JIM 212 - Statistical Methods [Kaedah StatistikJ

UNIVERSITI

SAINS

MALAYSIA

Peperiksaan Kursus Semasa Cuti Panjang Academic Session 200812009

Jun 2009

JIM 212 - Statistical Methods [Kaedah StatistikJ

Duration : 3 hour

[Masa:

³

jamJ

Please ensure that

this

examination paper contains

TWENTY FOUR

printed pages before you begin the examination.

Answer

ALL

questions.

You

may answer either

in

Bahasa Malaysia or in English.

Read the instructions carefully before answering.

Each question is

worth

100 marks.

[Sila

pastilran bahawa kertas peperilrsaan

ini

mengandungi

Dt]A PULUH

EMPAT muka surat yang bercetak sebelum anda memulakan peperilcsaan.

Jawab SEMUA

soalan. Anda

dibenarkan meniawab sama

ada dalam

Bahasa

Malavsia

atau Bahosa Inggeris.

Baca arahan dengan

teliti

sebelum anda menjawab soalan.

Setiap soalan diperuntukkan 100 markah.J

165

...2/-

-2 - tJrMzL2l

1. A random sample of eight drivers insured with a company and having similar

auto insurance

policies was selected.

Results based on

their driving

experiences

(in

years),

x

and monthly auto insurance premiums

(RM),y

are given below.

f x :90, ly = 474, l*y

^{= +lZS}

Z*' =1396, Zy' =29,642

a)

Compute the value of the correlation

coefficient

(25 marks)

b)

^{Test the} significance of the correlation coefficient equal to 0

ata

₌ 0.05 ^.

(25 marks)

c)

^Find the equation of the regression line.

(25 marks)

d)

^{Find the 95Yo} prediction interval

when

x = 10 years.

(25 marks)

2.

Three random samples of times (in minutes) that commuters zre stuck in

trafftc

are shown.

Citv

A

Citv B Citv C

6I

63 58 62 59

53 58 55 52 54

53 56 54 49 52 xr

.rf)

=60.6

= 4,3

7z

=54'4 sl=5'3

Is

_{= 52.8}

s? = 6'7

Xcu

_{= 55'93}

(a) At a =0.05,

^{is there a} difference

in

the mean times ^Elmong the three cities?

(40 marks)

(b) If

^the

null

hypothesis is rejected, test all possible pair wise comparisons for the mean

times.

Use

a=0.05.

(40 marks)

1"66

...3/-

- 3

- LlrMzrzl

(c) List

the

tlree

assumptions for the F test

for

comparing three or more means.

(20 marks)

3. (a)

Lack

of discipline

among school children has become concemed

to

Parents Teachers

Association. A random

sample

of 300

adults

was

selected, and these adults were asked

if they favor giving

more freedom

to

teachers

to punish

students

for lack of discipline. The results are

shown

here. Atq,=0.05, test the claim that the

two

'

atlributes gender and opinion of adults are dependent.

In Favor Against

No

Opinion

Men 93 70 T2

Women 87 32 6

(50 marks)

A

researcher wishes

to

see

if

the proportions

of

workers

for

each

type

of

job

_have

changed during the last 10 years.

A

sample

of

100 workers is selected, and the results are shown.

Ata

₌ 0.05 _, test the claim that the proportions have not changed.

Types

of

iobs Services Manufacturins Government Others

10 years ago JJ 13 11 ^a

Now

18

t2

8 2

(50 marks)

(a) A

student believes that the answers to his history true/false test are not random. Test the claim,

at d

₌

0.05.

The answers to the questions are shown.

TTTFFTTTFFFFFFT TTFFFTTTFTFFTTF

(40 marks) (b)

4.

...41-

4

- UrMzrzl

(b)

The incomplete result

of

an experiment using

Latin

square design

is

summarized

in

the analysis of variance table below.

Source

of

Variation

Sum

of

Squares

Degrees

of

Freedom Mean Squares

Treatments 330 ? ?

Rows 68 4 ^,)

Columns ^{2 ,)} 37.5

Error ^{2 ,) 2}

Total 676 24

(D

State the size of the

Latin

square design.

(ii)

Complete the analysis of variance table.

(iii) Test to

determine

whether there is

evidence

to indicate

differences among treatment means.

Use a

_{= 0.05}

(60 marks)

5. (a)

Six different actors were ranked by male and female patrons on the basis of diction and appearance.

The

data are shown here (1 is the highest

rating). At a

= 0.05, is there ^a relationship between the ratings?

Actors

ABC DEF

Males ^{6 aJ} 2 5 1 4

Females 4 5

I

6 ^aJ 2

(40 marks)

(b) The ANOVA table below shows the result of an experiment using

randomized complete block design.

Source d.f. SS

Treatment

Block Error

2 J 6

0.26 6.763

0.1 87

Total 11 7.21

Using

o:0.05,

test to determine whether there is sufficient evidence to indicate differences among,

(i)

treatment means,

(ii)

block means.

(60 marks)

...5t-

168

- _s

- urMzr2l

1.

Satu sampel rawak

terdiri

daripada lapan orang pemandu yang mempunyai

polisi

insurans

kereta yang sama dengan

sebuah

syarikat insurans dipilih. Hasil kajian

berdasarkan pengalaman memandu

(dalam

tahun),

x

dan

premium

bulanan insurans kereta

(RM), y

diberikan di bawah.

f x

⁼

90, ll = 474, \*1,=

^+lZS

Z*' =1396, Zy' =29,642 a)

^Hitung

nilai

pekali korelasi.

(25 markah)

b) Ujikan

^keertian pekali korelasi sama dengan 0 pada

a

= 0.05.

(25 markah)

c)

Dapatkan persamaan garis regresi.

(25 markah)

d)

Dapatkan selang ramalan

95%bagi

x = 10 tahun.

(25 markah)

2.

^{Tiga sampel rawak} menunjukkan jangka masa (dalam

minit)

orang yang berulang-alik terperangkap dalam kesesakan di jalan _raya.

Bandar

A

Bandar B Bandar C

61 63 58 62 59

53 58 55 52 54

53 56 54 49 52 xr = 60'6

s?

=

^4'3

7z

=54.4

s2z

=

^5'3

7t

=52'8

s? = 6'7

Xcu =55'93

(a)

Pada

a

₌ 0.05, adakah terdapat perbezaanmin jangka masa antara ketiga-tiga bandar?

(40 markah)

(b)

Jika hipotesis nol ditolak, jalankan ujian berpasangan untuk min jangka masa. Guna

a =0.05.

(40 markah) ...61-

169

- 6

- UrM2r2l

(c)

Senaraikan

ketiga-tiga

andaian

bagi ujian F untuk

membandingkan

tiga

atau lebih min.

(20 markah)

3. (a)

Kekurangan

disiplin

dikalangan

murid-murid

sekolah merisaukan Persatuan IbuBapa

Guru.

Satu sampel rawak yang

terdiri

daripada 300 orang dewasa

dipilih,

dan mereka ditanya sama ada bersetuju memberikan kebebasan kepada guru

untuk

menghukum

murid-murid yang mempunyai

masalah

disiplin.

Keputusan

kajian diberikan di bawah. Pada a

₌

0.05, uji

dakwaan bahawa kedua-dua

atribut iaitu jantina

dan

p endapat adalah berkait.

Menyokong Menentang Tiada pendapat

Lelaki

93 70

t2

Perempuan 87 32 6

(50 markah) Seorang

penyelidik ingin

mengetahui sama

ada kadar pekerja bagi

setiap jenis pekerjaan berubah dalam masa 10 tahun

lalu.

Satu sampel

terdiri

daripada 100 orang

pekerja dipilih, dan

keputusan

kajian ditunjukkan di bawah. Pada a =0.05, uji

dakwaan bahawa kadar tersebut tidak berubah.

Jenis

pekeriaan Perkhidmatan Pembuatan Kerajaan

Lain-lain

10 tahun lalu JJ 13

1l

^aJ

Sekarang ¹⁸

t2

8 2

(50 markah)

4. (a)

^Seorang

pelajar

percaya bahawa

jawapan kepada ujian

benar/salah

bagi

subjek sejarah adalah

tidak rawak. Uji

dakwaan tersebut pada

a

₌ 0.05

.

Jawapan-jawapan tersebut diberikan di bawah.

BBBSSBBBSSSSSSB BBSSSBBBSBSSBBS

(40 markah) (b)

170

1t

7-

(b)

^Keputusan

tidak lengkap ujikaji

menggunakan diberikan dalam jadual analisis varians di bawah.

urM2r2l

rekabentuk segiempat sama

Latin

5.

Sumber Hasil tambah kuasa

dua Dariah kebebasan

Min

kuasa dua

Olahan 330 ^{2 ,)}

Baris 68 4 ?

Lajur

^{,) 2} 37.5

Ralat ² ? ^,)

Jumlah 676 ^.A.L+

(D

Nyatakan saiz rekabentuk segiempat sama Latin yang digunakan.

(ii)

Lengkapkan jadual analisis varians di atas.

(iii) Uji untuk

menentukan sama ada terdapat perbezaan antara

min olahan.

Guna

a

₌ 0.05.

(60 markah)

(a)

Enam orang pelakon dipangkatkan oleh penaung-ponaung yang

terdiri

daripada

lelaki

dan wanita berdasarkan sebutan dan penampilan

mereka.

Data ditunjukkan di bawah

(1

adalah

pangkat tertinggi). Pada a =0.05,

^adakah

perkaitan

^{diantara pangkat-}

pangkat tersebut?

Pelakon

ABCDEF

Lelaki

6 ^aJ 2 5

I

4

Wanita 4

) I

6 ^aJ 2

(40 markah) Jadual

ANOVA

di bawah menunjukkan keputusan

ujikaji

rekabentuk blok lengkap rawakan.

Sumber d.k. ^SS

Olahan

Blok

Ralat

2

aJ 6

0.26 6.763 0.187

Jumlah 11 7.21

Pada or

:

0. 0 5, uj

i

sama ada ter dapat perbezaan antara,

(i) min

olahan,

(ii)

min blok.

(60 markah) .. .8/- (b)

L72

Formulas

-8- urM212l

1.

2.

4.

a_a

t=f

A=

5. $- "(zw)-(>*Xrv)

6. sest =

7. y' !

Z (n,-r) 10. 4=

"(>*')-(>*)'

-;7-a-T

s;,1- +-

|

[n,

ⁿⁱ

)

ta/2sestffi

Z n, (7,-Xor)'

(r"r)(r"')-{r4{rv)

"(z*')- (l

")'

8.

9.

k-l L (n,-t)',?

4=

szw =

t- -12

\x'-xi)

- aZy -bZxy

173

...91-

-9

-

urM2r2l

11.

7, -Ti

,!

----T-- ls;{tn

12. (a)

^,S,S* =

,=zr(*,-r)' =\.? -

\n-2 @

_Y

_.La

ln P

E,,,l

(b) ss, -r)

^{t2n^}

=L_,!i -

(c)

SS,y

14.

^ssR =

a(ss"r)= B'(^t^s*)

16.

B

=2.3026

Q=(n-a) rogs| (", -l) logsf

s;

_{' =}

--l-

_{fl- O}

i @,-

¹⁾

,,'

i=l

h=r+

# n,-l 1 -l n-a) 1)

=l^(r, (h,)'

=

2{*, -t) (v, - v) =L *,v, - (l')8,)

n

13.

15.

t-

I

h

Ia It

\,=r

174

...t0t-

-

10

- urM2tzl

17.

Two-Way

ANOVA

y,..=

_j=tft=1'

Li y,,o 't'"

V,.. =

?

_bn

i =L,2,...,a

!;=iiy,* y,=Y'it j=!,2,...,b

'J

'

i=t

k=t

An

Y, =frY* ,, =* {'r:l;,?, ',i

ab

n

., _\i s \i..,

'' -

i1-Ji"=lko=lYiik

sSr ob n,

u2

=I I Zy'tik-'"'

i=l

j=Ik=l "

abn

! Y-i.. 22 Y-...

SS,->."' ta i=l bn

abn

! v2.i. v2,.

SSP

D j=l an

= abn

2't21

ss.ar =rr + ,#->U.h

18.

Randomized Complete

Block

Designs

!, = Z !r, ,

b

i=lr2r'..ra

"

t. j-=I'U

., _ \_

a

1,

;

_Ir2r..,rb

,.j-Fsrij ' r-

ab o

^b

r,-Y\irr-Yy.=Iv

-v..

- 1 j"=t'ij A'i.- j"=l'.j

;

v

aDn

175

...tt/-

urM2r2l

11-

ab )

^v2

= I t v:.-""

i74

i-=1' U

_ab

& Yi. 2) Y:

=L- i4b

ab

2.

- - /.,/ y:

aab x +

^0.s)

- (n rz) s%

SS,A

s%

Ji rz R- rt,

_IT

oR

nr,('r+nr+t)

ItR

=

oR=

19. z=

11LL. L_

n(n +I)

20.

ws-

,(, +r)(zn +t)

24

6r.d2

rs=r

ryi

22.

zJ.

n(z(nt

...121-

t?6

-t2- lIrM2l2l

.07 .

.08

.09

Appendix

0.0 0.1 012 u.5 0.4 0.5 0.6 0.7.

0.8 0.9 1.0 1.1

1.3

14

1.5' 1.6

t.7

1.8 1.9 2.0 a.r 2.2 2.3 2.4 2.5 2,6 2.7 2.8 2.9 3.0

.0000 .0398 .0793

.tt79

7 <<i .1915 .2257 .2580 .2881 .3159 .3413

.3&3

.3849 .4032 .4192

.0M0 .0438 .0832 ,L2t7 t <ot .1950 .229t .2611 .29t0 .3186 .3438 .3665 .3869 .4049 .4207

.4332 ,4345

.4357

.4452 ,4463

^.4474

.4554 .45&

.4573

,464L4&9

^.46s6

.4713

. .4719

^,4726

.4772 .4778

^.4783

.4821 .4826

.4830

.4861 .4864

.4868

.4893 .4896

^.4898

.4918 .4920

,4922

.4938 .4940

.4s41

.4953 .4955

.49s6

,4965 .4966

^.4967

.4974 .4975

^.4976

,4981

' .4982

^.4982

,4987 .4987

.4987

.3554

.3577

.3770

,379(i

.3962

^1aR0

.4131

^.4147

.4279

.4292

.0319

^.0359

.wu

^.0753

.i103

^.1141

.1480

^.1517

.1844

^.L879

.2190

^.2224

.2s17

^.25!9

.2823

^.2852

.3106

^.3133

.3365

^.3389

.3s99

^,3621

.3810

^.3830

.3997

^.4015

.4162

^.4177

.4306

.4319

;4429

.4441

.4535

^.4545

.4625

^.4633

.4699

^.4706

.475r

^.4767

.4812

^.4817

,4854

^.4857

.4887

^.4890

.49t3

^.4916

.4934

^.4936

.495L

^,4952

.4963

.4964

,4973

^.4974

.4980

,

^.4981

.4986

.4986

.4990

^.4990

,04 ;05

.4. .03 .00

.0080

.fia

.08?1

;r25s .1q28 I .1985 .23U.

.2&2 .2939 .3212

.0.120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236

.0160 .0557 .0948

.1700 .2454 .2389

.nM

.2995

.32&

.3508 .3729 ,3925 .4099 .4251

.0199 .0596 .0987 ,1368 .I /JO .2088 .2422

.n34

,tu45 .3289 .3531 .3749 .3944 .4115 ,4265

.0239 .0636 .t026 .1406 .1772

-ztz)

.2454 .2764 .3051 .3315

.0n9

,0675 .1064 .1443 .1808

,2tst

.2486

.n94

.3078 .3340

.346r .3686 .3888 .4065 4222

.4370

.4382

.448r'.

^.4495

.4582

^.4591

,4664

^.4671

.47.32

.4738

.4788

^.4793

.4834

^.4838

.487t

^{' .4875}

.4901

' .490/.

.4925

^.4927

.4943

^.4945

.4957

^.4959

.4968

^.4969

.4977

^,4977

.4983

^.4984

.4988

^.4988

.4394 .4505 .4599 .4678 .4744 .{198 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .49't8 .4984 .4989

.4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .493t .4948 .4961 .497t .4979 .4985 .4989

.44t8 .4525 .46t6 .4693 .4756

"4808 .4850 ,4884 .4911 .4932 .4949 .4962 ..+, I Z .4979 .4985 .4989

Nore.' Us 0.4999 fu z valus abovc 3.09.

Somr,' Feddick Mostella od Robcrt E. K Roukc, Jrardy Srararicr, Tablc A-1 (Radin!, Ms.: Addison-Wesley, 1973).

Reprinted wi& pemission of thc copyright owncn.

0z

L77

...r3t-

-13- [JrM2r2]

99Vo 0.005 0.01

i

7 J

5 o

I

9

10'

11

13 1t 15 16

t7

18

l9 20 t1 ,,7

ta.

t<

26 ,)*, /z\ o

1,000 .816 .765 4t1

.718 .711 .706 .703 .700 .697 .695 ..694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .68s .684 .684 .684 .683 .674

3.078 1.886 1,638 1.533 1.476 1.440 t.415 1.397 I.JdJ I e1, L.363 1.356 1.350 1.345 L341

I.JJU 1 Itt 1.325 I ztl I ?',lo 1 ?1q 1.316 1.315 1.314

1.282:

o.Jlq 2,920 2.353 2.L32 2.015 1.9+t 1.895 1:860 I.6JJ 1.812 L.796 1.182 t.771 1.761 1.753

t:l46

1;140 L.I J+

1.729 1 1)<

r.727 1.7L7 't 114 I t1t 1.708 1.706 1,703 L,IV

' 1.645b

12.106 4.303

l rRt 2.776 .2.571

2.M7 2.365' 2.306 2.262 2.228 2.20L 2.179 2.L60 .2.r45 2.13r 2.120'

2.110 2,101 2.093 2.086 2.080 2.0't4 2.069 2.064 2.060 2.056

r n<t 2.O48 L960

31.821 6 0K<

4.541 3.747 3.365 J. I4J 2.995 2.896 2.821 2;7&

2.718 2.68r 2.650 2:624 2.602 2.583 2.s67

" ^<<t 2.539

r <tR 2.518 2.508 2,500 2.492 2.485 7.479 2.473 2.461 2.326',

63.657 9.925 5,841 4.6M 4.032 5,191 3.499 3,355 3,250 3.169 3.106

? /'r{{

3.012 2.977 2.947

r otl

2.898 2.8'18 2,86'.1 2.845 2.831 2.8t9 2,807 2.797 2387 2.771 2.763 2.576d

nThis valuc hu bcen roun&dTo l2&fu ^{the textbook'} elhis _{value has} brcn rcundcd to 1.65 i! thc tcxtbook'

"This valuc hs bccn roundtd to 2'33 i! lhc tcxtlooL dThis _vatuc hu bcen munded lo ?'58 h thc Ertboolc

so{rcti AdaPEd froE W. H, Bcycr, IlardDo ok ofTables for Pmba'iltry ^ond &aBlics'

i"i J.,'6ni ^pt"*' soca Rator' Fla', 1986' RePrintsd with pcmissio!'

Area

q

1"7 8

...r41-

0.99 0.ns ^{0.95 0.90 0.05 0.02s}

LJIM2r2l

0,01 0.005

-t4-

;T#:f.$:$i '"' :;.:;r',' d:.,:

Degrees of freetlom

1

I 3 t

2.706

^3.841

4,605

^5,991

6.251

^7.815

7-779

^9.488

9.236

^11.071 10.645 ^12.592 12.017 ^t4.067 13.362 ^15.507 14.684 ^16.919 15.987 ^18.307 L7.275 ^19.6'75 18.549 ^'2r.026 19,812 ^22.362 21.064 ^23.68s 22.307 ^24.996

n.542

^26.296

z4.i6s

^77.587

25.989 ^28.869

27.2M

^30.144

28.412 ^31.410

29.6t5

^{32.67 L} 30.813 ^33.924 32.007 ^35.t72 33.196 ^36.415 34.382 ^3'1.652 35.563 ^38.885

36J41

^40.1l3 37.916 ^41.337 39.087 ^42.557 40.256 ^43.773 51,805 ^55.758

63.t67

^67.5ps 74.397 ^79.082 85.527 ^90:531 96.578 ^101.879 107.565 ll3.14s 118.498 124.342 0.995

6-

7 8 9 10 ll

t2

13 1t

l5

1A LI

l8 l9 20

2l

zz ZJ at 25 26

28 29 30 40

<n 60 1n

80.

90 100

0.010 0.072 '0.207 0.412 a.676 0.989

t.3M

1.735 2.156 2.603 3.474 3.565 4.075 4.601 5.142 f,.ov / 6.26s 6.844 7.434 8.034 8.643 9.262 9.886 rn {rn It.l60

11,808 12.46r t3.L2l

13.787 20.707 27.991 35.534 43.2"t5

5r.t12

59,196 67328

0.001 0.004

^0.0i6

0.020 0.051 0.103

^0.211

0.115 0.216 0,352

^0.s84

0.297 0.484 0.?11

^1.064

0.554 0.831 Ll45

^1.610

0.872 r.237 I.635

^2.204

r.239 1.690 2.167

^2.833

t.646 2.180 2.733

^3.490

2.088 2:100 3.325

^4.168

2.558

^{3.247 .}

3.940

^4.865

3.053 3.816 4.575

^5.578

3.571 4.404 5.226

^6.304

4.107 5.009

^{-5.892 7.M2}

4.660 5.629 6.571

^7.790

5.229 6;262 7'261

^8'547

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7.633

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8.260

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8.89? 10.283 11.s91 ^13.240 9.542 10.982 12,338 ^14.042 10.196 11.689 13.091 ^14.848 10.856 12.401

,

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3't

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45.442 48,758 5L739 ^s5.329 53.540 5?.153 60.391 ^64.278 61.754 6s.647 ^{69.126 73.29r}

?0.065 74.222 71,929 ^82.358

5.024 6.635

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7.378 9.210

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9.348

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11.i43

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12.833

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16.013

18.475 20.278

1?.s35

^{20.090 2t.9ss} 19.023 21.666 ^23.589

20.483

^{23.209 25.188}

2r.920

^{24.725 26.757}

23.337

26.217 28.299

24.736

^{27.688 29.819}

26.tr9

^{29.141 31.319}

27.488

30.578 32.801

28.845 32.000

^34.267

30.191

33.409 ^35.718

3r.526

^{34.805 37.1s6}

32.852

36.191 ^38.s82 34.17A 37.566 ^39.997

35.479

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36.?81

40.289 ^42.796

38.076

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39.364

42.980 ^45.559

40.646

44.314 46.928

41.923 45.&2

^48.290

43.194

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44.461 48.n8

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49.588

^52'336

46.9.'19

50.892

^53.672 59.342

63.691

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7t.420

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83.298

88.379 ^9t.952 95.023 ^{100.425 104.215} 106.629 112.329 ^116.321 118.136 L24.Lt6 ^128.299 12956r 135.807 140.169 so!rcr., Donald B. Awcn, Hondbook of srorsricr I0rrs. Thc chi-squm Dislribution Table, o 1962 by Addison'wcslcy

publishing company, lnc. copydghr Btres8l @ 1990. Rcprinted by Pcmission ^of Pe6on Eduolion. Inc.

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