UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Monday, 1 June, 2015 9:00 am to 12:00 pm PAPER 33 APPLIED STATISTICS Attempt no more than FOUR questions, with at most THREE from Section A. There are SIX questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 SECTION A 1 Suppose that Y = (Y1, ... , Yn)T satisfies Y = X6 + E, where X is a known n xp matrix (p < n), 3 = (61, ... , Bp)T is unknown, E = (E1, . .. , En)" where E1, . .. , En are independent normally distributed random variables with mean zero and variance o2 (unknown), and where T denotes the transpose. Assume that XT X is invertible. Find 3 that minimises R(6) = (Y - X6)T (Y - X6). Define the fitted values Y and show that Y = HY for some suitable matrix H (which you should specify in terms of X) satisfying HT = H and HH = H. Define the residuals & and show that ê Ê = YTGY for some matrix G satisfying GT = G and GG = G. Find cov( E, Y). An investigation is carried out into the use of a windmill to generate electricity. There are 25 observations of the electrical output together with the corresponding values of the wind velocity. In the (edited) R output below, the electrical outputs and wind velocities are in outputand velocity, respectively. Figure 1 shows a plot of the data and various residual plots. For these plots, res1, res2 and res3 contain the residuals from models wind1.lm, wind2.lmand wind3.lm, respectively. The corresponding fitted values are in fit1, fit2and fit3. Write down the algebraic form of the model fitted in wind1.lm. Several numerical values in the output to the directive anova(wind1.lm)have been replaced by asterisks. Write down what these numerical values should be. Explain how the value of Multiple R-squaredis calculated. With reference to the plots, explain why the model wind2.lm is fitted to the data. Write down the algebraic form of the model fitted in wind2.lm. What hypothesis is being tested in the line marked (A)in the output? Carry out this test in detail, and give your conclusion. Compare the models wind2.lmand wind3.lm. What further information or plots would help you to compare the adequacy of these two models? > wind1.lm <- lm(output~velocity) > anova(wind1.lm) Df Sum Sq Mean Sq F value velocity * 8.9296 * 160.26 7.546e-12 Pr(>F) Residuals * 1.2816 0.0557 > summary(wind1.lm) <output omitted> Multiple R-squared: 0.8745 > velocity2 <- velocity*velocity > wind2.lm <- lm(output~velocity+velocity2) Part III, Paper 33
UNIVERSITY OF CAMBRIDGE 3 > summary(wind2.lm) Estimate Std. Error t value Pr(>|t|) (Intercept) -1.155898 0.174650 -6.618 1.18e-06 velocity 0.722936 0.061425 11.769 5.77e-11 velocity2 -0.038121 0.004797 -7.947 6.59e-08 (A) Residual standard error: 0.1227 on 22 degrees of freedom