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# chapter6 - Chapter 6 Inferences Regarding Locations of Two...

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Chapter 6 Inferences Regarding Locations of Two Distributions

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Comparing 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 ( μ 1 , σ 1 ) and ( μ 2 , σ 2 )) Measurements across groups are independent (different individuals in the 2 groups) Summary statistics obtained from the 2 groups: 2 2 2 1 1 1 : Size Sample : Dev. Std. : Mean : 2 Group : Size Sample : Dev. Std. : Mean : 1 Group n s y n s y
Sampling 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): 2 1 Y Y - ( 29 ( 29 2 2 2 1 2 1 2 2 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 n n n n Y Y V Y Y E Y Y Y Y Y Y σ σ σ σ σ σ μ μ μ + = + = = - - = = - - - -

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Small-Sample Test for μ 1 2 Normal Populations Case 1: Common Variances ( σ 1 2 = σ 2 2 = σ 2 ) Null Hypothesis : Alternative Hypotheses : 1-Sided: 2-Sided : Test Statistic: (where S p 2 is a “pooled” estimate of σ 2 ) 0 2 1 0 : = - μ μ H 0 2 1 : - μ μ A H 0 2 1 : - μ μ A H 2 ) 1 ( ) 1 ( 1 1 ) ( 2 1 2 2 2 2 1 1 2 1 0 2 1 - + - + - = + - - = n n s n s n s n n s y y t p p obs
Small-Sample Test for μ 1 2 Normal Populations Decision Rule: (Based on t -distribution with ν = n 1 + n 2 -2 df) 1-sided alternative • If t obs t α , ν ==> Conclude μ 1 - μ 2 0 • If t obs < t α,ν ==> Do not reject μ 1 - μ 2 = ∆ 0 2-sided alternative • If t obs t α/2 , ν ==> Conclude μ 1 - μ 2 0 • If t obs - t α/2,ν ==> Conclude μ 1 - μ 2 < ∆ 0 If - t α/2,ν < t obs < t α/2,ν ==> Do not reject μ 1 - μ 2 = ∆ 0

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Small-Sample Test for μ 1 2 Normal Populations Observed Significance Level ( P -Value) Special Tables Needed, Printed by Statistical Software Packages 1-sided alternative P=P(t t obs ) (From the t ν distribution) 2-sided alternative P= 2 P( t |t obs | ) (From the t ν distribution) If P -Value α , then reject the null hypothesis
Small-Sample (1- α29 100% Confidence Interval for μ 1 - μ 2 - Normal Populations Confidence Coefficient (1- α ) refers to the proportion of times this rule would provide an interval that contains the true parameter value μ 1 - μ 2 if it were applied over all possible samples Rule: Interpretation (at the α significance level): If interval contains 0, do not reject H 0 : μ 1 = μ 2 If interval is strictly positive, conclude that μ 1 > μ 2 If interval is strictly negative, conclude that μ 1 < μ 2 ( 29 + ± - 2 1 2 / 2 1 1 1 n n s t y y p α

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t -test when Variances are Unequal Case 2: Population Variances not assumed to be equal ( σ 1 2 σ 2 2 ) Approximate degrees of freedom Calculated from a function of sample variances and sample sizes (see formula below) - Satterthwaite’s approximation Smaller of n 1 -1 and n 2 -1 Estimated standard error and test statistic for testing H 0 : μ 1
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