Two-Sample Hypothesis Testing and Confidence Intervals in Statistics, Exams of Data Analysis & Statistical Methods

Download Two-Sample Hypothesis Testing and Confidence Intervals in Statistics and more Exams Data Analysis & Statistical Methods in PDF only on Docsity! STATISTICS 571 TA: Perla Reyes DISCUSSION 8 Review 1. Two Sample Testing and C.I (a) Paired Sample Suppose (X1, Y1), (X2, Y2), · · · , (Xn, Yn) are paired samples from (X, Y ) and E(X)= µ1, E(Y)= µ2. Suppose D = X − Y is a random sample from a N(µD, σ2D), where µD = µ1 − µ2. i. The test statistic for H0 : µD = d versus HA : µD 6= d is T = D̄ − d SD/ √ n ∼ tn−1 ii. A (1− α)% C.I. for µD is given by d̄− tn−1,α/2 sd√ n ≤ µD ≤ d̄ + tn−1,α/2 sd√ n (b) Independent Sample, assuming σ21 = σ 2 2 Suppose X1, X2, ..., Xn1 is a random sample from N(µ1, σ 2 1) and Y1, Y2, ..., Yn2 is a ran- dom sample from N(µ2, σ22). Suppose those two samples are independent and σ 2 1 = σ 2 2 = σ2. i. The test statistic for H0 : µ1 − µ2 = a versus HA : µ1 − µ2 6= a is T = (X̄ − Ȳ )− a Sp √ 1 n1 + 1n2 ∼ tn1+n2−2, where S2p = (n1 − 1)S21 + (n2 − 1)S22 n1 + n2 − 2 ii. A (1− α)% C.I. for µ1 − µ2 is x̄− ȳ − tn1+n2−2,α/2sp √ 1 n1 + 1 n2 ≤ µ1 − µ2 ≤ x̄− ȳ + tn1+n2−2,α/2sp √ 1 n1 + 1 n2 (c) Independent Sample σ21 6= σ22 Suppose X1, X2, ..., Xn1 is a random sample from N(µ1, σ 2 1) and Y1, Y2, ..., Yn2 is a ran- dom sample from N(µ2, σ22). Suppose those two samples are independent, but σ 2 1 6= σ22 i. The test statistic for H0 : µ1 − µ2 = a versus HA : µ1 − µ2 6= a is T = (X̄ − Ȳ )− a√ S21 n1 + S 2 2 n2 ∼ t with adf = (vr1 + vr2) 2 ( vr 2 1 n1−1) + ( vr22 n2−1) where vr1 = S21/n1 and vr2 = S 2 2/n2. ii. A (1− α)% C.I. for µ1 − µ2 is x̄− ȳ − tadf,α/2 √ s21 n1 + s22 n2 ≤ µ1 − µ2 ≤ x̄− ȳ + tadf,α/2 √ s21 n1 + s22 n2 (d) Test of equal variance (Levene’s Test) i. Determine the median of each sample ii. Calculate the absolute value of all deviates from the median iii. If, in either sample, there is an odd number of observations, delete exactly one “0” iv. Perform an independent sample T-test with variances assumed equal email: reyes@stat.wisc.edu 1 Office: 248 MSC M2:30-3:30 R3:30-4:30