MVA802S - MULTIVARIATE ANALYSIS - 1ST OPP - NOVEMBER 2023 :: NUST past examination papers between 2020 and 2024

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MVA802S - MULTIVARIATE ANALYSIS - 1ST OPP - NOVEMBER 2023

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MVA802S - MULTIVARIATE ANALYSIS - 1ST OPP - NOVEMBER 2023

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nAmlBIA UnlVERSITY

OF SCIEnCEAno TECHnOLOGY

FacultyofHealthN, atural

ResourceasndApplied

Sciences

Schoolof NaturalandApplied

Sciences

Departmentof Mathematics.

StatisticsandActuarialScience

13JacksonKaujeuaStreet

PrivateBag13388

Windhoek

NAMIBIA

T: •264612072913

E: msas@nust.na

W: www.nust.na

QUALIFICATION: Bachelor of Science Honours in Applied Statistics

QUALIFICATION CODE: 08BSHS

LEVEL: 8

COURSE CODE: MVA802S

COURSE NAME: MULTIVARIATE ANALYSIS

SESSION: NOVEMBER 2023

DURATION: 3 HOURS

PAPER: THEORY

MARKS: 100

EXAMINER

FIRST OPPORTUNITY EXAMINATION QUESTION PAPER

Dr D. B. GEMECHU

MODERATOR:

Prof L. PAZVAKAWAMBWA

INSTRUCTIONS

1. There are 6 questions, answer ALL the questions by showing all the necessary steps.

2. Write clearly and neatly.

3. Number the answers clearly.

4. Round your answers to at least four decimal places, if applicable.

PERMISSIBLE MATERIALS

1. Nonprogrammable scientific calculators with no cover.

THIS QUESTION PAPER CONSISTS OF 5 PAGES (Including this front page)

ATTACHMENTS

Two statistical distribution tables (z-and F-distribution tables)

2 Page 2

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Question 1 [12 Marks)

1.1. Briefly discuss multivariate statistical analysis and compare the multivariate techniques:

discriminant analysis and cluster analysis.

[3]

1.2. Briefly discuss a two-way MANOVA additive model. Your answer should include (the model,

three assumptions, hypothesis to be tested under two-way MANOVA and two of the most

common test statistics used to test the hypothesis).

[2+3+2+2]

Question 2 [10 Marks)

2. In assessment of an experiment of assessing plant growth, a researcher collected data on height

(y1), root length (y2 ), and width (y3 ) of three new species of plants in mm. The results obtained

were listed below:

Plant

Height length

Width

320

260

120

260

250

100

350

300

140

Assume that y~N 3 (µ, E) with unknown µ and unknown E. Then, using the matrices

approach, calculate the maximum likelihood estimate of population:

2.1. mean vector.

[2]

2.2. variance-covariance matrix.

[6]

2.3. the total sample variance.

[2]

Question 3 [34 Marks)

3.1. If y~Np(~, Ey) and x~Np(µx, Ex) are independent, then show thaty +x~Np(~ +

µx,Ey + Ex)- Hint use the uniqueness property of joint moment generating function.

[9]

3.2. Each Delicious Candy Company sore makes 4 sizes candy bars: mini (x1 ), regular (x2 ), fun (x3 )

and big size (x4 ). Assume the weights (in ounces) of the candy bars (xi, x2 , x3 , x4 ) follow

multivariate normal distribution with parameters:

(i) 2

µ=

and E = (~

!:~l

402

3.2.1. Find the conditional distribution of x 3 given (x2 , x4 ).

3.2.2. If y = 2x2 - 3x3 + x4 , then find P(y > 7)

[12]

[SJ

3.3. Suppose that Xi, x2, x3 and x4 are independent variables with the N (0, 2) distribution. Define

the following random variables:

=X1 +-Y21

Zz = X1 +Xz +2Y2

= where y1 ,y 2 ,y 3 and y4 have the N(O, 1) distribution and are independent of each other and

independent of Xi, x 2 , x 3 and x 4 . Find the variance-covariance matrix for the vector z' (z1 z2 z3 z4 )

(8)

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nAmlBIA UntVERSITY

OF SCIEnCEAnDTECHnOLOGY

FacultyofHealthN, atural

ResourceasndApplied

Sciences

Schoool f NaturalandApplied

Sciences

Departmentof Mathematics.

StatisticsandActuarialScience

13JacksonKaujeuaStreet

PrivateBag13388

Windhoek

NAMIBIA

T: +264612072913

E: msas@nust.na

W: www.nust.na

QUALIFICATION: Bachelor of Science Honours in Applied Statistics

QUALIFICATION CODE: 08BSHS

LEVEL: 8

COURSE CODE: MVA802S

COURSE NAME: MULTIVARIATE ANALYSIS

SESSION: NOVEMBER 2023

DURATION: 3 HOURS

PAPER: THEORY

MARKS: 100

EXAMINER

FIRST OPPORTUNITY EXAMINATION QUESTION PAPER

Dr D. B. GEMECHU

MODERATOR:

Prof L. PAZVAKAWAMBWA

INSTRUCTIONS

1. There are 6 questions, answer ALL the questions by showing all the necessary steps.

2. Write clearly and neatly.

3. Number the answers clearly.

4. Round your answers to at least four decimal places, if applicable.

PERMISSIBLE MATERIALS

1. Nonprogrammable scientific calculators with no cover.

THIS QUESTION PAPERCONSISTSOF 5 PAGES(Including this front page)

ATTACHMENTS

Two statistical distribution tables (z-and F-distribution tables)

4 Page 4

▲back to top

Question 1 (12 Marks]

1.1. Briefly discuss multivariate statistical analysis and compare the multivariate techniques:

discriminant analysis and cluster analysis.

[3]

1.2. Briefly discuss a two-way MANOVA additive model. Your answer should include (the model,

three assumptions, hypothesis to be tested under two-way MANOVA and two of the most

common test statistics used to test the hypothesis).

[2+3+2+2]

Question 2 (10 Marks)

2. In assessment of an experiment of assessing plant growth, a researcher collected data on height

(y1 ), root length (y 2 ), and width (y 3 ) of three new species of plants in mm. The results obtained

were listed below:

Plant

Height length

Width

320

260

120

260

250

100

350

300

140

Assume that y~N 3 (µ,I) with unknownµ and unknown I. Then, using the matrices

approach, calculate the maximum likelihood estimate of population:

2.1. mean vector.

[2]

2.2. variance-covariance matrix.

[6]

2.3. the total sample variance.

[2]

Question 3 134Marks)

3.1. If y~Np(lry, Iy) and x~Np(µx, Ix) are independent, then show that y + x~Np(lry +

µx, Iy + Ix)- Hint use the uniqueness property of joint moment generating function.

[9]

3.2. Each Delicious Candy Company sore makes 4 sizes candy bars: mini (x1), regular (x2 ), fun (x 3)

and big size (x4). Assume the weights (in ounces) of the candy bars (xi,x 2 ,x 3,x 4 ) follow

(D 1 multivariate normal distribution with parameters:

and E (

-1

3.2.1. Find the conditional distribution of x 3 given (x2, x 4 ).

[12]

= 3.2.2. If y 2x 2 - 3x 3 + x 4 , then find P (y > 7)

[5]

3.3. Suppose that x1 , x2 , x3 and x 4 are independent variables with the N (0, 2) distribution. Define

the following random variables:

Zz = X1 +x2 +2Y2

= X1 + Xz + X3 + Y23

= where y1, y2 , y3 and y4 have the N(O, 1) distribution and are independent of each other and

independent of x 1 , x 2 , x3 and x 4 . Find the variance-covariance matrix for the vector z' (z1 z2 z3 Z4)

(81

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Question 4116 Marks)

4. Bars of soap manufactured in each of two ways. The characteristics y 1 =lather and y2 = mildness

is measured. The summary statistics for 9 bars produced by each method (methods 1 and 2) are:

Assume that the observations are bivariate and follow multivariate normal distributions N (µi, :E),

for i = l and 2.

4.1. Compute the pooled covariance matrix

[3]

4.2. Conduct a test if there is any significant difference between the vector of expected mean

measurements of the two methods at 5% level of significance. Your answer should include

the following:

4.2.1. State the null and alternative hypothesis to be tested

[1]

4.2.2. State the test statistics to be used and its corresponding distribution

[2]

4.2.3. State the decision (rejection) rule and compute the tabulated value using an

appropriate statistical table

[3]

4.2.4. Compute the test statistics and write up your decision and conclusion

[7]

Question 5 (7 Marks)

5. As part of the study of love and marriage, a sample of husbands and wives were asked to respond

to these questions:

Question 1: What is the level of passionate love you feel for your partner?

Question 2: What is the level of passionate love that your partner feels for you?

Question 3: What is the level of companionate love that you feel for your partner?

Question 4: What is the level of companionate love that your partner feels for you?

The responses were recorded on the following 5-point scale. Thirty husbands and 30 wives gave

the responses in Table 3, where y1, =a 5-point- scale response to Question 1, y 2 =a 5-point-scale

response to Question 2, y 3 = a 5-point-scale response to Question 3, and y4 = a 5-point-scale

response to Question 4.

The R package output for the analysis of this data is given Table 1. Based on the output provided

Answer the following questions based on the output provided (use a = 0.05 level of significancy).

Your answer should include the hypothesis to be tested (H0 ), the test statistics, the p-value and

your decision and conclusions.

5.1. Is the husband rating wife profile parallel to the wife rating husband profile?

[3]

5.2. Test for coincident profiles at the same level of significance. Thus, are the groups at the

same level?

[2]

5.3. Test for level profiles.

[2]

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Table 1: Profile analysis software result

> library(profileR)

n >Y<-read.csv(file=mydata.csv", header=

>profres <- pbg(Y[,1:4], Y[,5], original.names= TRUE, profile.plot= TRUE)

>summary(profres)

Call:

pbg(data = Y[, 1:4], group = Y[, 5], original.names = TRUE, profile.plot = TRUE)

Hypothesis Tests:

$' Profiles are parallel'

Multivariate.Test

1 Wilks

2 Pillai

3 Hotelling-Lawley

4 Roy

$' Profiles have equal levels'

Statistic

0.878573

0.121427

0.13821

Approx.F

2.579917

num.df

den.df

p.value

0.062559

group

Residuals

$' Profiles are flat'

Sum Sq

0.234

8.869

Mean

0.2344

0.1529

F value

1.533

Pr(>F)

0.221

Hotteling T2 F

df1

df2

p-value

25.4415 8.18807

56 0.000131

Question 6 [21]

= 6.1. Let X' [Xi,X 2 , ••. ,Xp] have covariance matrix

with eigenvalue-eigenvector pairs

(},i,e1), (-12, e2 ), ••• , (Ap, ep) where -11 -12 ··• Ap 0. Let = e;x, Y2 = e~X, ..., Yp =

e~X be the principal components. Then show that If=iVar(~)= -11 + Az + ···+ Ap=

If=i Var(Xi)

[9}

6.2. At the beginning of the 20th century, one researcher obtained measurements on seven physical

characteristics for each of 3000 convicted male criminals. The characteristics he measured are:

X1 Length of head from front to back (in cm); X2 Head breadth (in cm); X3 Facebreadth (in cm);

X4 Length of left forefinger (in cm); X5 Length of left forearm (in cm); X6 Length of left foot (in

cm); X7 Height (in inches).

6.2.1. Basedon the correlation matrix provided, it the data suitable for PCA?

[Z]

6.2.2. Show that the value of -17 = 0.11

[Z]

6.2.3. What is the percentage of the total standardized variation attributed to the second

principal component?

[Z]

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6.2.4.

6.2.5.

Using the results of the principal components analysis, draw a scree plot. How many

principal components do you recommend to retain based on the scree plot?

(4)

Give a formula for computing the scores of the first principal component.

[2]

Table 2: The sample correlation matrix is:

1 0.402 0.395 0.301 0.305 0.399

0.34

1 0.618

0.15 0.135 0.206 0.183

1 0.321 0.289 0.363 0.345

1 0.846 0.759 0.661

1 0.797

0.8

1 0.736

Table 3: The eigenvalues and eigenvectors for the sample correlation matrix are:

Eigenvectors

Eigenvalues

0.285

0.211

0.294

0.435

0.453

0.434

3.82

-0.351

-0.643

-0.515

0.24

0.282

0.167

0.182

1.49

877

-0.246

-0.387

-0.113

-0.079

0.028

-0.027

0.65

-0.088

0.686

-0.693

0.126

0.127

0.023

-0.09

0.36

-0.076

-0.098

-0.112

-0.604

-0.024

-0.065

0.776

0.34

0.112

-0.01

0.029

0.33

0.27

-0.873

0.208

0.23

-0.023

0.02

-0.074

0.5

-0.787

0.024

0.352

A.7

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9 Page 9

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Table for a=.05

F c.os,df,d1f2J

I I I df2~dfl

I3 I4 I5 I6 I 7

I8 I9

I I ! I I I I I 1

161.4481 199.500 215.101 224.583 230.162 233.986 236.768 238.883 240.543

I I I I 2

18.513 1 19.000 19.164 19.2471 19.2961 19.3291 19.353 19.371 19.384

I I I 3

10.128 9.552

9.277

I I I I I 9.111 9.014 8.941

8.887

8.845

8.812

I I 4

7.7091 6.944

6.591

I I 6.388

6.2561 6.163 1 6.09421 6.041

5.998

I I 5

6.6081 5.786

5.409

I I 5.1921 5.050 4.950

4.8761 4.8181 4.772

I 10

241.882 1 243.906

19.3961 19.413

8.7861 8.745

5.9641 5.912

I 4.735 4.678

I I 6

5.9871 5.143

I I I 7

5.591 4.7371

I 5.3181 4.459

I I 5.111 4.256

I I 10

4.9651 4.103

I I 11

4.8441 3.982

4.757 4.533 1 4.3871 4.2841

4.347 4.120

4.0661 3.838

3.8631 3.633

3.7081 3.478

I 3.587

3.358

3.9721

3.6881

I 3.482

I 3.326

3.2041

3.8661

I 3.581

3.3741

3.2171

I 3.095

4.2071 4.1471 4.0991 4.060 1 3.999

3.7871

I 3.501

3.2931

3.1361

I 3.012

3.7261

I 3.438

3.2291

I 3.072

2.9481

3.6761

I 3.388

3.1781

I 3.020

2.8961

3.6371

3.3471

3.137 I

I 2.978

2.8541

3.575

3.284

3.073

2.913

2.788

I 12

I 13

I 14

I 15

I 16

I 4.7471 3.885

I 4.6671 3.806

I 4.600 1 3.739

I 4.5431 3.682 \\

I 4.4941 3.6341

3.490 1

I 3.411

3.259

3.179

3.3441 3.112

3.2871 3.056

3.2391 3.0071

3.106 1 2.996 1

I I 3.025 2.915

2.9581

I 2.901

2.8521

2.848 1

I 2.791

I 2.741

2.913

2.832

2.764

2.707

2.657

2.8491

2.767 I

2.796 1

2.7141

2.6991 2.6451

2.641 1 2.5871

2.s91 1 2.5371

2.753 1

I 2.611

I 2.602

2.687

2.604

2.534

2.5441 2.475

2.494 1 2.425

I 17

I 18

I 19

I 20

I 4.451 1 3.591 1 3.1971 2.9651

I I I 4.4141 3.555

3.160

2.9281

I I I 4.38 I

3.522

3.1271 2.8951

I I 4.351 1 3.493 3.0981 2.8661

2.810 1

I 2.773

I 2.740

I 2.71 I

2.6991

2.661 1

2.6281

2.5991

2.614

2.577

2.544 \\

2.5141

2.5481

I 2.s 10

I 2.477

I 2.44]

2.4941

I 2.456

I 2.423

2.3931

2.450 1

2.4121

2.3781

2.348 1

2.381

2.342

2.308

2.278