Chapter 6Comparing 2 Means - Independent SamplesSampling Distribution ofSmall-Sample Test for m1-m2 Normal PopulationsSlide 5Slide 6Small-Sample (1-a)100% Confidence Interval for m1-m2 - Normal Populationst-test when Variances are UnequalExample - Maze Learning (Adults/Children)Slide 10Example - Maze Learning Case 1 - Equal VariancesExample - Maze Learning Case 2 - Unequal VariancesSPSS Output(1-a)100% Confidence Interval for m1-m2Small Sample Test to Compare Two Medians - Nonnormal PopulationsExample - Levocabostine in Renal PatientsComputer Output - SPSSRank-Sum Test: Normal ApproximationExample - Maze LearningSlide 20Slide 21Inference Based on Paired Samples (Crossover Designs)Test Concerning mDSlide 24Example Antiperspirant FormulationsSlide 26Slide 27Small-Sample Test For Nonnormal DataSlide 29Example - Caffeine and EnduranceSlide 31Slide 32Slide 33Slide 34Sample Sizes for Given Margin of ErrorSample Size Calculations for Fixed PowerExample - Rosiglitazone for HIV-1 LipoatrophyData SourcesChapter 6Inferences Regarding Locations of Two DistributionsComparing 2 Means - Independent Samples•Goal: Compare responses between 2 groups (populations, treatments, conditions)•Observed individuals from the 2 groups are samples from distinct populations (identified by (1,1) and (2,2)) •Measurements across groups are independent (different individuals in the 2 groups)•Summary statistics obtained from the 2 groups:222111 :Size Sample :Dev. Std. :Mean :2 Group :Size Sample :Dev. Std. :Mean :1 GroupnsynsySampling Distribution of •Underlying distributions normal sampling distribution is normal•Underlying distributions nonnormal, but large sample sizes sampling distribution approximately normal•Mean, variance, standard error (Std. Dev. of estimator):21YY 2221212221212212121212121nnnnYYVYYEYYYYYYSmall-Sample Test for Normal Populations•Case 1: Common Variances (12 = 22 = 2)•Null Hypothesis:•Alternative Hypotheses: –1-Sided: –2-Sided:•Test Statistic:(where Sp2 is a “pooled” estimate of 2)0210: H021: AH021: AH2)1()1(11)(2122221121021nnsnsnsnnsyytppobsSmall-Sample Test for Normal Populations•Decision Rule: (Based on t-distribution with =n1+n2-2 df)–1-sided alternative•If tobs t, ==> Conclude •If tobs < t ==> Do not reject –2-sided alternative•If tobs t , ==> Conclude •If tobs -t ==> Conclude •If -t < tobs < t ==> Do not reject Small-Sample Test for Normal Populations•Observed Significance Level (P-Value)•Special Tables Needed, Printed by Statistical Software Packages–1-sided alternative •P=P(t tobs) (From the t distribution)–2-sided alternative•P=2P( t |tobs| ) (From the t distribution)•If P-Value then reject the null hypothesisSmall-Sample (1-100% Confidence Interval for Normal Populations•Confidence Coefficient (1-) refers to the proportion of times this rule would provide an interval that contains the true parameter value if it were applied over all possible samples•Rule:•Interpretation (at the significance level):–If interval contains 0, do not reject H0: 1 = 2–If interval is strictly positive, conclude that 1 > 2–If interval is strictly negative, conclude that 1 < 2 212/2111nnstyypt-test when Variances are Unequal•Case 2: Population Variances not assumed to be equal (1222)•Approximate degrees of freedom–Calculated from a function of sample variances and sample sizes (see formula below) - Satterthwaite’s approximation–Smaller of n1-1 and n2-1•Estimated standard error and test statistic for testing H0: 1=2: 11 :df site'Satterthwa :StatisticTest :error standard Estimated2222212121222212122212121212122212121nnsnnsnsnsnsnsyyyySEyytnsnsYYSEobsExample - Maze Learning (Adults/Children)• Groups: Adults (n1=14) / Children (n2=10) • Outcome: Average # of Errors in Maze Learning Task• Raw Data on next slideAdults (i=1) Children (i=2)Mean 13.28 18.28Std Dev 4.47 9.93Sample Size 14 10• Conduct a 2-sided test of whether true mean scores differ• Construct a 95% Confidence Interval for true differenceSource: Gould and Perrin (1916)Example - Maze Learning (Adults/Children)Name Group Trials Errors AverageH 1 41 728 17.76W 1 25 333 13.32Mac 1 33 453 13.73McG 1 31 528 17.03 Group n Mean Std DevL 1 41 335 8.17 1 14 13.28 4.47R 1 48 553 11.52 2 10 18.28 9.93Hv 1 24 217 9.04Hy 1 32 711 22.22F 1 46 839 18.24Wd 1 47 473 10.06Rh 1 35 532 15.20D 1 69 538 7.80Hg 1 27 213 7.89Hp 1 27 375 13.89Hl 2 42 254 6.05McS 2 89 1559 17.52Lin 2 38 1089 28.66B 2 20 254 12.70N 2 49 599 12.22T 2 40 520 13.00J 2 50 828 16.56Hz 2 40 516 12.90Lev 2 54 2171 40.20K 2 58 1331 22.95Example - Maze LearningCase 1 - Equal Variances)2.1,2.11(20.600.5)99.2(074.200.5:%95EXCEL) (From 1091.|)67.1|(2 :value074.2||:67.199.200.510114122.728.1828.13:22.715.5221014)93.9)(110()47.4)(114(22,025.22CITPPttRRtTSsobsobspH0: HA: 0 ( = 0.05)No significant difference between 2 age groupsExample - Maze LearningCase 2 - Unequal Variances )36.2,36.12(36.700.5)36.3(19.200.5:%9519.2||:49.136.300.510)93.9(14)47.4(28.1828.13:63.1196.1046.1279)86.9(13)43.1(86.943.186.910)93.9(43.114)47.4(63.11,025.22222*22222121CIttRRtTSnSnSobsobsH0: HA: 0 ( = 0.05)No significant difference between 2 age groupsNote: Alternative would be to use 9 df (10-1)SPSS OutputGroup
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