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Unformatted text preview: 4/27/11 FR OM FO BE RE Four Steps to Hypothesis Tes5ng 1. State a H0 and H1 2. Determine criteria for rejec5ng the null hypothesis 3. Obtain a random sample of size n from the popula5on and compute a test sta,s,c. 4. Made Decision: If value of test sta5s5c is more extreme than cri5cal values of test sta5s5c, reject H0; otherwise, fail to reject H0 FR BE OM FO RE ZTest zX = X ! "X FR O BE M FO RE zX = X ! "X zX > zcritical
Reject H0 zX < zcritical
Retain H0 or Fail to Reject H0 One Sample Test Requires knowing popula5on and 1 4/27/11 FR BE OM FO RE Truth There is an Effect H1 is True FR OM FO BE RE Truth There is an Effect H1 is True There is No Effect H0 is True There is No Effect H0 is True Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT FR BE OM FO RE Distribu5on of Sample Means FR O BE M FO RE Truth There is an Effect H1 is True There is No Effect H0 is True 2.5% Decisions High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 2.5 2.5 Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT zcri5cal = 1.96 zcri5cal = +1.96 2 4/27/11 p(Type I Error if H0 True) = p(Type I Error if H1 True) = 0 p(Type I Error  H0 True) = p(Type I Error  H1 True) = 0 FR O BE M FO RE Truth There is an Effect H1 is True p(Type I Error  H0 True) = p(Type I Error  H1 True) = 0 Decisions There is No Effect H0 is True Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT Condi5onal Probability 3 4/27/11 FR BE OM FO RE Truth There is an Effect H1 is True There is No Effect H0 is True p(Type II Error  H0 True) = 0 p(Type II Error  H1 True) = Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT FR BE OM FO RE Truth There is an Effect H1 is True Truth There is No Effect H0 is True There is an Effect H1 is True There is No Effect H0 is True Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT 4 4/27/11 Truth There is No Effect H0 is True There is an Effect H1 is True Truth There is No Effect H0 is True There is an Effect H1 is True Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT, p = 1 Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 POWER, p = 1 Today Sta5s5cal Power The power of a test is the probability that the test will correctly reject a false null hypothesis. Power is the probability of detec,ng a real effect. Sta5s5cal Power Visualizing Sta5s5cal Power Factors Affec5ng Sta5s5cal Power 5 4/27/11 Truth There is No Effect H0 is True There is an Effect H1 is True Distribu5on of X if H0 is True Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 POWER, p = 1 Distribu5on of zX if H0 is True Distribu5on of zX if H0 is True 0 0 Popula5on mean ! "X 6 4/27/11 Distribu5on of Sample Means Distribu5on of zX if H0 is True 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 0 2.5 2.5 zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 Distribu5on of zX if H1 is True? Distribu5on of zX if H0 is True Distribu5on of zX if H0 is True 0 0 zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 7 4/27/11 Sta5s5cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna5ve Hypothesis (H1) treatment without treatment Sta5s5cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna5ve Hypothesis (H1) treatment without treatment zcri5cal = 1.96 0 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True zcri5cal = +1.96 8 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Z X 0 = 100 words Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Z X 0 = 100 words = 120 words Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 9 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Check Your Understanding Sta5s5cal power is the __________________________. A probability of retaining H0 if H0 is actually false probability of rejec5ng H0 if H0 is actually false probability of retaining H0 if H0 is actually true probability of rejec5ng H0 if H0 is actually true Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 B C D Check Your Understanding Sta5s5cal power is the __________________________. A B C D probability of retaining H0 if H0 is actually false probability of rejec5ng H0 if H0 is actually false probability of retaining H0 if H0 is actually true probability of rejec5ng H0 if H0 is actually true Z X Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 10 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Propor5on of zX under H1 where zX < zcri5cal Power Distribu5on of zX if H0 is True Effect Size Z X 0 = 100 words Z X = 80 words 0 = 100 words zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Factors That Affect Power Size of the effect in the popula5on Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 11 4/27/11 zX = X ! "X zX = X ! "X Es5ma5ng ( alterna5ve null ) Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size zX = X ! "X Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 12 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Factors that Affect Power Size of the effect in the popula5on Variability in the popula5ons Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 140 words zcri5cal = 1.96 zcri5cal = +1.96 zX = X ! "X zX = X ! "X 13 4/27/11 ! !X = n ! !X = n Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 14 4/27/11 Factors that Affect Power Size of the effect in the popula5on Variability in the popula5ons Sample Size zX = X ! "X X ! zX = "X ! !X = n 15 4/27/11 Distribu5on of zX if H1 is True ! !X = n Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Factors that Affect Power Size of the effect in the popula5on Variability in the popula5ons Sample Size Alpha level Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 16 4/27/11 Distribu5on of ZSta5s5c zX = X ! "X 2.5% zX > zcritical High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 2.5 2.5 zcri5cal = 1.96 zcri5cal = +1.96 ! = 0.05 Distribu5on of ZSta5s5c Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power 5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 5% Z 2.5 2.5 X 0 = 100 words = 120 words zcri5cal = 1.64 zcri5cal = +1.64 zcri5cal = 1.96 zcri5cal = +1.96 ! = 0.05 ! = 0.10 17 4/27/11 Distribu5on of ZSta5s5c Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power 5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 5% Z X 0 = 100 words = 120 words 2.5 2.5 zcri5cal = 1.64 zcri5cal = +1.64 ! = 0.10 zcri5cal = 1.64 zcri5cal = +1.64 ! = 0.10 Distribu5on of ZSta5s5c Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power 0.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 0.5% Z 2.5 2.5 X 0 = 100 words = 120 words zcri5cal = 2.575 zcri5cal = +2.575 zcri5cal = 1.64 zcri5cal = +1.64 ! = 0.10 ! = 0.01 18 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Factors that Affect Power Size of the effect in the popula5on Variability in the popula5ons Sample Size Alpha level Direc,onal Hypotheses Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 2.575 zcri5cal = +2.575 ! = 0.01 Sta5s5cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna5ve Hypothesis (H1) treatment without treatment Direc,onal Sta5s5cal Hypotheses Null Hypothesis (H0) treatment without treatment Alterna5ve Hypothesis (H1) treatment < without treatment 19 4/27/11 Distribu5on of ZSta5s5c NonDirec5onal (Two Tailed) Distribu5on of ZSta5s5c Direc,onal (one tailed) 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5 2.5 2.5 2.5 zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.64 ! = 0.05 ! = 0.05 Direc,onal Sta5s5cal Hypotheses Null Hypothesis (H0) treatment without treatment Alterna5ve Hypothesis (H1) treatment < without treatment Direc,onal Sta5s5cal Hypotheses Null Hypothesis (H0) treatment without treatment Alterna5ve Hypothesis (H1) treatment > without treatment 20 4/27/11 Distribu5on of ZSta5s5c Direc,onal (one tailed) Distribu5on of ZSta5s5c Direc,onal (one tailed) 5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5 2.5 2 1.5 High Probability Samples if H0 is True 1 .5 .5 1 1.5 2 5% 2.5 2.5 zcri5cal = 1.64 zcri5cal = +1.64 ! = 0.05 ! = 0.05 Direc,onal Sta5s5cal Hypotheses Null Hypothesis (H0) treatment without treatment Alterna5ve Hypothesis (H1) treatment > without treatment Z X 0 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power = 100 words = 120 words zcri5cal = 1.96 zcri5cal = +1.96 ! = 0.05
NonDirec5onal (Two Tailed) 21 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Direc5onal Tests Must specify before running study Effects in the other direc5on must be unimportant, uninteres5ng, or conceptually unlikely Poten5al for abuse makes them suspect to many researchers Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = +1.64 ! = 0.05
NonDirec5onal (Two Tailed) Distribu5on of ZSta5s5c Direc,onal (one tailed) Distribu5on of ZSta5s5c Direc,onal (one tailed) High Probability Samples if H0 is True 2.5 2 1.5 1 .5 .5 1 1.5 2 5% High Probability Samples if H0 is True 2.5 2 1.5 1 .5 .5 1 1.5 2 5% 2.5 2.5 zcri5cal = +1.64 zcri5cal = +1.64 zX = 2.4 ! = 0.05 ! = 0.05 22 4/27/11 Distribu5on of ZSta5s5c Direc,onal (one tailed) test changed to nondirec5onal (two tailed) test 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 5% 2.5 2.5 zcri5cal = 1.96 zX = 2.4 zcri5cal = +1.64 ! = 0.05
true ! = 0.0725 Distribu5on of ZSta5s5c NonDirec5onal (Two Tailed) Distribu5on of ZSta5s5c NonDirec5onal (Two Tailed) 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 2.5 2.5 2.5 2.5 zcri5cal = 1.96 zcri5cal = +1.96 zcri5cal = 1.96 zcri5cal = +1.96 ! = 0.05 ! = 0.05 zX = 1.7 23 4/27/11 Distribu5on of ZSta5s5c Nondirec,onal (two tailed) test changed to direc5onal (one tailed) test Direc5onal Tests Must specify before running study Effects in the other direc5on must be unimportant, uninteres5ng, or conceptually unlikely Poten5al for abuse makes them suspect to many researchers 5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 5% 2.5 2.5 zcri5cal = 1.64 zcri5cal = +1.64 zX = 1.7 true ! = 0.10 ! = 0.05 Factors that Affect Power Size of the effect in the popula5on Variability in the popula5ons Sample Size Alpha level Direc,onal Hypotheses Check Your Understanding Which of the following is guaranteed to not increase power? A B C D E Decreasing the alpha level Increasing sample size Decreasing the popula5on variability Increasing the size of the effect All of the above will increase power 24 4/27/11 Check Your Understanding Which of the following is guaranteed to not increase power? A B C D E Decreasing the alpha level Increasing sample size Decreasing the popula5on variability Increasing the size of the effect All of the above will increase power Distribu5on of ZSta5s5c 5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 5% 2.5 2.5 zcri5cal = 1.64 zcri5cal = +1.64 ! = 0.10 Distribu5on of ZSta5s5c Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Propor5on of zX under H1 where zX > zcri5cal Power 0.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 0.5% Z 2.5 2.5 X 0 = 100 words = 120 words zcri5cal = 2.575 zcri5cal = +2.575 zcri5cal = 1.64 zcri5cal = +1.64 ! = 0.10 ! = 0.01 25 4/27/11 Distribu5on of zX if H1 is True Distribu5on of zX if H0 is True Effect Size Factors that Affect Power Size of the effect in the popula5on Variability in the popula5ons Sample Size Alpha level Direc,onal Hypotheses Propor5on of zX under H1 where zX > zcri5cal Power Z X 0 = 100 words = 120 words zcri5cal = 2.575 zcri5cal = +2.575 ! = 0.01 Truth There is No Effect H0 is True There is an Effect H1 is True For Next Time (Re)Read Chapter 8 Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 POWER, p = 1 26 ...
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This note was uploaded on 01/10/2012 for the course PSYC PSYC 60 taught by Professor ? during the Winter '09 term at UCSD.
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