2011-Psych 60-Lecture 15

2011-Psych 60-Lecture 15 - 4/29/11 What's your preference?...

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Unformatted text preview: 4/29/11 What's your preference? Which classroom lighEng condiEon do you most prefer? Some issues to consider: Your ability to take notes Your ability to see the slides clearly Your ability to stay awake/focus A B C Fluorescent Lights On Fluorescent Lights Off (incandescent lights only) No Preference Last Time StaEsEcal Power Visualizing StaEsEcal Power Factors AffecEng StaEsEcal Power 1 4/29/11 Truth There is No Effect H0 is True There is an Effect H1 is True F BE ROM FO RE F BE ROM FO RE DistribuEon of zX 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- Z X 0 = 100 words zcriEcal = -1.96 zcriEcal = +1.96 F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Z X 0 = 100 words Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 zcriEcal = -1.96 zcriEcal = +1.96 2 4/29/11 F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Z X 0 = 100 words = 120 words Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 zcriEcal = -1.96 zcriEcal = +1.96 F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size Factors That Affect Power Size of the effect in the populaEon F BE ROM FO RE ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 3 4/29/11 F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Power ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words Z X 0 = 100 words = 140 words zcriEcal = -1.96 zcriEcal = +1.96 zcriEcal = -1.96 zcriEcal = +1.96 Factors that Affect Power Size of the effect in the populaEon Variability in the populaEons F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 4 4/29/11 DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size F BE ROM FO RE Factors that Affect Power Size of the effect in the populaEon Variability in the populaEons Sample Size F BE ROM FO RE ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size F BE ROM FO RE ProporEon of zX under H1 where zX > zcriEcal Power ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 zcriEcal = -1.96 zcriEcal = +1.96 5 4/29/11 Factors that Affect Power Size of the effect in the populaEon Variability in the populaEons Sample Size Alpha level F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 ! = 0.05 F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Power ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words Z 0 = 100 words = 120 words zcriEcal = -1.64 zcriEcal = +1.64 ! = 0.10 X zcriEcal = -2.575 zcriEcal = +2.575 ! = 0.01 6 4/29/11 Factors that Affect Power Size of the effect in the populaEon Variability in the populaEons Sample Size Alpha level Direc)onal Hypotheses F BE ROM FO RE StaEsEcal Hypotheses Null Hypothesis (H0) treatment = without treatment AlternaEve Hypothesis (H1) treatment without treatment F BE ROM FO RE Direc)onal StaEsEcal Hypotheses Null Hypothesis (H0) treatment without treatment AlternaEve Hypothesis (H1) treatment > without treatment F BE ROM FO RE F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words zcriEcal = -1.96 zcriEcal = +1.96 ! = 0.05 Non-DirecEonal (Two Tailed) 7 4/29/11 F BE ROM FO RE DistribuEon of zX if H1 is True DistribuEon of zX if H0 is True Effect Size ProporEon of zX under H1 where zX > zcriEcal Power Z X 0 = 100 words = 120 words zcriEcal = +1.64 ! = 0.05 DirecEonal (One Tailed) Today StaEsEcal Power Sundry Topics in Hypothesis TesEng P-Values Standardized Effect Sizes AssumpEons of the Z-Test CalculaEng StaEsEcal Power CalculaEng StaEsEcal Power 8 4/29/11 PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon Treatment $ All Individuals in the USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 = ? hours = 5 hours DistribuEon of zX if H0 is True HypotheEcal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 z Sample Time on Facebook during sixth month for each person 0 Sample Sta)s)cs X = 16 s = 5.1 DistribuEon of zX if H0 is True DistribuEon of zX if H0 is True z X 0 15 z X -1.96 0 15 +1.96 9 4/29/11 PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon Treatment $ All Individuals in the USA using Facebook PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon Treatment $ All Individuals in the USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 = ? hours = 5 hours in the USA = 15 hours = 100 = 5 hours = 15 = ? hours = 5 hours HypotheEcal HypotheEcal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Random Sample n = 100 Treatment $ Random Sample n = 100 Sample Sta)s)cs X = 16 s = 5.1 Sample Time on Facebook during sixth month for each person Sample Time on Facebook during sixth month for each person Sample Time on Facebook during sixth month for each person PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon All Individuals in the DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True HypotheEcal in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = 16 hours = 5 hours Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 z Sample Time on Facebook during sixth month for each person X -1.96 0 15 +1.96 10 4/29/11 = 15 x = 0.5 X = 16 zX = X ! "X null = 15 x = 0.5 alt = 16 zalt = alt ! null !X zX = 16 !15 0.5 zalt = 16 !15 0.5 zX = 2 zalt = 2 DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True z X -1.96 0 15 +1.96 z X -1.96 0 15 +1.96 11 4/29/11 DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True z X -1.96 0 15 +1.96 16 z X -1.96 0 15 +1.96 16 DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power z X -1.96 0 15 +1.96 16 z X -1.96 0 15 +1.96 16 12 4/29/11 CalculaEng Power Step 1: Find criEcal value in original X units DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power z X -1.96 0 15 +1.96 16 Find this value null = 15 x = 0.5 zcriEcal = 1.96 Xcritical = null + zcritical! X null = 15 x = 0.5 zcriEcal = 1.96 Xcritical = null + zcritical! X Xcritical = 15 + 1.96 ! 0.5 13 4/29/11 null = 15 x = 0.5 zcriEcal = 1.96 Xcritical = null + zcritical! X DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power Xcritical = 15 + 1.96 ! 0.5 z X -1.96 0 15 +1.96 16 15.98 Xcritical = 15.98 CalculaEng Power Step 1: Find criEcal value in original X units CalculaEng Power Step 1: Find criEcal value in original X units Step 2: Forget completely about the distribuEon of zX given Ho : true 14 4/29/11 DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power DistribuEon of zX if H1 is True, where treatment = 16 ProporEon of zX under H1 where zX > zcriEcal Power z X -1.96 0 15 +1.96 16 15.98 z X 15.98 16 CalculaEng Power Step 1: Find criEcal value in original X units Step 2: Forget completely about the distribuEon of zX given Ho : true CalculaEng Power Step 1: Find criEcal value in original X units Step 2: Forget completely about the distribuEon of zX given Ho : true Step 3: Forget that you're doing anything complicated. Find shaded region as if a normal z-score problem. 15 4/29/11 DistribuEon of zX if H1 is True, where treatment = 16 ProporEon of zX under H1 where zX > zcriEcal Power z X 15.98 16 z X 15.98 0 16 p(X > 15.98)? Check Your Understanding z X z X 15.98 0 16 15.98 0 16 A B p(X > 15.98) is greater than 0.50 p(X > 15.98) is less than 0.50 16 4/29/11 Check Your Understanding Finding Area Under the Normal Curve 1. 2. 3. 4. Sketch distribuEon Shade in area to be found Restate problem in terms of z Use proporEons from Unit Normal Table to find area in shaded region F BE ROM FO RE z X 15.98 0 16 A B p(X > 15.98) is greater than 0.50 p(X > 15.98) is less than 0.50 p(X > 15.98)? p(X > 15.98)? P(Z > ?)? z X 15.98 0 16 z X ? 0 16 15.98 17 4/29/11 alt = 16 x = 0.5 Xcrit = 15.98 z= Xcritical ! alternative "X alt = 16 x = 0.5 Xcrit = 15.98 z= Xcritical ! alternative "X z= 15.98 ! 16 0.5 alt = 16 x = 0.5 Xcrit = 15.98 X ! alternative z = critical "X z= z= p(X > 15.98)? P(Z > -0.04)? 15.98 ! 16 0.5 z X -0.04 0 16 -0.02 = -0.04 0.5 15.98 18 4/29/11 Unit Normal Table 19 4/29/11 p(X > 15.98)= 0.5160 P(Z > -0.04)= 0.5160 DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power = .5160 z X -0.04 0 16 z X -1.96 0 15 +1.96 16 15.98 15.98 CalculaEng Power Step 1: Find criEcal value in original X units Step 2: Forget completely about the distribuEon of zX given Ho : true Step 3: Forget that you're doing anything complicated. Find shaded region as if a normal z-score problem. Check Your Understanding The previous study would have cost at least $60,000. (100 subjects x $100 each x 6 months). The power of the study was .516, meaning that even if there was a true 1 hour difference in Facebook usage (and = 5), the chance of detecEng the result was only 51.7%. Is this an experiment worth doing considering the 48.3% chance of finding nothing? A B Yes. Viva science! No. One should always consider the power of an experiment before undertaking it 20 4/29/11 Check Your Understanding The previous study would have cost at least $60,000. (100 subjects x $100 each x 6 months). The power of the study was .516, meaning that even if there was a true 1 hour difference in Facebook usage (and = 5), the chance of detecEng the result was only 51.7%. Is this an experiment worth doing considering the 48.3% chance of finding nothing? PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon Treatment $ All Individuals in the USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 = 16 hours = 5 hours HypotheEcal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 A B Yes. Viva science! No. One should always consider the power of an experiment before undertaking it Sample Time on Facebook during sixth month for each person PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon All Individuals in the Check Your Understanding True or False: If the true populaEon mean aqer treatment was = 17 hours, power would be higher than if the true populaEon mean aqer treatment was = 16 hours (assuming everything else stays the same) A B True False HypotheEcal in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = 17 hours = 5 hours Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Sample Time on Facebook during sixth month for each person 21 4/29/11 Check Your Understanding True or False: If the true populaEon mean aqer treatment was = 17 hours, power would be higher than if the true populaEon mean aqer treatment was = 16 hours (assuming everything else stays the same) A B True False DistribuEon of zX if H1 is True, where treatment = 16 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power = .5160 z X -1.96 0 15 +1.96 16 15.98 DistribuEon of zX if H1 is True, where treatment = 17 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal CalculaEng Power Step 1: Find criEcal value in original X units Step 2: Forget completely about the distribuEon of zX given Ho : true Step 3: Forget that you're doing anything complicated. Find shaded region as if a normal z-score problem. z X -1.96 0 15 +1.96 17 15.98 22 4/29/11 alt = 17 x = 0.5 Xcrit = 15.98 z= Xcritical ! alternative !X alt = 17 x = 0.5 Xcrit = 15.98 z= Xcritical ! alternative !X z= 15.98 !17 0.5 alt = 17 x = 0.5 Xcrit = 15.98 z= Xcritical ! alternative !X z= 15.98 !17 0.5 DistribuEon of zX if H1 is True, where treatment = 17 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal z X -1.96 0 15 +1.96 17 15.98 -1.02 z= = -2.04 0.5 P(Z > -2.04)= 23 4/29/11 Unit Normal Table 24 4/29/11 DistribuEon of zX if H1 is True, where treatment = 17 DistribuEon of zX if H0 is True ProporEon of zX under H1 where zX > zcriEcal Power = .9793 z X Today StaEsEcal Power Sundry Topics in Hypothesis TesEng P-Values Standardized Effect Sizes AssumpEons of the Z-Test CalculaEng StaEsEcal Power -1.96 0 15 +1.96 17 15.98 p(Z > -2.04)= 0.9793 P-value (for a hypothesis test) Sundry Topics in Hypothesis TesEng P-Values The p-value is the probability of obtaining a random sample with the observed effect, or an effect more extreme, if the null hypothesis is true. 25 4/29/11 Conveying Results The effect of money on individuals' usage of Facebook was invesEgated. Money was found to staEsEcally significantly increase Eme spent on Facebook, z = 2.0, p < 0.05. Conveying Results The effect of money on individuals' usage of Facebook was invesEgated. Money was found to staEsEcally significantly increase Eme spent on Facebook, z = 2.0, p = 0.046. Conveying Results The effect of money on individuals' usage of Facebook was invesEgated. Money was found to staEsEcally significantly increase Eme spent on Facebook, z = 2.0, p = 0.046. Exact p-value DistribuEon of Sample Means X = 15 ! X = 0.5 0.025 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.025 -2.5 13.75 2.5 16.25 zcriEcal = -1.96 zcriEcal = +1.96 Xobserved = 16 zx = +2 26 4/29/11 DistribuEon of Sample Means X = 15 ! X = 0.5 DistribuEon of Sample Means X = 15 ! X = 0.5 0.025 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.025 0.025 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.025 -2.5 13.75 2.5 16.25 -2.5 13.75 2.5 16.25 zcriEcal = -1.96 zcriEcal = +1.96 Xobserved = 16 zx = +2 zcriEcal = -1.96 zcriEcal = +1.96 Xobserved = 16 zx = +2 P value = p( |Z| > |zx| ) P value = p( |Z| > 2 ) DistribuEon of Sample Means X = 15 ! X = 0.5 DistribuEon of Sample Means X = 15 ! X = 0.5 0.023 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.023 0.023 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.023 -2.5 13.75 2.5 16.25 -2.5 13.75 2.5 16.25 Xobserved = 14 zx = -2 P value = p( |Z| > 2 ) Xobserved = 16 zx = +2 Xobserved = 14 zx = -2 P value = p( |Z| > 2 ) Xobserved = 16 zx = +2 27 4/29/11 DistribuEon of Sample Means X = 15 ! X = 0.5 DistribuEon of Sample Means X = 15 ! X = 0.5 0.023 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.023 0.023 High Probability Samples if H0 is True -2 14 -1.5 14.25 -1 14.5 -.5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 0.023 -2.5 13.75 2.5 16.25 -2.5 13.75 2.5 16.25 Xobserved = 14 zx = -2 P value = p( Z < -2 ) + p(Z > 2) Xobserved = 16 zx = +2 Xobserved = 14 zx = -2 P value = p( |Z| > |zx| ) Xobserved = 16 zx = +2 Conveying Results The effect of money on individuals' usage of Facebook was invesEgated. Money was found to staEsEcally significantly increase Eme spent on Facebook, z = 2.0, p = 0.046. P-value (for a hypothesis test) The p-value is the probability of obtaining a random sample with the observed effect, or an effect more extreme, if the null hypothesis is true. 28 4/29/11 P-value (for a hypothesis test) The p-value is the probability of obtaining a random sample with the observed effect, or an effect more extreme, if the null hypothesis is true. NOT the probability that H0 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 POWER, p = 1- Truth There is No Effect H0 is True p = 1 or p = 0 There is an Effect H1 is True p = 1 or p = 0 Retain H0 TYPE II ERROR, p = Reject H0 POWER, p = 1- P-value (for a hypothesis test) The p-value is the probability of obtaining a random sample with the observed effect, or an effect more extreme, if the null hypothesis is true. NOT the probability that H0 is true. Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = 29 4/29/11 Sundry Topics in Hypothesis TesEng Standardized Effect Sizes StaEsEcal Significance Whether the observed effect is likely or unlikely to occur when the null hypothesis is true StaEsEcal Significance Whether the observed effect is likely or unlikely to occur when the null hypothesis is true PracEcal Significance Whether the observed effect is meaningful or valuable 30 4/29/11 Cohen's d Cohen's d is a measure of effect size that describes the effect in terms of how much variability scores have in general Standardized Effect Size Indicators Cohen's d Cohen's d X ! untreated " Cohen's d = treated ! untreated " Estimated Cohen's d = 31 4/29/11 Cohen's d X ! untreated " Cohen's d d = 0.2 d = 0.5 d = 0.8 Small Effect Medium Effect Large Effect Estimated Cohen's d = PopulaEon Standard DeviaEon NOT the Standard Error. PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon All Individuals in the PopulaEon All Individuals in the All Individuals USA using Facebook PopulaEon All Individuals in the in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = ? hours = 5 hours in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = ? hours = 5 hours HypotheEcal HypotheEcal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Random Sample n = 100 Treatment $ Random Sample n = 100 Sample Sta)s)cs X = 16 s = 5.1 Sample Time on Facebook during sixth month for each person Sample Time on Facebook during sixth month for each person Sample Sta)s)cs X = 16 s = 5.1 Sample Time on Facebook during sixth month for each person 32 4/29/11 = 15 = 5 X = 16 Estimated Cohen's d = X ! untreated " = 15 = 5 X = 16 Estimated Cohen's d = X ! untreated " estimated d = 16 ! 15 5 = 15 = 5 X = 16 Estimated Cohen's d = X ! untreated " Cohen's d d = 0.2 d = 0.5 d = 0.8 Small Effect Medium Effect Large Effect 16 ! 15 estimated d = 5 estimated d = 1 = 0.20 5 33 4/29/11 Cohen's d X ! untreated " Estimated Cohen's d = AssumpEons of a Test Sundry Topics in Hypothesis TesEng Assump)ons of the Z-Test AssumpEons of a test refer to condiEons of nature or the specific research situaEon that need to be true in order for the test results to be valid. 34 4/29/11 AssumpEons of the Z-test Random Sampling Any discrepancy between the mean of the sample and the populaEon mean is due to sampling error only Z-Test zX = X! "X F BE ROM FO RE ObservaVons are "Independent" Each individual in the sample provides new informaEon ObservaVons are "IdenVcally Distributed" Each individual is coming from the same populaEon The value is unchanged by the treatment The effect of the treatment is simply to add or subtract some value to each person's score One Sample Test Requires knowledge of populaEon and The sampling distribuVon is normally distributed The proporEons in the z-table will be correct when we find p-values or lookup criEcal values for a given alpha Z-Test X! zX = "X One Sample Test Requires knowledge of populaEon and F BE ROM FO RE For Next Time Read Chapter 8 (again... or for the first Eme) Chapter 9 Do Homework 5 35 ...
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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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