∙ we tend to focus on three kinds of rejection

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Unformatted text preview: ∙ We tend to focus on three kinds of rejection rules for a properly chosen constant c , called a critical value : T c T c | T | c 21 ∙ But others are possible, such as an asymmetric rejection rule in place of | T | c : reject H if T c L or T c U . ∙ Asymmetric rules are sometimes preferred when testing hypotheses about a parameter that must be strictly postive: 0. 22 EXAMPLE : Suppose in a Normal ,1 population, we want to test H : ≤ 0 against H 1 : 0. We can obtain a random sample of size of size n . We have an unbiased estimator of , X ̄ . It seems reasonable to base the rejection rule on how large X ̄ is. ∙ Let x ̄ denote the average for the actual data we observe. ∙ If we get x ̄ ≤ 0, that is no evidence against H : ≤ 0. 23 ∙ The harder question is what to do if x ̄ 0. Should we reject H if, say, x ̄ .02? What about x ̄ 1.30? We need to determine a rule once we have chosen the size of the test. To do that, we need to choose a statistic whose distribution we can compute under the null hypothesis. ∙ Using sd X ̄ 1/ n , we can use the standardized sample average: T X ̄ / sd X ̄ X ̄ / 1/ n n X ̄ 24 ∙ Note how the sample size n enters T directly, along with the sample average. For a positive outcome on X ̄ , T increases with n . ∙ Because T has the same sign as X ̄ , negative values of T are no evidence against H : ≤ 0. We only will reject in favor of H 1 : 0 if T is positive and “sufficiently large.” 25 ∙ In other words, we choose a rejection rule of the form T c for a constant c to be chosen by us. ∙ To find a rejection rule that allows us to control the size of the test, note that if 0, T has a standard normal distribution: T Normal 0,1 . ∙ If 0 then E T 0. Therefore, of all the null values, 0 is the least favorable case because it is the null value we are most likely to reject when the null is true. 26 ∙ Suppose we designate .05, that is, we want to make a Type I error in only 5% of the random samples we draw. (It is very common to choose a size of 5%, but it is not written in stone.) Then we need to find c such that P Z c .05 where Z Normal 0,1 . (Remember, T has a standard normal distribution if 0.) 27 ∙ From the standard normal distribution we can find that c ≈ 1.65. ∙ So the rejection rule that gives a 5% (or .05) size is: “Reject H in favor of H 1 if T 1.65.” 28 area = .05 .1 .2 .3 .4 phi(z) c = 1.65 One-Sided Rejection Rule, alpha = .05 29 ∙ A test against a one-sided alternative is also called a one- tailed test ....
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∙ We tend to focus on three kinds of rejection rules for...

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