I- lUniversiti $ Malaysia PAHANG En

FACULTY OF INDUSTRIAL SCIENCES & TECHNOLOGYFINAL EXAMINATION

COURSE : APPLIED STATISTICS

COURSE CODE : BUM2413/BSU1023/BCT20531BPF3313/ BKU20321BAM3022

LECTURER : DR ROSLINAZAIRIMAII BINTI ZAKARIA NOR HAFIZAH BINTI MOSLIM AZLYNA BINTI SENAWI MOHD RASHID BIN AB HAMID NOR AZILA BINTI CHE MUSA NOOR FADHILAH BINTI AHMAD RADI

DATE : 5 JUNE 2012

DURATION : 3 HOURS

SESSION/SEMESTER : SESSION 2011/2012 SEMESTER II

PROGRAMME CODE : BSB/BSK1BAAIBAE/BCNIBCG1BCS/BPPIBPT/ BPS/BFFIBFMIBKCIBKGIBKB/BMMIBM1/ BMBIBMF/BMAJBEE/BEP/BEC

INSTRUCTIONS TO CANDIDATES

1. This question paper consists of SEVEN (7) questions. Answer all questions. 2. All answers to a new question should start on new page. 3. All the calculations and assumptions must be clearly stated. 4. Candidates are not allowed to bring any material other than those allowed by

the invigilator into the examination room.

EXAMINATION REQUIREMENTS: 1. Statistical Table 2. Scientific Calculator

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

This examination paper consists of FIFTEEN(15) printed pages including front page.

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QUESTION 1

An article in the Journal of Strain Analysis compares Karlsruhe and Lehigh methods for predicting the shear strengths for steel plate girders. Data of these two methods, are

shown in Table 1.

GirderKarlsruhe Method

Lehigh Method

Gi 1.186 1.067 G2 1.151 0.992 G3 1.322 1.063 G4 1.339 1.062 G5 1.200 1.062 G6 1.402 1.178 G7 1.365 1.037 G8 1.537 1.086 G9 1.559 1.052

Table 1: Shear strengths for steel plate girders of two methods

(a) Find the mean and standard deviation for the difference of methods in Table 1. (3 Marks)

(b) Find a 98% confidence interval for the mean difference in shear strengths between Karlsruhe and Lehigh methods.

(4 Marks)

(c) Is there any mean difference between the two methods? By assuming the data is normally distributed, test the hypothesis at 5% level of significance.

(7 Marks)

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QUESTION 2

The variability in the thickness of the oxide layers is a critical characteristic of the semiconductor wafers. Low variability of the oxide thickness is desirable for subsequent processing steps. Two different mixtures of gases are being studied to determine whether

one is superior in reducing the variability of the oxide thickness. Twenty one wafers are etched in each gas. For gas A, the mean of oxide thickness is 10.05 angstroms and standard deviation is 1.96 angstroms while for gas B the mean is 13.22 angstroms and

standard deviation is 2.13 angstroms.

(a) Find a 98% confidence interval for population mean for mixture of gas B and give the interpretation of the parameter estimate.

(5 Marks)

(b) Determine whether the two mixtures of gases have different in variability of oxide layers thickness at the 0.1 level of significance.

(8 Marks)

(c) Can we conclude that the mean mixture of gas A is not more than gas B at 10% significance level? Your assumption of the condition of population variances is

based on your answer in (b).(8 Marks)

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QUESTION 3

A researcher wishes to see whether there is any difference in the weight gains of athletes following one of three special diets. Athletes are randomly assigned to three groups and placed on three different diets for six weeks. The weight gains (kg) are shown below.

Type of diet Weight Gains (kg) Diet 2 3 4 2 Diet 5 6 5 7 4 3 Diet 4 1 2 1 1 3

Table 2: Weight gains (kg) for different type of diets

(a) How many treatments involved in the experiment? (1 Mark)

(b) Based on the data in Table 2, is there any treatment effect between the type of diets at 5% level of significance?

(16 Marks)

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QUESTION 4

(a) What is the different between simple linear regression and multiple linear regression?

(1 Mark)

(b) A fitted simple linear regression equation is given by j 12.9 + 2.34x where n=1O, S,=929.98 and S=389.93.

(i) Calculate the value of correlation coefficient, r and comment on the value.

(4 Marks)

(ii) Complete the ANOVA table below and test the hypothesis that the linearity of the regression line at a = 0.05 significance level.

Source of variation Sum of squares

Degrees of freedom

Mean of squares

F lest

Regression 1

Residual

Total 929.98

Table 3: ANOVA(12 Marks)

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QUESTION 5

A study was performed to investigate the car performance for car models produced by the U.S., Japan, Germany and Sweden between 1978 and 1979. The fuel mileage in kilometers per liter is believed to be related to car weight (in thousand kilograms), drive-ratio and horsepower. A multiple regression analysis is conducted to determine the multiple linear regression equation which gives the best fit to the data. The following

show the Excel outputs of the multiple regression analysis for the study.

SUMMARY OUTPUT

Regression Statistics Multiple R 0.9031 RSquare 0.8155 Adjusted R Square 0.8104 Standard Error 2.9508 Observations 38!

ANOVAdf SS MS F SignficanceF

Regression 1 1293.5156 1293.5155 159.1610 8.88939E-15! Residual 36 292 5752 8.12711 Total 37 1586.0908;

Coefficients Standard Error tStat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 4871 19537 249311 0.0000; 447452 52 6697 447452 52.1665.7:

Weight -8.36. 0.6630 -12.5159 0.0000 -9.7093! -7.0199 -9.7093. -7.0199

SUMMARY OUTPUT:

Regression Statistics Multiple 0.4172 ... . . RSquare 0.1741 Adjusted RSquare 01511 Standard Error 6.0323 . Observations . 38. .

ANOVAdf SS MS F Significance F

Regression 1: 276.1009 276.1009 7.5876: 0.0092 .. . Residual 36 13099899 353886 Total 37 15860908

Coefficients . Standard Error tStat I P-value Lower 95% . Upper95% Lower95.0% Upper 95.0% Intercept 8.44 6.0065 1.4046 0.1687 -3.7453 20.5181 -3.7453 20.6181! Drive Ratio 5.28 1.9158 2.7546. 0.0092 1.39171 9.1524. 1.3917 9.1624

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SUMMARY OUTPUT

Regression Statistics 'MultipleR 0. 8713 RSquare 0.7591 Adjusted RS re 0.7524 Standard Error 3.2.1576: Observations 38

:ANOVA df i SS MS F Significance

Regression 1 1204.0530: 1204.0530 113.45

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SUMMARY OUTPUT

Regression Statistics Multiple R 0.9G951 RSquare 0.82721 C Adjusted R Square 0.8173: Standard Error 27986 Observations 38 .

ANOVAdf SS MS F Significance

Regression 2. 1311 9559 655.9830i 83.7553 14 4.55362&14

Residual 35 2741249 78321 Total 37 1586.0908.

Coefficients Standard Error tStat P-value Lower 95016 Upper95% Lower 95.0% Upper 95.0%

Intercept 4894 1.9240 254379 0 0000 45.0361: 52.8475 450361 528479

Weight -6.06. 1.6338 -3.7119 0.0007 -9.3814: -2.7477 -9.3814 2.7477

Horsepower -0.07 0.0437 -1.5348;: 0.1338, -0.1557 0.0216 -0.15571 0.0216.

SUMMARY OUTPUT

Regression Statistics Multiple 08793 RSquare 07732 Adjusted R Square 07602 Standard Error 3.2051 Observations 38. .

ANOVAdf i SS MS F SignficanceF:

Regression 2 1226.3751 613.1875 59.6626 5.29113E-12 Residual . 35 359.7157 10.2776 : Total 37 1586.0908

Coefficients iStandard Error tStat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

Intercept 54.63 5 7676 9.4714 00000 42 9182 56 3359 42 9182 663359

Drive Ratio 1 86 1.2597E -1.4737 : 0.1495 -4.4140, 0.7009,: 44140 07009

Horsepower i -0.24. 0.0247 -9.6157 0.0000 -0.2872 -0.1871 -0.2872. -0.1871.

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SUMMARY OUTPUT

Regression Statistics Multiple R 0.9482 Rsquare 0.8991 Adjusted R Square 0.8902 Standard Error 2.1595 Observations 38

ANOVAdf SS MS F Significance F

Regression 3 1426.0575 475.3525 1 100.9914i 5.26413E17 Residual 34 150.0333 4.7069: Total 3T 1585.0908

Coefficients : Standard Error tStat P4'alue Lower 95% Upper 9556 Lower 95.0% 1 Upper95.0% Intercept 70.28 4.5838: 15.3326i 0.0000 60.96611 79.5969 60.9561 79.5969 Weight 928 14255 65133 00000 121813 63875 121813 63876 Drive Ratio 472 09595 49234 0.0000 ! -6.6736; -2.7739 -6.67361 27739 Horseoower -0.04: 0.0342; -1.2432 : -0.11211 -0.1121; 0.0270

Based on the given Excel outputs, answer the following.

(a) State the response and predictor variables? (2 Marks)

(b) Extract the relevant informations and summarize your results in a table. (7 Marks)

(c) Hence, determine the best regression equation for predicting the mileage (kilometer per liter) value.

(2 Marks)

IN

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QUESTION 6

The goals scored per match by MyKid football team gave the following results:

Number of goals per match 0 1 2 3 1 4 5 6 7 Number of matches IL_L 18 29 1 18 1 10 7 3 1

Table 4: Goals scored in all football matches

Test whether the number of goals per match follows a Poisson distribution at 10%

significance level.(12 Marks)

QUESTION 7

In a study of the television viewing habits of children, a developmental psychologist selects a random sample of 300 primary students; 100 boys and 200 girls. Each student is

asked which of the following TV programs they preferred most; Word World, Dibo the

Gift Dragon or Mickey Mouse Clubhouse. Results are shown in Table 5 below.

Viewing Preferences

Word World Dibo the Gift DragonMickey Mouse

Clubhouse Row total

Boys 50 30 20 100 Girls 50 80 70 200 Column total 100 110 90 300

Table 5: TV program viewing preferences

At a = 0.01, is there enough evidence to support the claim that the proportions of

viewing preferences for boys are equal for each of the three TV programs?(8 Marks)

END OF QUESTION PAPER

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Appendix - Table Of Formulas

Confidence Intervals, Sample Sizes and Hypothesis Testing

Confidence intervals for p Hypothesis testing for p

I .x_za/2 7; X+Zai7=J Ztt/T

X - Z , X + Za/2 7J Ztest =

Xta12vJ_ , X+ta/2v_j=J ttest = where v=n-1

Confidence intervals for du - 4u2 Hypothesis testing for A . -

2i2),u0

ZtestJcY22 cT

Foro-^o: Foro^o:

(i 2)u0 ____

Zte L2 2

fl 2 111 fl2

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For o # For o #2)u0

( F2 2

I (

X2)tai2v /L+Lttest 12 2

41+ Vi 2 V'i '2

/ 2 2\ 2 1 2 2s\2 n1 n2 )

where v =In1 n2

where v = 2 212 1fl2 S2 2 1 1

ni) n2) n) n1 -1 n2 -1 n1-1 n2-1

For o

= o- : For o = o:

( x2)P0 )Z 12 sj-_-+_1_ J

Ztest =

For cr = For cr = o:

J 2)ta12v(yi

test

-

p ri^ + n2

where v-n1 +n2 -2 where v=n1+n2-2 Pooled estimator, s

/(n-1)s+(n2-1)s =n1+n2-2

Confidence Interval for PD Hypothsis Testing for PD

d _z/iTd,,,Fn

ttest =

d-,u D where v = n -1 SD/'.I

-S D SD) t\ Za//=s +Za/T)

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-

( n-I f' a/2,n1 Confidence intervals for ir Hypothesis testing for 7r

[JP(1_P) , p+ai2ifr(1_]

-_______

Ztest - 1-7r0)

Confidence intervals for 7t1 - 7t2 Hypothesis testing for it1 - it2

(p, p2 )r0 = ___________ If ,r

0 1 Zt.,it1 (1irk )

92(1-7r2)

[(p1 _p2)Za12(1 ) +

= o, Z. =

FPP (I --PP + n2

where pp = x1 + x2fl1 + fl2

Confidence intervals for a2 Hypothesis testing for a2

((n_1 )s 2 (n _l)s2 I I

2 2 (n-1)s 2 2 I ' I k

X. /2, Z1-a/2,v ) Zest 2co

where v=n-1

Confidence intervals for Hypothesis testing fora2 U2

1 where v1 =nj 1 ' fa12 I

-

.test - 2 2 2 V2 V1

S2 faizvi,v2 S2 ) v2= fl2 1 S2

Sample sizes

n = [' 0 J 2 n=p(1_p)2-J

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Analysis of Variance (ANOVA) One-way ANOVA Two-way ANOVA

SST = k 1 SST = >jx __x2

1=1 j1 k=1 a r

H jl 1 a 1 SSA =--x2 ___x2

br 1=1 ' abr

SSB= - x. ___x2 ar 1 ar

11 fl 1 iab 1 SSAB=--x2 -------x2 SSA SSB

' r 1=1 j1

abr

SSE = SSTSS(Tr) SSE = SST SSA SSBSSAB

Goodness of Fit Test and Contingency Tables

Goodness of Fit Test Test using Contingency Tables

(0 E)1.

E.n,.

fl 2' i Xtest _ L4 i

r c(O_E) ZtSt

1=1

Free distribution DoF; v = ki Ejj Hypothesized distribution DoF; V =k

- p 1 where v_(r - 1)(c 1)

Simple Linear Regression and Correlation

Simple Linear Regression and Correlation SXY r-

.'jsxxs.y [Exi)t i J ,, ______ [nJ2

S,=x1y1 ' S=Jx ' S,=y 1=1

i=1 n 1=1 fl 1=1

Regression line equation: 5' = /3 + x where /3 = and fl = - xx

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Hypothesis Testing for Intercept, '80 Hypothesis Testing for Slope, /3

A s.e(flo) !+T

y2

ttest FMSR-

_i-i A s.e(i)

FMSR- n

Sum of Squares Regression, SSR Mean Square Residual, MSReS

SSR/31SXY MSRes_[ n-2

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