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Quiz 3_columns

# Quiz 3_columns - Inference on the Means of 2 Populations...

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Inference on the Means of 2 Populations, Variances Known Assumptions: Both samples are random samples, independent, and are normal (if not normal, C.L.T. apply) E(Xbar1 – Xbar2) = E(Xbar1) – E(Xbar2) = μ1-μ2 V(Xbar1-Xbar2)=V(Xbar1)+V(Xbar2)= + σ12n1 σ22n2 = - -( - ) + Z X1 X2 μ1 μ2 σ12n1 σ22n2 Hypothesis Testing on the Difference in Means, Variances Known Null hypothesis: μ1-μ2 = delta_0 Test stat: = - - + Z0 X1 X2 ∆0σ12n1 σ22n2 Alternative Hyp: != delta_0—P-value: P above Z_0 and P below –Z_0, P = 2[1-phi(|Z_0|)] Rejection: Z_0 > Z_a/2 or Z_0 < -Z_a/2 Hyp > Delta_0—P-value: P above Z_0, P = 1-phi(Z_0), reject Z_0 > Z_a Hyp < Delta_0—P-value: P below Z_0, P = phi(Z_0), reject Z_0 < Z_a Type II Error and Choice of Sample Size Sample Size for 2-sided Alt Hyp. On Difference in Means, Variances known, n1=n2 = + ( + )( - ) n Zα2 Zβ2 σ12 σ22 Δ Δσ 2 For one-sided: = + ( + )( - ) n Zα Zβ2 σ12 σ22 Δ Δ0 2 = - - + - - - - β ϕZα2 Δ Δ0σ12n1 σ22n2 ϕ Zα2 Δ + Δ0σ12n1 σ22n2 Confidence Intervals on the Difference in Means, Variances Known - - / + - - x1 x2 zα 2σ12n1 σ22n2 μ1 μ2 x1 + / + x2 zα 2σ12n1 σ22n2 Sample Size for a specified E on the Difference in Means, and variances known when n1=n2 = ( + ) n Zα2E2 σ12 σ22 Inference on the Means of Two Populations, Variances Unknown Hypothesis Testing on the Difference in Means Pooled estimator of σ 2 = S p 2 : = - +( - ) + - Sp2 n1 1S12 n2 1 S22n1 n2 2 = - -( - ) + T X1 X2 μ1 μ2 Sp1n1 1n2 Alt. Hyp: != delta_0: P-value: Sum of P above t_0 and below –t_0. Reject if t_0>

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Quiz 3_columns - Inference on the Means of 2 Populations...

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