Assignment 2 Solutions - Regression Analysis | ISYE 6414, Assignments of Systems Engineering

Download Assignment 2 Solutions - Regression Analysis | ISYE 6414 and more Assignments Systems Engineering in PDF only on Docsity! Solution to homework 2 2.1 (a) Yes, the conclusion is warranted at α = 5% since the confidence interval does not include 0. (b) No problem. Since the range of X does not extend to zero. 2.4 (a) i. 99% C.I. for β1 b1 ± t0.995,118s{b1} = 0.0388± 2.618× 0.01277 = (0.0054, 0.07226) ii. At a significance level of 1%, GPA increases between 0.0054 and 0.07226 for each additional unit of entrance test score. iii. No, CI does not include zero. iv. If the confidence interval includes zero, it may not be meaningful to relate GPA to entrance exam score. (b) i. H0 : β1 = 0 , Ha : β1 6= 0 ii. Reject H0 if |t∗| > t(0.995, 118) = 2.618 iii. |t∗| = b1s{b1} = 0.0388 0.01277 = 3.0384 > 2.618 Thus, reject H0. There exists a linear relationship between entrance exam score and GPA at α = 0.01. (c) i. p-value = 2× P{t118 > 3.0384} = 0.002929 ii. Since p−value < 0.01, reject H0. It supports the conclusion in part (b). 2.6 (a) i. 95% C.I. for β1 = b1±t0.975,8s{b1} = 4±2.306×0.469 = (2.9185, 5.0815) ii. The number of the ampules broken increases between 2.9185 and 5.0815 for each additional number of transferred shipment route at a significance level of 5%. (b) i. H0 : β1 = 0 , Ha : β1 6= 0 ii. Reject H0 if |t∗| > t(0.975, 8) = 2.306 iii. |t∗| = b1s{b1} = 4 0.469 = 8.5288 > 2.306 Thus, reject H0. There exists a linear relationship between the number of transferred shipment route and the broken ampules at α = 0.05. iv. p-value = 2× P{t8 > 8.5288} = 0.000027 (c) i. 95% C.I. for β0 b0 ± t0.975,8s{b0} = 10.2± 2.306× 0.6633 = (8.6704, 11.7296) 1 ii. With a 95% confidence level, the number of broken ampules lies be- tween 8.6704 and 11.7296 when there is no transferred shipment route. (d) i. H0 : β0 ≤ 9 , Ha : β0 > 9 ii. Reject H0 if |t∗| > t(0.975, 8) = 2.306 iii. |t∗| = b0s{b0} = 10.2−9 0.6633 = 1.809 < 2.306 Thus, fail to reject H0. We conclude the number of of Y does not exceed 9 when X = 0 at α = 0.025. iv. p-value = P{t8 > 1.809} = 0.054 (e) i. δ = 20.5 = 4 From Table B.5 for α = 0.05, df = 8, δ = 4, power = 0.94. ii. δ = 11−90.75 = 2.67 From Table B.5, for α = 0.05, df = 8, when δ = 2, power = 0.42 and when δ = 3, power = 0.75. By interpolation, δ = 2.67, power = 0.42 + 2.67−23−2 × (0.75− 0.42) = 0.64. 2.13 (a) i. s(Ŷh) = √ 0.3883× ( 1120 + (28−24.725)2 2379.925 ) = 0.07 Ŷh = 2.11 + 0.0388× 28 = 3.1964 95% C.I. for E[Yh|X = 28] Ŷh ± t0.975,118s{Ŷh} = 3.1964± 1.9803× 0.07 = (3.06, 3.34) ii. If many independent samples are taken where the level of X(entrance score) are the same as in the data set, 95% of the intervals will contain true value of E{Yh} (b) i. s(pred) = √ 0.3883× (1 + 1120 + (28−24.725)2 2379.925 ) = 0.6271 95% P.I. for Yh(new) = 3.1964± 1.9803× 0.6271 = (1.95, 4.44) ii. If many independent samples are taken where the level of X(entrance score) are the same as in the data set, 95% of the intervals will contain true value of predicted GPA when X=28. (c) Yes, PI is wider than CI. Since PI has additional variance, PI should be wider than CI. (d) i. W 2 = 2F (0.95, 2, 118) = 6.146, W = 2.479 95% C.B. for Yh(new) = 3.1964± 2.479× 0.07 = (3.02, 3.37) ii. Yes, CB must encompass the entire regression line where the confidence interval apply only at the single level Xh. 2