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CONFIDENTIAL CS/OCT2010/QMT500

UNIVERSITI TEKNOLOGI MARAFINAL EXAMINATION

COURSECOURSE CODEEXAMINATIONTIME

STATISTICS FOR ENGINEERINGQMT500OCTOBER 20103 HOURS

INSTRUCTIONS TO CANDIDATES1 . This question paper cons ists of five (5) questions.2. Answer ALL questions in the Answer Booklet. Start each question on a new page.

Do not bring any material into the examination room unless permission is given by theinvigilator.Please check to make sure that this examination pack consists of:i) the Que stion Paperii) an Ans we r Booklet - provided by the Facultyiv) a two - page Append ix 1 (Key Formulas)v) Statistical Table - provided by the Faculty

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SOThis examination paper consists of 6 printed pag es

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CONFIDENTIAL 2 CS/OCT 2010/QMT500

QUESTION 1a) A machine makes electrical resistors having a mean resistance of 40 ohms and astandard deviation of 2 ohm s. Assum e that resistance follows a norm al distribution.

i) Find the percentage of resistors that will have resistance s greater than 44 ohms.(3 marks)ii) Find the resistance excee ded by 85% of the resistors. (4 marks)iii) A box contains 20 resistors. If a box is randomly cho sen , wha t is the probability thatthe mean resistance of this samp le is less than 40.5 ohms?

(3 marks)iv) An electronic de vice has 6 of these resistors fitted to it. The device is consideredmalfunctions if at least one of the resistances exceeds 44 ohms . Compute theprobability that a randomly chosen electronic device is considered m alfunctions.(3 marks)

b) The following table shows the num ber of errors found in a random sam ple of 85 softwareproducts. The probabilities pi associated with each num ber of errors are partly shown.Number of errors

012345678

Number of softwareproducts3142025146201

Pi0.04980.14940.2240.2240.1680.1008

Complete the m issing values for the above table. Do the data provide sufficientevidence to conclude that the number of errors has a Poisson distribution with u = 3?Test at a = 0.01. (7 marks)

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CONFIDENTIAL 3 CS/OCT2010/QMT500

QUESTION 2

a) Two factors, implant dose and furnace position, in a manufacturing process for anintegrated circuit were studied to determine their effect on the resistivity of the wafer.The data collected are shown below.

Dose1

2Total

Position1 2 315.514.314.226.227.526.3124

14.813.915.324.924.024.1117

21 .319.420.326.126.225.7139

Total149

231380

i) Wha t is the experimental design used? How many treatment combinations arethere? (1mark)ii) Complete the following ANO VA table.Source of variationDosePositionDose x PositionErrorTotal

df Sum of squares42 .1111

451.9778

Mean Square F

(6marks)iii) Is there any interaction effects at the 1 % significance level between dose and theposition of the furnac e? (5 marks)

b) The m anufacturer of a small battery-powered tape recorder decides to include fouralkaline batteries with its product. Two battery suppliers are being considered; each hasits own brand (brand 1 and brand 2). The supervising inspector of incoming qualitywants to know if the average lifetimes of the two brands are the same. Based on pastexperience, she believes that the battery lifetimes follow a normal distribution. A sampleexperiment is conducted: each of ten batteries (five of each brand) is connected to atest device that places a small drain on the battery power and records the batterylifetime. T he following results (in hours) are obtained:Brand 1Brand 2 4330 4826 3837

41315134

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i) Construct the 95 % confidence interval on the ratio of the varian ces of lifetimes ofthe battery for the two brands. Interpret the con fidence interval obtained. (4 marks)

ii) Based on the results in i) construct the 95% confiden ce interval on the differencebetween the average lifetimes of the two brands. Can the supervising inspectorof incoming quality conclude that the average lifetimes of the two brands areequal? (4 marks)

QUESTION 3a) A shipment of computer keyboards is inspected for defects before being accepted. Theprobability that a com puter keybo ard is defective is 0.04.

i) If 9 com puter keyb oards are inspected, wha t is the probab ility that one or twokeyboards are d efective? (4 marks)ii) Find the probability that at most 7 computer keyb oards from 50 inspected aredefective. (3 marks)Suppose one crate contains 100 computer keyboards.iii) How many com puter keyboa rds in a crate would you expect to be defective, andwhat is the standard deviation? (2 marks)iv) Approximate the probability that from 5 to 10 computer keyboards in a crate aredefective. (3 marks)

b) The following table shows the summ ary of promotion exercises in a large multinationalcompany.Promotion (years)GraduatestatusLocal 127 55 61Overseas 150 65 42

Test at the 5% level of significance whether the promotion exercises depends on thegraduate status. (8 marks)

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QUESTION 4a) The following data shows the number of years people smo ked and the percentage oflung damage they sustained.

Years, xDamage, y

2220 1414 3154 3663 917 4171 1923i) Write the least squares line to estimate percentage of lung damag e from num ber ofyears smok ing. Explain the meaning of the estimated regression coefficients.(3 marks)

ii) Estimate the percentage of lung damage for a person who has been smoking for 30years. (1 mark)iii) Calculate the coefficient of correlation and explain its va lue. (2 marks)iv) At the 1 % significance level, can we conclude that the linear regression model issignificant? (8 marks)

b) A certain process produces 10 0-meter-long sheets of vinyl com pose d of simulated woodgrain top layer and a black bottom layer. A blemish (nonconformity) occurs when theblack layer shows through or the wo od grain pattern is not distinct. A 1 0-meter-sample isobtained by trimming the end of a roll, at which the number of blemishes is observedand recorded. It is believed that the number of blemishes per sample follows a Poissondistribution with an average of two blemishes per 10-meter sample. Determine theprobability that:i) There will be at least two blemishes observed in a 10-meter sample. (1 m ark)ii) There will be at most eight blemishes observed if a 30-meter sample is used.(2 marks)iii) If there are five 1 0-meter samples taken indepen dently, wha t is the probability thatexactly two of the will have at least two b lemishes obs erved? (3 marks)

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QUESTION 5

a) The R&D de partment of an industry imposed that the mean life of the light bulbsproduced should exceed 4000 hours and with a standard deviation of less than 150hours before it could be supp lied to the m arkets. A sample 15 bulbs were tested and thelength of life are as follows (in hours):

4300 4302 4415 4483 4301 4446 4478 43193985 4483 4377 4401 4346 4261 4353i) Estimate the mean life of the light bulbs using a 98% confidence interval.

(4 marks)ii) Do the data indicate that the industry is able to produce the light bulbs with standarddeviation of less than 150 hours? Test at the 1 % significance level.

(6 marks)iii) Using the above resu lts, is the industry ready to supply the light bulbs? Explain youranswer. (2 marks)

b) Mental measurem ents of young children are often made by giving them blocks andtelling them to build a tower as tall as possible. One experiment of block building wasrepeated a mon th later, with the times (in seconds) listed in the table below:ChildFirsttrialSecondtrial

13030

2196

31914

4248

52914

617852

74214

82022

91217

10398

1 11411

128130

131714

143117

155215

Can we conclude that the child performance is better in the second trial? Test ata = 0 .01 . (8 marks)

END OF QUESTION PAPER


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CONFIDENTIAL APPENDIX 1(1)K E Y F O R M U L A S

CS/OCT 2010/QMT500

Binomial probability formula

Poisson probability formula

p(x) = P(X = x) = 'rT p x ( 1 _ p ) n - x . x = 0 1 i 2 n

e~^ u xp(x ) = P(X = x ) = - c - ; x = 0 , 1 , 2,x !

C O N F I D E N C E I N T E R V A L SParameter & description Two-sided (1 - ot)100% confidenc e intervalMean p. of a normal distribution,variance a 2 unknown x t a/2

f \SvVny

df = n - 1

Difference in means of twonormal distributions ^ - \i2,variances a? = a 2 2 and unknown( x i - x 2 ) t a / 2 s p ^ + ^ ;

df = r\-\ + n 2 - 2 , sD = .( ^ - 1 ) 8 ^ + ( n 2 - 1 ) s 2 2n-) + n 2 - 2

Difference in means of twonormal distributions H I - \x2, 2 2S-| s 22 _ 2variances c-\ * a 2 and unknown ( x i - x 2 ) t a / 2 l M -+n 1 n 2

d f _ ( s i 2 / n 1 + s 2 2 / n 2 f( s i V n J | ( s 2 2 / n 2 f

n-i - 1 n 9 - 1

Mean difference of two normaldistributions forpaired sam ples, \xdd t a/2 df = n - 1where n is no. of pairs

Variance a of a normaldistribution ( n - l ) s 2 ( n - l ) s ;2 ' 2

X a / 2 X 1 - a / 2df = n - 1

Ratio of the variances a-|2/a22 oftwo normal distributions S 1 s i2 FS 2 r a / 2 ; v i , V2 S 22 r a /2 ; V2,v-| ; Vi = n! - 1 , v2 = n 2 - 1

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CONFIDENTIAL APPENDIX 1(2) CS/OCT2010/QMT500

H Y P O T H E S I S T E S T I N GNull Hypothesis Test statisticH 0 : \i = \x0a2 unknown

_ x - n oicalc S df = n - 1

H 0 : Hi - ^ 2 = Dof = o22 and unknown_ (x :-x 2)-D .^calc i i1 1

n 1 n 2

df = n, + n 2 - 2 , s p = ( r M - W + t o - Wn-\ + n 2 - 2Ho : m - H2 = Dvariances of * o-22 and unknown

t _ ( x i - x 2 ) - DIcalc I n-i n->

d f _ ( s i 2 / n 1 + s 2 2 / n 2 fs i 2 / n i f , (s22/n2Jn i - 1 n? - 1

H 0 : no = D tcaic = ; df = n - 1whe re n is no. of pairsV n ~

H 0 : a = r j 0 (or a = o 0 ) ( n - l ) sX c a i c ^ ^ ^ ^ d f = n " 1

O O i - CJ22 _ 2 "calc , 2 ' Vi = n-i - 1 , v2 = n2 - 1

T E S T I N G S I G N I F I C A N C E O F R E G R E S S I O N(Ana lys is of Var iance appro ach)

Total sum of squaresssY= x ( y i - y ) 2=Sy f -i=1 i=1

Regression sum of squares s sR = z ( y i - y ) 2 = P i Si=1 xy

where S x y = EXJVJ -Ex,

V . i = 1 U=1i=1


2010 oct qmt500

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