In the first phase of a study of the expenditure of transporting astronauts via Elon Musk's SpaceX space program versus NASA's space program was surveyed. Expenditure data on X1 = fuel, X2 = repair, and X3 = capital expenditure in billions of dollars, were all measured on a per-1000 mile basis, for n1 = 36 Elon Musk and n2 = 23 NASA trips. The survey gave sample mean vectors (in billions of dollars) of 1=[12.219

8.113

9.590];

2=[10.106

10.762

18.168]. The survey data gave the 2 sample variance-covariance matrices as, 1=[223.0134 12.3664 2.9066

12.3664 17.5441 4.7731

2.9066 4.7731 13.9633];

2=[4.3623 0.7599 2.3621

0.7599 25.8512 7.6857

2.3621 7.6857 46.6543]. The following pooled variance covariance matrix Spooled and associated inverse were also reported as follows, =[15.8112 7.8550 2.6959

7.8550 20.7458 5.8960

2.6959 5.8960 26.5750] [(11+12)]1=[1.0939 0.4084 0.0203

0.4084 0.8745 0.1525

0.0203 0.1525 0.5640]. a) Test for differences in the mean cost vectors. Set = 0.05. State your null hypothesis and carry out the appropriate T2 test using Spooled. Use the appropriate F3,55 critical value. (4 marks) b) Did you reject H0 in part a)? If the hypothesis of equal cost vectors was rejected, find the linear combination of the mean components most responsible for the rejection. (3 marks) c) Construct the 95% simultaneous confidence intervals for the pairs of mean components 1j - 2j for j = 1,2,3. (4 marks) d) Which costs, if any, appear to be quite different? (1 mark) e) How is the pooled sample variance covariance matrix, Spooled calculated from S1 and S2? Give the mathematical equation. (1 mark) MATH1309 MULTIVARIATE ANALYSIS Assignment 2 (125 marks) 2 f) Comment on the validity of the assumption that 1 = 2 used in the above analysis. (1 mark) g) Use the large sample variant of the T2 test assuming unequal population variance covariance matrices, i.e. 1 2, to carry out the test of H0. Use the appropriate 2 critical value. (4 marks) h) Did you reject H0 in part g) above? Justify your answer. (2 marks) Critical values of F distribution (you choose the correct one): F3,55 F3,55(0.10) = 2.18 F3,55(0.05) = 2.76 F3,55(0.025) = 3.34 F3,55(0.01) = 4.13