Each of a sample of 49 unemployed men was given a mental health examination using the General Health Questionnaire (GHQ). The GHQ is widely recognized measure of present mental health , with lower values indicating better mental health. The mean and standard deviation of the GHQ scores were x = 10. 94 and s = 5. 10, ? respectively. (a). Specify the appropriate null and alternative hypothesis if we wish to test the research hypothesis that the mean GHQ score for all unemployed men exceeds 10. Is the test one-tailed or two-tailed? (b). If we specify ? = 0. 05, what is the appropriate rejection region for this test? c). Conduct the test, and state your conclusion clearly in the language of this exercise. Find the p-value of the test. (Ans. H0 : µ = 10; Ha : µ > 10; One-tailed test; Rejection region: Z > 1. 645; Test score: Z = 1. 29; Do not reject H0 , GHQ score does not exceeds 10; p-value = 0. 0985) Example. A consumer protection group is concerned that a ketchup manufacturer is ? lling its 20-ounce family-size containers with less than 20 ounces of ketchup. The group purchases 49 family-size bottles of this ketchup, weigh the contents of each, and ? nds that the mean weight is 19. 6 ounces, and the standard deviation is equal to 0. 22 ounces. (a). Do the data provide su? cient evidence for the consumer group to conclude that the mean ? ll per family-size bottle is les than 20 ounces? Test using ? = 0. 05. (b). Find the p-value of the your test in part (a). (Ans. H0 : = 20; Ha : < 20; Rejection Region is Z < ? 1. 645 (one-tailed test); test score Z = ? 4. 45; Reject H0 at ? = 0. 05, su? cient evidence to say that the mean ? ll per family-size bottle is less than 20 ounces; p-value = 0) Example. State University uses thousands of ? uorescent light bulbs each year.
The brand of bulb it currently uses has a mean life of 900 hours. A manufacturer claims that its new brands of bulbs, which cost the same as the brand the university currently uses, has a mean life of more than 900 hours. The university has decided to purchase the new brand if, when tested, the test evidence supports the manufacturer’s claim at the . 10 signi? cance level. Suppose 99 bulbs were tested with the following results: x = 919 hours, s = 86 hours. Find the rejection region for the test of interest to the State University. ? (Ans. Rejection Region: Z > 1. 28) 1 . Small sample (n ? 30) test for H0 : µ = µ0 (known). t= This test requires that the sampled population is normal. x ? µ0 ? s v n Example. A random sample of n observations is selected from a normal population to test the null hypothesis that µ = 10. Specify the rejection region for each of the following combinations of Ha , ? , and n. (a). Ha : µ = 10, ? = 0. 01, n = 14. (Ans. t < ? 3. 012, or t > 3. 012) (b). Ha : µ < 10, ? = 0. 025, n = 26. (Ans. t < ? 2. 06) Example. According to advertisements, a strain of soybeans planted on soil prepared with a speci? d fertilizer treatment has a mean yield of 475 bushels per acre. Twenty farmers who belong to a cooperative plant the soybeans. Each uses a 40-acre plot and records the mean yield per acre. The mean and variance for the sample of 20 farms are x = 462 and s2 = 9070. ? Specify the null and alternative hypothesis used to determine if the mean yield for the soybeans is di? erent than advertised. (Ans. H0 : µ = 475; Ha : µ = 475) Example. A psychologist was interested in knowing whether male heroin addicts’ assessments of self-worth di? er from those of the general male population.