Solved: A plan for an executive travelers club has been

Chapter 8, Problem 8.41

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A plan for an executive travelers club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. 310 CHAPTER 8 Tests of Hypotheses Based on a Single Sample EXERCISES Section 8.3 (3544) among the 20 that survive. If the desired significance level is .05, c must satisfy B(c; 20, .9) .05. From Appendix Table A.1, B(15; 20, .9) .043, whereas B(16; 20, .9) .133. The appropriate rejection region is therefore x 15. If the accelerated test results in x 14, H0 would be rejected in favor of Ha, necessitating a modification of the proposed warranty. The probability of a type II error for the alternative value p .8 is (.8) P(H0 is not rejected when X Bin(20, .8)) P(X 16 when X Bin(20, .8)) 1 B(15; 20, .8) 1 .370 .630 That is, when p .8, 63% of all samples consisting of n 20 cans would result in H0 being incorrectly not rejected. This error probability is high because 20 is a small sample size and p .8 is close to the null value p0 .9. a. Using this data, test at level .01 the null hypothesis that the companys premise is correct against the alternative that it is not correct. b. What is the probability that when the test of part (a) is used, the companys premise will be judged correct when in fact 10% of all current customers qualify?