introduction to statistics and probability

CONFIDENTIAL CS/OCT2003/QMT161

UNIVERSITI TEKNOLOGI MARAFINAL EXAMINATION

COURSE

COURSE CODE

DATE

TIME

FACULTY

SEMESTER

PROGRAMME / CODE

INTRODUCTION TO STATISTICS ANDPROBABILITY

QMT161

7 OCTOBER 2003

3 HOURS (2.15 p.m - 5.15 p.m)

Information Technology and Quantitative Sciences

June 2003- November 2003

Diploma in Computer Science / CS110Diploma in Actuarial Science / CS112Bachelor of Science (Hons) (Intelligence System) / CS223Bachelor of Science (Hons) (Business Computing) / CS224Bachelor of Science (Hons) (Data Communication andNetworking) / CS225Bachelor of Science (Hons) (Information SystemEngineering) / CS226

INSTRUCTIONS TO CANDIDATES

1. This question paper consists of two (2) sections : SECTION A (8 Questions)SECTION B (5 Questions)

2. Answer ALL questions from Section A and Section B. Answers to each question from SectionB should start on a new page.

3. Calculators can be used.

4. Do not bring any materials into the examination room unless permission is given by theinvigilator.

5. Please ensure that this examination pack consists of:i) the Question Paperii) an Answer Booklet - provided by the Facultyiii) Appendix 1-1 page (Formulae)iv) Appendix 2-1 page (Standard Normal Table)

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

This examination paper consists of 8 pagesCONFIDENTIAL

CONFIDENTIAL CS/OCT2003/QMT161

SECTION A (40 MARKS)

Answer ALL questions.

QUESTION 1

The number of goals scored per game by a national footballer in the year of 2001 and 2002were as follows:

Number ofgoals

Number ofgames

0

9

1

6

2

3

3

1

4 or more

1

Calculate:a) the meanb) the median and explain the meaning of the value obtained.

QUESTION 2

(5 marks)

The table below shows a summary of the distributions of Statistics scores for 2 groups ofstudents in the final examination.

MeanMedianStandard deviationNumber of students

Group A60.562.012.528

Group B65.063.015.030

a) Find the Pearson's coefficient of skewness for group A and determine the shape ofthe distribution.

b) Find the mean score obtained for both groups.

(5 marks)

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CONFIDENTIAL 3 CS/OCT 2003/QMT161

QUESTION 3

Given that events A and B are independent and that P( A ) = - and P( A u B ) = - ,8 3

finda) P( B )b) P( B1 / A').

(5 marks)

QUESTION 4

A random variable X has the following probability distribution

P(X = x) =k,

where k is a constant.a) Find the value of k.

b) Calculate P( |X -1 < 4 ) .

(5 marks)

QUESTION 5

On average, the demands for a particular electronic device at a warehouse are made 4times per day. Find the probability that on a given day this device isa) demanded at least 4 times,b) not demanded at all.

(5 marks)

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QUESTION 6

X is normally distributed with mean 300 and standard deviation, a. Find the value of a ifP (295 < X < 305) = 0.1587.

(5 marks)

QUESTION 7

A tile manufacturer packaged its tiles in cartons. Each carton consists of 100 boxes. Theprobability that a box is found to be broken is 0.03. Using an appropriate approximation, findthe probability that a carton contains more than 2 broken boxes.

(5 marks)

QUESTION 8

A basket contains oranges of small, medium and large sizes in the ratio of 3 : 4 : 5. Thesizes are graded according to their weights. Three oranges are taken randomly withoutreplacement. Find the probability thata) all three are of the same size,b) at least two large oranges are selected.

(5 marks)

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CONFIDENTIAL CS/OCT2003/QMT161

SECTION B (60 MARKS)

Answer ALL questions.

QUESTION 9

The following frequency distribution shows the daily production (in '000 units) of SyarikatMaju Sdn. Bhd. in June 2003.

Production

15

25

30

35

40

45

( ' 000 units )

-24

-29

-34

-39

-44

-49

Number of days

7

9

6

3

3

2

a) Calculate the mean and standard deviation of the above data.(51/2 marks)

b) Find the modal production and explain the meaning of the value obtained.(31/2 marks)

c) From its record, the company obtained the following statistics regarding itsproduction in December 2002.

Mean = 35080 units.

Standard deviation = 9030 units.

Using an appropriate measure, determine which month (June or December) has aless consistent production. Justify your answer.

(3 marks)

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QUESTION 10

The probability density function of a continuous random variable X is given by

f(x) =

' 2(x + 1)5m ,0 ,

1 j»- V jf* 4, - I < X < I

1 <x<3elsewhere

where m is a constant.

a) Prove that m = 1/10. (3 marks)

1b) Find the probability that X is greater than — . (3 marks)

c) Calculate the expected value of X. (3 marks)

d) Sketch the graph of the probability density function, f(x). (3 marks)

QUESTION 11

A distribution manager can hire a truck from three transporting firms for the delivery of hisgoods. Of the trucks hired 30% are from firm A, 50% from firm B and 20% from firm C. Theprobabilities that the trucks from firms A, B and C will not deliver on time are 15%, 9% and

5% respectively.

a) Find the probability that a truck chosen at random will be from firm B and will deliveron time.

(4 marks)

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CONFIDENTIAL 7 CS/OCT 2003/QMT161

b) Find the probability that a truck hired will not deliver on time.(4 marks)

c) Suppose a late delivery is made, what is the probability that it came from firm A?(4 marks)

QUESTION 12

The volume of soy sauce in a bottle produced by a food manufacturer is normally distributedwith mean 350 ml and standard deviation 10 ml.

a) Find the probability that a randomly selected bottle of soy sauce contains less than340 ml.

(3 marks)

b) If a shopkeeper orders 1000 bottles of soy sauce from the manufacturer, estimate thenumber of bottles with the volume less than 340 ml.

(2 marks)

c) Past records show that on average, there is one bottle of soy sauce that has volumemore than k ml in a batch of 100 bottles produced. Find the value of k.

(3 marks)

d) A bottle of soy sauce is rejected if its volume differs from the mean by more than15 ml. Find the probability that a bottle of soy sauce will be rejected.

(4 marks)

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QUESTION 13

It is known that 75% of mice inoculated with a serum are protected from a certain disease. If8 inoculated mice are chosen randomly, find

a) the average number of mice that do not contract the disease,

b) the probability that none contracts the disease,

c) the probability that at least 2 mice do not contract the disease,

d) the probability that not more than 3 mice contract the disease.

(3 marks)

(21/2 marks)

(31/2 marks)

(3 marks)

END OF QUESTION PAPER

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CONFIDENTIAL APPENDIX 1 CS/OCT 2003/QMT161

Sample Measurements

Yfx1. Mean, x = ±=>—

n

2. Median, x = Lm +-Yfm-1

. C

3. Mode, x=Lm o ++ A

. C

4. Standard Deviation, s =n-1

fx2-n

5. Coeficientof Variation, CV = — x 100x

6. Pearson's Measure of Skewness = Mean-Mode 3(Mean-Median)

where

nL

m-1

A2

C

S tan dard Deviation S tan dard Deviation

total frequencylower median class boundarylower modal class boundarycumulative frequencies for the classes before the median classfrequency of median class(modal class frequency) - (frequency for the class before the modalclass)(modal class frequency) - (frequency for the class after the modalclass)size of the class

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CONFIDENTIAL APPENDIX 2 CS/OCT2003/QMT161

Areas in Upper Tail of the Normal Distribution

The function tabulated is 1 - <P(z) where <fl(z) is the cumulative distributiDa function of a standardised Normal variable, 2.03

1 r 2Thus I - 0{z) = -y== I *"* is the probability that a standardised Normal variatc sdected at random wiU be greater/than a

value of i =

*-,"a

0.00.1OJ20.30.40_50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.22.42.52.62.72.32.93.03.13J23.33.43.53.63.73.83.94.0

5.0-}

.00

.5000

.4602

.4207

.382134463085.2743.2420.2119.1841.1587.1357.1151.0968.0808.0668.0548.0446.0359.0287.02275.01786.01390.01072.00820.00621.00466.00347.00256.00187.00135.00097.00069.00048.00034.00023.00016.000108.000072.000048.000032

i- o.aoo QOO

.01

.4960

.4562

.41683783.34093050J2.709.2389.2090.1814.1562-.1335.1131:0951.0793.0655.0537.0436.0351.0281.07777.01743.01355.01044.00798.00604.00453.00336.00248.00181.00131.00094.00066..00047.00032.00022.00015.000104.000069.000046

2867

.02

.4920

.4522

.4129374533723015.2676.2358.2061.1788.1539.1314.1112.0934.0778.0643.0526.0427.0344.0274.02169.01700.01321.01017.00776.00587.00440.00326.00240.00175.00126.00090.00064.00045.00031.00022.00015.000100.000067.000044

5:5'-

.03

;4880.4483.40903707J336J2981J2643.2327.2033.1762.1515.1292.1093.0918.0764.0630.0516.0418.0336.0268.02118.01659.01287.00990.00753.-00570.00427.00317.00233.00169.00122.00087.00062.00043.00030.00021.00014.000096.000064.000042

*• 0.000 000

.04

.4840

.4443

.4052

.3669

.3300

.2946

.2611

.2296

.2005

.1736

.1492

.1271

.1075

.0901

.0749

.0618

.0505

.0409

.0329

.0262

.02068

.01618

.01255

.00964

.00734

.00554

.00415

.00307

.00226

.00164

.00118

.00084

.00060

.00042

.00029

.00020

.00014

.000092

.000062

.000041

0190

.05

.4801

.4404

.401336323264J2912.2578J2266.1977.1711.1469.1251.1056.0885.0735.0606.0495.0401.0322.0256.02018.01578.01222.00939.00714.00539.00402.00298.00219.00159-.00114.00082.00058.00040.00028.00019.00013.000088.000059.000039

.06

.4761

.4364397435943222.2877JZ546.2236.1949.1685.1446.1230.1038.0869.0721.0594.0485.0392.0314.0250.01970.01539.01191.00914.00695.00523.00391.00289.00212.00154.00111.00079.00056.00039.00027.00019.00013.000085.000057.000037

6.0 -v

.07

.4721

.4325

.393635573192.2843.2514J22Q6.1922.1660.1423.1210.1020.0853.0708.0582.Q475.0384.0307.0244.01923-.01500.01160.00889.00676.00508.00379.00280.00205.00149.00107.00076.00054.00038.00026.00018.00012.000082.000054.000036

0.000:0000

.08

.4.681

.4286

.389735203156.2810.2483.2177.1894.1635.1401.1190.1003.0838.0694.0571.0465.0375.0301.0239.01876.01463.01130.00866.00657.U0494.00368.00272.00199.00144.00104.00074.00052.00036.00025.00017.00012.000078.000052.000034

01.0.

.09

.4641

.4247385934833121.2776J2451J2148

-.1867.1611.1379,1170.0985.0823.0681.0559.0455.0367.0294.0233,01831.01426.01101.00842.00639.00480.00357.00264.00193.00139.00100-.00071.00050.00035.00024.00»17.00011.000075000050000033

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