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Tut 3 Soln e-Learning_13092013085622591

Tut 3 Soln e-Learning_13092013085622591 - ST3131 Regression...

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ST3131 Regression Analysis AY21013/14 Semester 1 Tutorial 3 ST3131 Regression Analysis CYM 1
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Question 1 (i) 𝑋𝑋𝑋 = 𝑛 𝑥 1𝑖 𝑖 𝑥 2𝑖 𝑖 𝑥 1𝑖 𝑖 𝑥 1𝑖 2 𝑖 𝑥 1𝑖 𝑥 2𝑖 𝑖 𝑥 2𝑖 𝑖 𝑥 1𝑖 𝑥 2𝑖 𝑖 𝑥 2𝑖 2 𝑖 = 11 66 22 66 506 346 22 346 484 and 𝑋𝑋𝑦 = 𝑦 𝑖 𝑖 𝑥 1𝑖 𝑦 𝑖 𝑖 𝑥 2𝑖 𝑦 𝑖 𝑖 = 33 85 142 CYM 2 ST3131 Regression Analysis
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Question 1 (i) The least squares estimate 𝛽 ̂ = 𝑋𝑋𝑋 −1 𝑋𝑋𝑦 It is given that 𝑋𝑋𝑋 −1 = 1 28644 125188 24332 11704 24332 4840 2354 11704 2354 1210 Hence 𝛽 ̂ = 1 28644 125188 24332 11704 24332 4840 2354 11704 2354 1210 33 85 142 = 14.0 2.0 0.5 That is, 𝛽 ̂ 0 = 14.0, 𝛽 ̂ 1 = 2.0, and 𝛽 ̂ 3 = 0.5 CYM 3 ST3131 Regression Analysis
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Question 1 (ii) Test H 0 : 𝛽 1 = 𝛽 2 = 0 (i.e. 𝑦 = 𝛽 0 + 𝜖 ) against H 1 : 𝛽 1 0 and/or 𝛽 2 0 (i.e. 𝑦 = 𝛽 0 + 𝛽 1 𝑥 1 + 𝜖 or 𝑦 = 𝛽 0 + 𝛽 2 𝑥 2 + 𝜖 or 𝑦 = 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝜖 Let 𝐹 = 𝑀𝑀𝑅 𝑀𝑀𝑀 Reject H 0 at the 5% significance level if the observed F -value > F 0.05 (2,8) = 4.46 (or p-value < 0.05) CYM 4 ST3131 Regression Analysis
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Question 1 (ii) From the data, we have SST = 𝑦 𝑖 2 𝑖 − 𝑛𝑦 2 = 289 – 11(3) 2 = 190 with 10 d.f. SSR = 𝛽 ̂ 𝑋𝑋𝑦 − 𝑛𝑦 2 =122 with 2 d.f. SSE = SST – SSR = 68 with 8 d.f. CYM 5 ST3131 Regression Analysis
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Question 1 (ii) We summarize the results in the following ANOVA table ANOVA table Since F obs = 7.176 > F 0.05 (2,8) = 4.46 (or p -value < 0.05), we conclude that the overall regression is statistically significant. CYM 6 ST3131 Regression Analysis Source SS d.f. MS F -ratio p -value Regression 122 2 61 7.176 0.0164 Error 68 8 8.5 Total 190 10
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Question 1 (iii) 𝑅 2 = 𝑟 𝑦⋅12 2 = 𝑀𝑀𝑅 𝑀𝑀𝑆 = 122 190 = 0.6421 About 64% of the variation in Y is explained by x 1 and x 2 .
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