mean2 - Comparison of 2 Population Means • Goal To...

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Unformatted text preview: Comparison of 2 Population Means • Goal: To compare 2 populations/treatments wrt a numeric outcome • Sampling Design: Independent Samples (Parallel Groups) vs Paired Samples (Crossover Design) • Data Structure: Normal vs Non-normal • Sample Sizes: Large ( n 1 , n 2 >20) vs Small Independent Samples • Units in the two samples are different • Sample sizes may or may not be equal • Large-sample inference based on Normal Distribution (Central Limit Theorem) • Small-sample inference depends on distribution of individual outcomes (Normal vs non-Normal) Parameters/Estimates (Independent Samples) • Parameter: • Estimator: • Estimated standard error: • Shape of sampling distribution: – Normal if data are normal – Approximately normal if n 1 , n 2 >20 – Non-normal otherwise (typically) 2 1- μ μ 2 1 Y Y- 2 2 2 1 2 1 n S n S + Large-Sample Test of μ 1-μ 2 • Null hypothesis: The population means differ by ∆ (which is typically 0): • Alternative Hypotheses: – 1-Sided: – 2-Sided: • Test Statistic: 2 1 : ∆ =- μ μ H 2 1 : ∆- μ μ A H 2 1 : ∆ ≠- μ μ A H 2 2 2 1 2 1 2 1 ) ( n S n S y y z obs + ∆-- = Large-Sample Test of μ 1-μ 2 • Decision Rule: – 1-sided alternative • If z obs ≥ z α ==> Conclude μ 1-μ 2 ∆ • If z obs < z α ==> Do not reject μ 1-μ 2 = ∆ – 2-sided alternative • If z obs ≥ z α/2 ==> Conclude μ 1-μ 2 ∆ • If z obs ≤- z α/2 ==> Conclude μ 1-μ 2 < ∆ • If - z α/2 < z obs < z α/2 ==> Do not reject μ 1-μ 2 = ∆ 2 1 : ∆- μ μ A H 2 1 : ∆ ≠- μ μ A H Large-Sample Test of μ 1-μ 2 • Observed Significance Level ( P-Value) – 1-sided alternative • P=P(z ≥ z obs ) (From the std. Normal distribution) – 2-sided alternative • P= 2 P( z ≥ |z obs | ) (From the std. Normal distribution) • If P-Value ≤ α, then reject the null hypothesis 2 1 : ∆- μ μ A H 2 1 : ∆ ≠- μ μ A H Large-Sample (1- α29 100% Confidence Interval for μ 1-μ 2 • 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: ( 29 2 2 2 1 2 1 2 / 2 1 n S n S z y y + ±- α Large-Sample (1- α29 100% Confidence Interval for μ 1-μ 2 • For 95% Confidence Intervals, z .025 =1.96 • Confidence Intervals and 2-sided tests give identical conclusions at same α-level: – If entire interval is above ∆ , conclude μ 1-μ 2 ∆ – If entire interval is below ∆ , conclude μ 1-μ 2 < ∆ – If interval contains ∆ , do not reject μ 1-μ 2 ≠ ∆ Example: Vitamin C for Common Cold • Outcome: Number of Colds During Study Period...
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This note was uploaded on 07/28/2011 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.

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mean2 - Comparison of 2 Population Means • Goal To...

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