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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Exercises 621 Exercises 11.71 Consider the general linear model Y = β 0 + β 1 x 1 + β 2 x 2 +···+β k x k + ε, where E(ε) = 0 and V (ε) = σ 2 . Notice that ˆβ i = a ′ ˆβ, where the vector a is defined by { 1, if j = i, a j = 0, if j ≠ i. Use this to verify that E( ˆβ i ) = β i and V ( ˆβ i ) = c ii σ 2 , where c ii is the element in row i and column i of (X ′ X) −1 . 11.72 Refer to Exercise 11.69. a Is there evidence of a quadratic effect in the relationship between Y and x? (Test H 0 : β 2 = 0.) Use α = .10. b Find a 90% confidence interval for β 2 . 11.73 The experimenter who collected the data in Exercise 11.68 claims that the minimum value of E(Y ) occurs at x = 1. Test this claim at the 5% significance level. [Hint: E(Y ) = β 0 + β 1 x + β 2 x 2 has its minimum at the point x 0 , which satisfies the equation β 1 + 2β 2 x 0 = 0.] 11.74 An experiment was conducted to investigate the effect of four factors—temperature T 1 , pressure P, catalyst C, and temperature T 2 —on the yield Y of a chemical. a The values (or levels) of the four factors used in the experiment are shown in the accompanying table. If each of the four factors is coded to produce the four variables x 1 , x 2 , x 3 , and x 4 , respectively, give the transformation relating each coded variable to its corresponding original. T 1 x 1 P x 2 C x 3 T 2 x 4 50 −1 10 −1 1 −1 100 −1 70 1 20 1 2 1 200 1 b Fit the linear model to the following table of data. Y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + ε x 4 +1 −1 x 3 x 3 −1 1 −1 1 −1 x 2 −1 22.2 24.5 24.4 25.9 x 1 1 19.4 24.1 25.2 28.4 −1 22.1 19.6 23.5 16.5 +1 x 2 1 14.2 12.7 19.3 16.0 c Do the data present sufficient evidence to indicate that T 1 contributes information for the estimation of Y ? Does P? Does C? Does T 2 ? (Test the hypotheses, respectively, that β 1 = 0, β 2 = 0, β 3 = 0, and β 4 = 0.) Give bounds for the p-value associated with each test. What would you conclude if you used α = .01 in each case?

www.downloadslide.com 622 Chapter 11 Linear Models and Estimation by Least Squares 11.75 Refer to Exercise 11.74. Find a 90% confidence interval for the expected yield, given that T 1 = 50, P = 20, C = 1, and T 2 = 200. 11.76 The results that follow were obtained from an analysis of data obtained in a study to assess the relationship between percent increase in yield (Y ) and base saturation (x 1 , pounds/acre), phosphate saturation (x 2 , BEC%), and soil pH (x 3 ). Fifteen responses were analyzed in the study. The least-squares equation and other useful information follow. ŷ = 38.83 − 0.0092x 1 − 0.92x 2 + 11.56x 3 , S yy = 10965.46, SSE = 1107.01, ⎡ 10 4 (X ′ X) −1 ⎢ = ⎣ 151401.8 2.6 100.5 −28082.9 2.6 1.0 0.0 0.4 100.5 0.0 8.1 5.2 −28082.9 0.4 5.2 6038.2 a Is there sufficient evidence that, with all independent variables in the model, β 2 < 0? Test at the α = .05 level of significance. b Give a 95% confidence interval for the mean percent increase in yield if x 1 = 914, x 2 = 65 and x 3 = 6. ⎤ ⎥ ⎦ . 11.13 Predicting a Particular Value of Y by Using Multiple Regression In Section 11.7, we considered predicting an actual observed value of Y in the simple linear regression, setting the single independent variable x = x ∗ . The solution was based heavily on the properties of error = Y ∗ − Ŷ ∗ , where Ŷ ∗ = ˆβ 0 + β 1 x ∗ was observed to be a predictor of the actual value of Y and an estimator for E(Y ) as well. The same method will be used in this section to provide the corresponding solution in the multiple linear regression case. Suppose that we have fit a multiple linear regression model Y = β 0 + β 1 x 1 + β 2 x 2 +···+β k x k + ε and that we are interested in predicting the value of Y ∗ when x 1 = x ∗ 1 , x 2 = x ∗ 2 ,..., x k = x ∗ k . We predict the value of Y ∗ with Ŷ ∗ = ˆβ 0 + ˆβ 1 x ∗ 1 + ˆβ 2 x ∗ 2 +···+ ˆβ k x ∗ k = a′ ˆβ, where ⎡ ⎤ 1 x ∗ 1 a = x2 ∗ . ⎢ ⎥ ⎣ . ⎦ x ∗ k

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