SAT802S - SAMPLING THEORY - 2ND OPP - JANUARY 2025 :: NUST past examination papers between 2021 and 2025

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SAT802S - SAMPLING THEORY - 2ND OPP - JANUARY 2025

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SAT802S - SAMPLING THEORY - 2ND OPP - JANUARY 2025

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nArnl BIA un IVERSITY

OF SCIEnCE Ano TECHnOLOGY

FacultyofHealthN, atural

ResourceasndApplied

Sciences

Schoolof NaturalandApplied

Sciences

Departmentof Mathematics,

Statisticsand Actuarial Science

13JacksonKaujeuaStreet

Private Bag13388

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: SAMPLING THEORY

COURSECODE: SAT802S

DATE: JANUARY 2025

SESSION: 1

DURATION: 3 HOURS

MARKS: 100

EXAMINER:

MODERATOR:

SECOND OPPORTUNITY/ SUPPLEMENTARY: QUESTION PAPER

Mr. Jan Johannes Swartz

Prof. Opeoluwa Oyedele

INSTRUCTIONS:

1. Answer all questions on the separate answer sheet.

2. Please write neatly and legibly.

3. Do not use the left side margin of the exam paper. This must be allowed for the

examiner.

4. No books, notes and other additional aids are allowed.

5. Mark all answers clearly with their respective question numbers.

PERMISSIBLE MATERIALS:

1. Non-Programmable Calculator

ATTACHMENTS

1. Z - Table

2. T-Table

This paper consists of 4 pages including this front page

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Question 1 [25 marks]

1.1. What is meant by the sampling distribution of a statistic?

[S]

1.2. Select all the 20 samples of size three from the population of six students in Table 2.1,

below without replacement.

1.2.1 From each sample, find the 95% confidence limits for the population mean of

the math scores with the known population variance and its estimates; use the

normal deviate Z = 1.96 in both cases.

[10]

1.2.2 Compare the average of the confidence widths obtained with the estimates of

variance with the exact width for the case of known variance.

[10]

Table 2.1. SAT verbal and math scores.

Student

Total

Mean

Variance

rr2

C.V (%)

Verbal

520

690

500

580

530

-180

8300

550

4866.67

5840

76.-12

13.89

Math

)';

670

720

650

720

560

700

4020

670

3066.67

8680

60.66

9.05

C.V. = cocflicicnL of variaLion.

Question 2 [28 marks]

2.1. Provide and explain four basic criteria for the acceptability of a sampling method? [8]

2.2. The investigator samples 10 one-acre plots by simple random sampling and counts the

number of trees (y) on each plot. She also has aerial photographs of the plantation from

which she can estimate the number of trees (x) on each plot of the entire plantation. Hence,

she knows µx = 19.7 and since the two counts are approximately proportional through the

origin, she uses a ratio estimate to estimate µY

Sampling Theory (SAT802S)

2nd Opportunity January 2025

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Table 1: To estimate the average number of trees per acre on a 1000- acre plantation

Plot

mean

Actual 110. per acre Y

22.10

Aerial est.imate X

20.80

Yi - 1"Xi

0.5625

0.1250

0.7,500

-2.5625

0.2500

-1.12.50

3.1250

1.312,5

l.[)000

-3.9375

2.2.1. Study Figures 1 and 2 and discuss the suitability of using ratio estimates.

[5]

'----,---

~--- ,,.-

Figure 1: Scatter plot.

Figure 2: Regression output

2.2.2. Construct the approximate 95% confidence interval for µY

[15]

Question 3 [17 marks]

3.1. The New York Times of February 25, 1994, summarized the results of a survey

conducted by Klein Associates, Inc. on 2000 lawyers on sexual advances in the office.

Between 85 and 98% responded to the questions in the survey; 49% of the responding

women and 9% of the responding men agreed that some sorts of harassment exist in the

offices. Assume that the population of lawyers is large and there are equal numbers of

female and male lawyers, and ignore the nonresponse; that is, consider the respondents to

be a random sample of the 2000 lawyers.

3.1.1 Find the standard errors for females and males.

Sampling Theory (SAT802S)

[5]

2nd Opportunity January 2025

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3.2. To estimate the percentage of people that carries a viral infection which produces AIDS,

128 people are examined and 72 of them are found to be infected. Calculate the standard

error of the estimated proportion and compute a 95% confidence interval for the

population proportion?

[7]

3.3. If no information of P (proportion} is provided when determining the sample size of a

population, find the error of the estimation e for n = 2000, Consider a= 0.05 for both

cases.

[5]

Question 4 [30 marks]

4.1.

[10]

Let there exist a populati,on

l.J = {l, 2, 3} '.Vit.h tbe following design:

Give t. nc first-order inch ··ion proba.1.Jilitics. Give the ,.t,niance-covaria;ncc

.:na-

t:rix .6.. of :indicator ·variabl<~s for inclusion in the sm:np e. Give the variauce

-rnatrix of chc~ unbiased est.irnat,or for the total.

4.2. Between the 100 computer corporations in Namibia, the average of employee sizes for

the largest 10 and smallest 10 corporations were known to be 300 and 100, respectively.

For a sample of 20 from the remaining 80 corporations, the mean and standard deviation

were 250 and 110, respectively. For the total employee size of the 80 corporations, find the

a} Estimate of the total,

[3]

b} Standard error of the estimate, and

[3]

c} 95% confidence limits.

[5]

4.3. Write a short description on the importance of the normal distribution in sampling

theory

[5]

4.4. List 4 properties of the normal probability distribution.

[4]

**************************END OFEXAMINATION*****************************

Sampling Theory (SAT802S)

2nd Opportunity January 2025

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Standard Normal Probabilities

Table entry for z is the area under the standard normal curve

to the le~ of z.

.00

-3.4 .0003

-3.3 .0005

-3.2 .0007

-3.1 .0010

-3.0 .0013

-2.9 .0019

-2.8 .0026

-2.7 .0035

-2.6 .0047

-2.5 .0062

-2.4 .0082

-2.3 .0107

-2.2 .0139

-2.1 .0179

-2.0 .0228

-1.9 .0287

-1.8 .0359

-1.7 .0446

-1.6 .0548

-1.5 .0668

-1.4 .0808

-1.3 .0968

-1.2 .1151

-1.1 .1357

-1.0 .1587

-0.9 .1841

-0.8 .2119

-0.7 .2420

-0.6 .2743

-0.5 .3085

-0.4 .3446

-0.3 .3821

-0.2 .4207

-0.1 .4602

-0.0 .5000

.01

.0003

.0005

.0007

.0009

.0013

.0018

.0025

.0034

.0045

.0060

.0080

.0104

.0136

.0174

.0222

.0281

.0351

.0436

.0537

.0655

.0793

.0951

.1131

.1335

.1562

.1814

.2090

.2389

.2709

.3050

.3409

.3783

.4168

.4562

.4960

.02

.0003

.0005

.0006

.0009

.0013

.0018

.0024

.0033

.0044

.0059

.0078

.0102

.0132

.0170

.0217

.0274

.0344

.0427

.0526

.0643

.0778

.0934

.1112

.1314

.1539

.1788

.2061

.2358

.2676

.3015

.3372

.3745

.4129

.4522

.4920

.03

.0003

.0004

.0006

.0009

.0012

.0017

.0023

.0032

.0043

.0057

.0075

.0099

.0129

.0166

.0212

.0268

.0336

.0418

.0516

.0630

.0764

.0918

.1093

.1292

.1515

.1762

.2033

.2327

.2643

· .2981

.3336

.3707

.4090

.4483

.4880

.04

.0003

.0004

.0006

.0008

.0012

.0016

.0023

.0031

.0041

.0055

.0073

.0096

.0125

.0162

.0207

.0262

.0329

.0409

.0505

.0618

.0749

.0901

.1075

.1271

.1492

.1736

.2005

.2296

.2611

.2946

.3300

.3669

.4052

.4443

.4840

.OS

.0003

.0004

.0006

.0008

.0011

.0016

.0022

.0030

.0040

.0054

.0071

.0094

.0122

.0158

.0202

.0256

.0322

.0401

.0495

.0606

.0735

.0885

.1056

.1251

.1469

.1711

.1977

.2266

.2578

.2912

.3264

.3632

.4013

.4404

.4801

.06

.0003

.0004

.0006

.0008

.0011

.0015

.0021

.0029

.0039

.0052

.0069

.0091

.0119

.0154

.0197

.0250

.0314

.0392

.0485

.0594

.0721

.0869

.1038

.1230

.1446

.1685

.1949

.2236

.2546

.2877

.3228

.3594

.3974

.4364

.4761

.07

.0003

.0004

.0005

.0008

.0011

.0015

.0021

.0028

.0038

.0051

.0068

.0089

.0116

.0150

.0192

.0244

.0307

.0384

.0475

.0582

.0708

.0853

.1020

.1210

.1423

.1660

.1922

.2206

.2514

.2843

.3192

.3557

.3936

.4325

.4721

.08

.0003

.0004

.0005

.0007

.0010

.0014

.0020

.0027

.0037

.0049

.0066

.0087

.0113

.0146

.0188

.0239

.0301

.0375

.0465

.0571

.0694

.0838

.1003

.1190

.1401

.1635

.1894

.2177

.2483

.2810

.3156

.3520

.3897

.4286

.4681

.09

.0002

.0003

.0005

.0007

.0010

.0014

.0019

.0026

.0036

.0048

.0064

.0084

.0110

.0143

.0183

.0233

.0294

.0367

.0455

.0559

.0681

.0823

.0985

.1170

.1379

.1611

.1867

.2148

.2451

.2776

.3121

.3483

.3859

.4247

.4641

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Standard Normal Probabilities

Table entry for z is the area under the standard normal curve

to the le~ of z.

.00

.01

.02

.03

.04

.OS

.06

.07

.08

.09

0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359

0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753

0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141

0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517

0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621

1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830

1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015

1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177

1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817

2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857

2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890

2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916

2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952

2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964

2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974

2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981

2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990

3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993

3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995

3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997

3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

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Numbers in each row of the table are values on a t-distributioo with

(df) degrees of freedom tor selected right-tail (greater-than) probabilities (p).

df/p· 0.40

0.25

1 j 0.324920 1.000000

2 0.288675 0.816497

3 0.276671 0.764892

4 0.270722 0.740697

5 0.267181 0.726887

6 0.264835 0.717558

7 0.263167 0.711142

8 0.261921 0.706387

9 0.260955 0.702722

10 0.260185 0.699812

1t • 0.259556 0.697445

12 . 0.259033 0.695483

13 0.258591 0.693829

14 0,258213 0.692417

15 0.257885 0.691197

16 0.257599 0.690132

17 0.257347 0.689195

18 0.257123 0.688364

19 0.256923 0.687621

20 0.256743 0.686954

21 0.256580 0.686352

22 0.256432 0.685805

23 0.256297 0.685306

24 0.256173 0.684850

25 ' 0.256060 0.684430

26 i 0.255955 0.684043

27 ' 0.255858 0.683685

28 0.255768 0.6B3353

29 0.255684 0.683044

30 0.255605 0.6B2756

z 0.253347 0.674490

Cl --

0.10

0.05

0.025

0.01

3.077mM 6.313752 12.70620 31.82052

1.885618 2.919986 4.30265 6,96456

1.637744 2.353363 3.18245 4.54070

1.53:3206 2.131847 2.77645 3.74695

1.475884 2.015048 2.5705B 3.36493

1.439756 1.943180 2.44691 3.14267

1.414924 1.894579 2.36462 2.99795

1.396815 1.859548 2.30600 2.89646

1.383029 1.833113 2.26216 2.82144

1.372.184 1.812461 2.22814 2.76377

1.363430 1.795885 2.20099 2.71808

1.356217 1.782288. 2.17881 2.6Bl00

1.350171 1.770933 2.16037 2.65031

1.3,15030 1.761310 2.14479 2.62449

1.340606 1.753050 2.13145 2.60248

1.336757 1.745884 2.11991 2..58349

1.333379 1.739607 2.10982 2.56693

1.330391 1.734il64 2.10092 2.55238

1.3277213 1.729133 2.09302 2.53948

l.325341 1.72471a 2.08596 2.52798

1.32318:S 1.720743 2.07961 2.51765

1.321237 1.717144 2.07387 2.50832

1.319460 1.713872 2.06866 2.49987

1.31783'6 1.710882 2.0539{) 2.49216

1.316345 1.708141 2.05954 2.48511

l.314972 1.705618 2.05553 2.47863

1.31370'3 1.703288 2.05183 2.47266

1.312527

1.3114J-4

1.310415

1.701131

1.699127

1.697261

2.048•11 2.467M

I 2.04523 2.46202

2.04227 2.45726

1.281552 1.644854 Ul5996 2.32635

80%

90%

95%

98%

0.005

0.0005

6.3.6567~ 636.6192

9.92484 31.5991

5.84091 12.9240

4.60409 8.6103

4.03214 6.8688

3.70743 5.9588

3.49948 5.4079

3.35539 5.0ill3

3,24984 4.7809

3.16927 4.5869

3.10581 4.4370

3.05454 43178

3.01228 4.2208

2.97684 4.1405

2.94671 4.0728

2.92078 4.0150

2.89823 3.9651

2.87844 3.9216

2.86093 3.8834

2.84534 3,6495

2.83136 3.8193

2.81876 3.7921

2.80734 3.7676

2.79694 3.7454

2.78744 3.7251

2,77871 3.7066

2.77068 3.6896

2.76326 3,6739

2.75639 3,6594

2.75000 3.6460

2.57583 3.2905

99%

99.9%