# 9.1-9.2,_9.5_-_Two_Independent_Means,_Variances.pptx - STAT...

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STAT 312 Chapter 9 - Inferences Based on Two Samples 9.1 - Z -Tests and Confidence Intervals for a Difference Between Two Population Means 9.2 - The Two-Sample T -Test and Confidence Interval 9.3 - Analysis of Paired Data 9.4 - Inferences Concerning a Difference Between Population Proportions 9.5 - Inferences Concerning Two Population Variances
Consider two independent populations… Null Hypothesis H 0 : μ 1 = μ 2 , i.e., μ 1 μ 2 = 0 (“No mean difference") Test at signif level α POPULATION 1 and a random variable X , normally distributed in each. POPULATION 2 Classic Example : “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control X 2 ~ N ( μ 2 , σ 2 ) 1 σ 1 2 σ 2 X 1 ~ N ( μ 1 , σ 1 ) Random Sample, size n 1 Random Sample, size n 2 Sampling Distribution =? μ 0 2 X 1 X
Consider two independent populations… Null Hypothesis H 0 : μ 1 = μ 2 , i.e., μ 1 μ 2 = 0 (“No mean difference") Test at signif level α POPULATION 1 and a random variable X , normally distributed in each. POPULATION 2 Classic Example : “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control X 2 ~ N ( μ 2 , σ 2 ) 1 σ 1 2 σ 2 X 1 ~ N ( μ 1 , σ 1 ) Random Sample, size n 1 Random Sample, size n 2 Sampling Distribution =? μ 0 2 2 2 2 ~ , X N n 1 2 ~ ???? X X 1 1 1 1 ~ , X N n
Mean( X Y ) = Mean( X ) – Mean( Y ) Consider two independent populations… Null Hypothesis H 0 : μ 1 = μ 2 , i.e., μ 1 μ 2 = 0 (“No mean difference") Test at signif level α POPULATION 1 and a random variable X , normally distributed in each. POPULATION 2 Classic Example : “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control X 2 ~ N ( μ 2 , σ 2 ) 1 σ 1 2 σ 2 X 1 ~ N ( μ 1 , σ 1 ) Random Sample, size n 1 Random Sample, size n 2 Sampling Distribution =? Recall from Stat 311 (§3.3, slide 28): and if X and Y are independent Var( X Y ) = Var( X ) + Var( Y ) μ 0 2 2 2 2 ~ , X N n 1 2 ~ ????, ???? X X N 1 1 1 1 ~ , X N n
Consider two independent populations… Null Hypothesis H 0 : μ 1 = μ 2 , i.e., μ 1 μ 2 = 0 (“No mean difference") Test at signif level α POPULATION 1 and a random variable X , normally distributed in each. POPULATION 2 Classic Example : “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control X 2 ~ N ( μ 2 , σ 2 ) 1 σ 1 2 σ 2 X 1 ~ N ( μ 1 , σ 1 ) Random Sample, size n 1 Random Sample, size n 2 Sampling Distribution =? Mean( X Y ) = Mean( X ) – Mean( Y ) and if X and Y are independent Var( X Y ) = Var( X ) + Var( Y ) μ 0 Recall from Stat 311 (§3.3, slide 28): 2 2 2 2 ~ , X N n 1 2 1 2 ~ , ???? X X N 1 1 1 1 ~ , X N n
Consider two independent populations… Null Hypothesis H 0 : μ 1 = μ 2 , i.e., μ 1 μ 2 = 0 (“No mean difference") Test at signif level α POPULATION 1 and a random variable X , normally distributed in each. POPULATION 2 Classic Example : “Randomized Clinical Trial”… Pop 1 = Treatment, Pop 2 = Control X 2 ~ N ( μ 2 , σ 2 ) 1 σ 1 2 σ 2 X 1 ~ N ( μ 1 , σ 1 ) Random Sample, size n 1 Random Sample, size n 2 Sampling Distribution =? Mean( X Y ) = Mean( X ) – Mean( Y ) and if X and Y are independent Var( X Y ) = Var( X ) + Var( Y ) μ 0 Recall from Stat 311 (§3.3, slide 28): 2 2 2 2 ~ , X N n 1 2 1 2 ~ , ???? X X N 1 1 1 1 ~ , X N n
Consider two independent populations… Null Hypothesis H 0 : μ 1 = μ 2 , i.e., μ 1 μ 2 = 0 (“No mean difference") Test at signif level α POPULATION 1 and a random variable X