lecture 12 Hypothesis Testing(pt.2)

# lecture 12 Hypothesis Testing(pt.2) - 4 Outcomes of a...

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Unformatted text preview: 4 Outcomes of a Hypothesis Test True Situation H0 True H0 False Retain H0 Experimenter ’s Decision Correct Acceptance Miss Correct Rejection Reject H0 False Alarm 1 4 Outcomes of a Hypothesis Test True Situation H0 True H0 False Retain H0 Experimenter ’s Decision Reject H0 Pr = 1 - α Type II Error Pr = β Power Pr = 1 - β Type I Error Pr = α 2 α and β: Conditional Probabilities • α = the probability of rejecting H THAT H0 is true GIVEN THAT H0 is false 0 GIVEN 0 • β = the probability of failing to reject H • Knowing α (by itself) provides no information about β, and knowing β (by itself) provides no information about α 3 Power • 1 - β = Power • The probability of rejecting H H0 is false 0 given that • Tells us how conﬁdent we can be that our statistical analysis would have led us to correctly endorse a true H1 • Inﬂuenced by a number of factors, most critically the actual, but unknown, value of, for example, µ (i.e., β and 1 - β themselves are always hypothetical probabilities) 4 Determining β • Calculation of β and 1 - β depends on the speciﬁcation of the following: • 1) The shape and parameters of the • 2) The level of signiﬁcance for the hypothesis test (α) sampling distribution under H0 (µ0, σ2, n) • 3) The shape and parameters of a speciﬁc sampling distribution under H1 (µ1, σ2, n) 5 Calculating z for β • 1) Specify µ (generally chosen to reﬂect the smallest nontrivial effect) 1 • 2) Identify X by transforming zcrit back to the original units of X - if the hypotheses are directional, choose the value ± zcrit that matches the speciﬁc value of µ1 being evaluated crit Xcrit = zcrit(σ/√ + µ0 n) • 3) Solve the following equation zβ = (Xcrit - µ1)/(σ/√ n) 6 From zβ to β If µ1 falls in between µ0 and Xcrit so that: 1) zβ is positive and µ1 > µ0, or 2) zβ is negative and µ1 < µ0 then β = .5 + the proportion in Column B If Xcrit falls in between µ0 and µ1 so that: 1) zβ is positive and µ1 < µ0, or 2) zβ is negative and µ1 > µ0, then β = the proportion in Column C 7 The researcher studying IQ (µ = 100, σ = 15) has decided that the smallest meaningful difference between San Diegans’ IQs and the national average would be 10. If she again uses a sample size of n = 16 and α = .05, what would be the power of the hypothesis test for µ1 = 110? 8 In testing the new educational curriculum, the state has decided that their budget crisis is severe enough that anything short of a 5 point reduction in test scores (µ = 200, σ = 50) would be sufficient to adopt it for widespread use. Calculate the power of the originally planned two-tailed hypothesis test (n = 100 and α = .05) to detect a true µ1 = 195. 9 • Calculations involving β are typically carried out in order to determine the sample size necessary to have a probabilistically acceptable chance of rejecting H0 for a given µ1 (e.g., what value of n would yield 1 - β = .8) Power Curves 10 Factors Affecting β • α - more conservative values (e.g., .01 • µ-µ • 0 instead of .05, or a nondirectional hypothesis instead of a directional one) increase β and reduce 1 - β - larger true effect sizes reduce β and increase 1 - β σ2 - larger variances increase β and reduce 1 -β X • n - increasing the sample size reduces σ , thereby reducing β and increasing 1 - β 11 Factors Affecting α and β • Experimenter Choice • The researcher chooses α, which in turn affects β • Appropriateness of the test statistic • The nominal values of α and β (the values speciﬁed/inferred by the researcher) will only equal the true probabilities of making a Type I or Type II error, respectively, if the assumptions of the hypothesis test are met 12 http://www.stat.sc.edu/~ogden/javahtml/power/power.html 13 ...
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