Only 5.5

Provide Example 5.3.

5.1. (a) Evaluate 72, for testing Ho: m' = [7, 11], using the data 2 12 8 9 X = 6 9 10 (b) Specify the distribution of 72 for the situation in (a). (c) Using (a) and (b), test Ho at the a = .05 level. What conclusion do you reach? 5.2. Using the data in Example 5.1, verify that 72 remains unchanged if each observation x;, j = 1,2, 3; is replaced by Cx;, where c=[: 4] Note that the observations Cx/ = x/ 1 - * ; 2 x ; 1 + X ; 2_ yield the data matrix (6 -9) (10 -6) (8-3) L(6 + 9 ) (10 + 6) (8 + 3) 5.3. (a) Use expression (5-15) to evaluate 72 for the data in Exercise 5.1. (b) Use the data in Exercise 5.1 to evaluate A in (5-13). Also, evaluate Wilks' lambda. 5.4. Use the sweat data in Table 5.1. (See Example 5.2.) (a) Determine the axes of the 90% confidence ellipsoid for u. Determine the lengths of these axes. (b) Construct Q-2 plots for the observations on sweat rate, sodium content, and potassium content, respectively. Construct the three possible scatter plots for pairs of observations. Does the multivariate normal assumption seem justified in this case? Comment. 5.5. The quantities x, S, and S are given in Example 5.3 for the transformed microwave- radiation data. Conduct a test of the null hypothesis Ho: u' = [.55, .60] at the a = .05 level of significance. Is your result consistent with the 95% confidence ellipse for u pic- tured in Figure 5.1? Explain.Example 5.3 (Constructing a confidence ellipse for mu) Data for radiation from microwave ovens were introduced in Examples 4.10 and 4.17. Let X1 = V measured radiation with door closed and X2 measured radiation with door openFor the n = 42 pairs of transformed observations, we nd that i: .564 s .0144 .0112 .603 ' \" .0112 .0146 ' . s" _ 203.013 163.391 -163.391 200223 The eigenvalue and eigenvector pairs for S are = .026. 61 = [.204, .210] .12 = .002, e5 = [210 .204] The 95% condence ellipse for :3. consists of all values (n.1, n2) satisfying _ 203.013 -163.391 .564 n. \"[564 ' "'1' '503 \"'11 [-163.39] 200.223] [.603 - n2] S2_(41) F2. l 05) _ 40 sci or. since 13.44.05) = 3.23, 4200301810564 #1): + 4200022810603 3211 84(163.391)(564 m}(.603 m) s 6.62 13'. To see whether p.' = [.562, .539] is in the confidence region, we compute 42(203.018)(.564 .562)2 + 42(200228) (.603 539)2 . 84(163391)(.564 562)(.603 .539) = 1.30 s 6.62 We conclude that p' = [.562, .589] is in the region. Equivalently, a test of Ho: ,2 = [:22] would not be rejected in favor of H1: p at [:23] at the a = .05 level of significance. The joint condence ellipsoid is plotted in Figure 5.1. The center is at i' = [.564, .603], and the half-lengths of the major and minor axes are given by 2. ]mn 1) _ , 2(_4_1) _ )'l \"(n _ p) Fp.n-'p(a ) 42(40) _(3 '23) , {ii(n - 1):: ._ _(__241) respectively. The axes lie along ei = [ .704, .710] and ez = [ .710 .704] when these vectors are plotted with x as the origin. An indication of the elongation of the con- dence ellipse is provided by the ratio of the lengths of the major and minor axes. This ratio is and P(" 1) 2W no: - p) F\""\"(a) WT. .161 l n - 1 V)? .045 ZV :n_ pgfp .n-P(a) 2 Figure S.l A 95% confidence . ellipse for ,u. based on microwave- E . radiation data. 1116 lengih of the major axis is 3.6 times the length of the minor axis. l