individuals who said their condition had improved differed depending on whether they received the serum?

1. a Test by using the X² statistic. Use α = .05.
2. b Test by using the Z test of Section 10.3 and α = .05. Compare your result with that in part (a).
3. c Give bounds for the attained significance level associated with the test in part (a).