ST5210 Multivariate Data Analysis 2022/23 Semester 1 Tutorial 4
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Department of Statistics and Applied Probability
2022/23 Semester 1
ST5210 Multivariate Data Analysis
Tutorial 4
Submit a softcopy in pdf format with filename “name student_number Tut4.pdf” to the LumiNUS before 7:00pm on 23/10/22. Show all your working leading to the answers or conclusions. R should be used to do Questions 3, 4 and 5, but not for Questions 1 and 2. Late submission and photos (e.g. jpg files) will not be accepted. You may convert your hard copy solution into a pdf file using any scan Apps. Show all your workings leading to the answers or conclusions. Please be reminded that copy others’ solutions is not allowed.
Question 1
Independent observations on two measurements (p = 2) are collected for each of the two treatments. The observations are given as follows
Treatment 1: ( 6(6)) , (9(2)) , (9(4)). The summary statistics are 
1 = (8(4)) and S1 = ( 
) Treatment 2: (5(6)) , (8(10)) , (11(8)). The summary statistics are 
2 = (8(8)) and S2 = (3(4) 9(3))
(a) Test H0 : u1 = u2 at the significance level a = 0.05. State the assumptions that you made.
(b) Find a 95% confidence region for the vector 6 , where 6 = u1 − u2 .
Question 2
Independent observations on two measurements (p = 2) are collected for each of the three treatments. The observations are given as follows
Treatment 1: (7(6)) , (9(5)) , (5(7)). The summary statistics are 
1 = (7(6)) and S1 = ( 
) Treatment 2: (3(3)) , (5(1)) , (4(2)). The summary statistics are 
2 = (4(2)) and S2 = ( 
) Treatment 3: (3(2)) , (4(6)) , (5(4)). The summary statistics are 
3 = (4(4)) and S3 = (1(4) 1(1)) The summary statistics for all the observations are 
= (5(4)) and S = (
)
(a) Compute SSCPtrt, SSCPres, SSCPtot and hence verify that SSCPtrt + SSCPres = SSCPtot .
(b) Use the answers in part (a) to construct the corresponding MANOVA table. Compute the test statistic
Λ = 
.
(c) When p = 2 and a ≥ 2, it can be shown that under H0 : u1 = ⋯ = ua , the statistic (
) (
)
has an F-distribution with 2(a − 1) and 2(∑g(a)=1 ng − a − 1) degrees of freedom,
where ng is the number of observations in group g. Use this result to test H0 : u1 = u2 = u 3 at the significance level α = 0.05.
Question 3
Each of 15 students wrote an informal and a formal essay. The variables recorded were the number of words and verbs:
y1 = number of words in the informal essay
y2 = number of verbs in the informal essay
x1 = number of words in the formal essay
x2 = number of verbs in the formal essay
E (X ). Test H0 : ud = 0 against H1 : ud ≠ 0.
Question 4
Twenty engineer apprentices and 20 pilots were given six tests. The variables were
y1 = intelligence
y2 = form relations
y3 = dynamometer
y4 = dotting
y5 = sensory motor coordination
y6 = perseveration
The data are given in the file “t4q4.txt” (group = 1 for engineer apprentices and 2 for pilots). (a) Test H0 : u1 = u2 against H1 : u1 ≠ u2 , where
u i = E((Y1 Y2 Y3 Y4 Y5 Y6)T) for Group i with i = 1, 2.
61, ⋯ , 66 .
Question 5
A study compared judges’ scores on fish prepared by three methods. Twelve fish were cooked by each method, and several judges tasted fish samples and rated each on four variables: y1 = aroma, y2 = flavour, y3 = texture, y4 = moisture. The data are given in the data file “t4q5.txt” . Each entry is an average score for the judges on that fish. Compare the three methods using MANOVA. That is, to test H0 : u1 = u2 = u 3 against H1 : H0 is not true, where u i = E((Y1 Y2 Y3 Y4)T ) for Group i with i = 1, 2, 3.
2022-10-20