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Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 13.13 Analysis of Variance Using Linear Models 703 To test H 0 : β 1 = 0, we construct the t statistic (see Section 11.12): t = ˆβ 1 − 0 s √ −.75 = c 11 .7656 √ 1/3 =−1.697. Because we are interested in a two-tailed test, the associated p-value is 2P(t < −1.697) = 2P(t > 1.697), where t is based on 10 df. Thus, using Table 5, Appendix 3, we obtain .05 < P(t > 1.697)

704 Chapter 13 The Analysis of Variance www.downloadslide.com { 1, if treatment B is used, x 4 = 0, otherwise, { 1, if treatment C is used, x 5 = 0, otherwise. We want to test the hypothesis that there are no differences among treatment means, which is equivalent to H 0 : β 3 = β 4 = β 5 = 0. Thus, we must fit a complete and a reduced model. (See Section 11.14.) For the complete model, we have ⎡ ⎤ ⎡ ⎤ 10.1 1 1 0 1 0 0 12.2 1 0 1 1 0 0 11.9 1 0 0 1 0 0 11.4 1 1 0 0 1 0 12.9 1 0 1 0 1 0 12.7 1 0 0 0 1 0 Y = and X = . 9.9 1 1 0 0 0 1 12.3 1 0 1 0 0 1 11.4 1 0 0 0 0 1 ⎢ 12.1 ⎥ ⎢ 1 1 0 0 0 0⎥ ⎣ ⎦ ⎣ ⎦ 13.4 1 0 1 0 0 0 12.9 1 0 0 0 0 0 A little matrix algebra yields, for this complete model, SSE C = Y ′ Y − ˆβ ′ X ′ Y = 1721.760 − 1721.225 = .535. The relevant reduced model is Y = β 0 + β 1 x 1 + β 2 x 2 + ε, and the corresponding X matrix consists of only the first three columns of the X matrix given for the complete model. We then obtain ⎡ ⎤ 12.225 ˆβ = (X ′ X) −1 X ′ Y = ⎣ −1.350 ⎦ .475 and SSE R = Y ′ Y − ˆβ ′ X ′ Y = 1721.760 − 1716.025 = 5.735. It follows that the F ratio appropriate to compare these complete and reduced models is F = (SSE R − SSE C )/(k − g) SSE C /(n − [k + 1]) = (5.735 − .535)/(5 − 2) (.535)/(12 − 6) = 1.733 .0892 = 19.4. The tabulated F for α = .05, ν 1 = 3, and ν 2 = 6 is 4.76. Hence, if we choose α = .05, we reject the null hypothesis and conclude that the data present sufficient evidence to indicate that differences exist among the treatment means. The associated p-value is given by P(F > 19.4). Table 7, Appendix 3, establishes that p-value < .005. The applet F-Ratio Probabilities and Quantiles, applied with 3 numerator and

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