Question 1 115Marks)
1.1. Briefly explain Principal components analysis (PCA)and state three assumptions of PCA [5]
1.2. State three reason why multivariate approach to hypothesis testing instead of univariate
approach in inference about multivariate mean vectors.
[3]
1.3. Briefly discuss a One-Sample Profile Analysis. Your answer should include (Definition of profile
analysis, assumptions of the variable, hypothesis to be tested, the contrast matrix, the test
statistics and the rejection rule).
[7]
Question 2 (10 Marks)
2. The following data represent measurements of blood glucose levels on three occasions
(y1, y 2 and y 3 ) for 4 women patients who gave consent to participate on the study. The results
obtained are listed below:
Individual
Y1
Yz
60
69
62
2
56
53
84
3
62
75
68
4
73
70
64
Then compute
2.1". the sample mean vectory.
[3]
2.2. the sample variance-covariance matrix, S.
[S]
2.3. the total sample variance.
[2]
Question 3 [23 Marks]
= 3.1. Given that y~Np(µy, !:y) a random variable z is defined as a linear combination of y
= = (y1 ,y 2 , ... ,yp)' as zi a1yi 1 + a 2 Yiz + · ·+ap Yip ,for i 1, 2, ... , n, then show
that z = a'y, where a' = (a 1 a 2 ••· ~) and y is the sample mean vector of the p-variables.
[S]
3.2. Suppose that test 1 (x1 ) and test 2 (x2) scores of MVA students that follow a bivariate normal
= = = distribution with parameters mean µ1 70 and µ2 60, the standard deviations cr1
10 and cr2 = 15, and p = 0.6.
3.2.1. Express a given information in the form of matrix notation, thus what would beµ and
!:?
[~
3.2.2. If a student is selected randomly, then find the probability that
3.2.2.1. the score of a randomly selected student is above 75 on test 2?
[4]
3.2.2.2. the score of a randomly selected student is above 75 on test 2 given that the student
scored 80 on Test 1.
[6]
3.2.2.3. the sum of the score of a randomly selected student on both tests is above 150. [3]
3.2.2.4. the students performance in test 1 is better than test 2.
[3]
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