Elements of a Test of Hypothesis 1. Null Hypothesis (H0 ) - A statement about the values of population parameters which we accept until proven false. 2. Alternative or Research Hypothesis (Ha )- A statement that contradicts the null hypothesis. It represents researcher’s claim about the population parameters. This will be accepted only when data provides suﬃcient evidence to establish its truth. 3. Test Statistic - A sample statistic (often a formula) that is used to decide whether to reject H0 . 4. Rejection Region- It consists of all values of the test statistic for which H0 is rejected. This rejection region is selected in such a way that the probability of rejecting true H0 is equal to α (a small number usually 0.05). The value of α is referred to as the level of signiﬁcance of the test. 5. Assumptions - Statements about the population(s) being sampled. 6. Calculation of the test statistic and conclusion- Reject H0 if the calculated value of the test statistic falls in the rejection region. Otherwise, do not reject H0 . 7. P-value or signiﬁcance probability is deﬁned as proportion of samples that would be unfavourable to H0 (assuming H0 is true) if the observed sample is considered unfavourable to H0 . If the p-value is smaller than α, then reject H0 . Remark: 1. If you ﬁx α = 0.05 for your test, then you are allowed to reject true null hypothesis 5% of the time in repeated application of your test rule. 2. If the p-value of a test is 0.20 (say) and you reject H0 then, under your test rule, at least 20% of the time you would reject true null hypothesis. 1. Large sample (n > 30) test for H0 : µ = µ0 (known). Z= x − µ0 ¯ σ √ n
Example. A study reported in the Journal of Occupational and Organizational Psychology investigated the relationship of employment status to mental health. 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 α = 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 ﬁlling 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 ﬁnds that the mean weight is 19.86 ounces, and the standard deviation is equal to 0.22 ounces. (a). Do the data provide suﬃcient evidence for the consumer group to conclude that the mean ﬁll per family-size bottle is les than 20 ounces? Test using α = 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 α = 0.05, suﬃcient evidence to say that the mean ﬁll per family-size bottle is less than 20 ounces; p-value = 0) Example. State University uses thousands of ﬂuorescent 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 signiﬁcance level. Suppose 99 bulbs were tested with...
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