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 twotailed 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 • µµ •
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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 Winter '08
 Ard
 Null hypothesis, Type I and type II errors, xcrit

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