Introduction To Statistics And Probability

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CS/OCT2003/QMT161

UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION

COURSE

INTRODUCTION TO STATISTICS AND PROBABILITY

COURSE CODE

QMT161

DATE

7 OCTOBER 2003

TIME

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

FACULTY

Information Technology and Quantitative Sciences

SEMESTER

June 2003- November 2003

PROGRAMME / CODE

Diploma in Computer Science / CS110 Diploma in Actuarial Science / CS112 Bachelor of Science (Hons) (Intelligence System) / CS223 Bachelor of Science (Hons) (Business Computing) / CS224 Bachelor of Science (Hons) (Data Communication and Networking) / CS225 Bachelor of Science (Hons) (Information System Engineering) / 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 Section B 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 the invigilator.

5.

Please ensure that this examination pack consists of: i) the Question Paper ii) an Answer Booklet - provided by the Faculty iii) 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 pages CONFIDENTIAL

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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 2002 were as follows:

Number of goals

0

1

2

3

4 or more

9

6

3

1

1

Number of games

Calculate:

a)

the mean

b)

the median and explain the meaning of the value obtained. (5 marks)

QUESTION 2

The table below shows a summary of the distributions of Statistics scores for 2 groups of students in the final examination. Group A Mean Median Standard deviation Number of students

60.5 62.0 12.5 28

Group B 65.0 63.0 15.0 30

a)

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

b)

Find the mean score obtained for both groups. (5 marks)

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QUESTION 3

Given that events A and B are independent and that

P( A ) = 8

and P( A u B ) = - , 3

find a)

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 4 times per day. Find the probability that on a given day this device is

a)

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 if P (295 < X < 305) = 0.1587.

(5 marks)

QUESTION 7

A tile manufacturer packaged its tiles in cartons. Each carton consists of 100 boxes. The probability that a box is found to be broken is 0.03. Using an appropriate approximation, find the 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. The

sizes are graded according to their weights. Three oranges are taken randomly without

replacement. Find the probability that a)

all three are of the same size,

b)

at least two large oranges are selected. (5 marks)

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SECTION B (60 MARKS)

Answer ALL questions. QUESTION 9

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

a)

Production ( ' 000 units )

Number of days

15 -24

7

25 -29

9

30 -34

6

35 -39

3

40 -44

3

45 -49

2

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 its production in December 2002. Mean

= 35080 units.

Standard deviation

= 9030 units.

Using an appropriate measure, determine which month (June or December) has a less 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 ' 2(x + 1)

f(x) =

5 m ,

0,

,

V jf* 4 -1 I <j»- X < I

1 <x<3

elsewhere

where m is a constant.

a)

Prove that m = 1/10.

(3 marks)

b)

1 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 his goods. Of the trucks hired 30% are from firm A, 50% from firm B and 20% from firm C. The probabilities 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 deliver on time. (4 marks) CONFIDENTIAL

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b)

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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 distributed with mean 350 ml and standard deviation 10 ml. a)

Find the probability that a randomly selected bottle of soy sauce contains less than

340 ml. (3 marks)

b)

If a shopkeeper orders 1000 bottles of soy sauce from the manufacturer, estimate the number 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 volume more 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 than 15 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. If 8 inoculated mice are chosen randomly, find a)

the average number of mice that do not contract the disease, (3 marks)

b)

the probability that none contracts the disease, (21/2 marks)

c)

the probability that at least 2 mice do not contract the disease, (31/2 marks)

d)

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

END OF QUESTION PAPER

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APPENDIX 1

CS/OCT 2003/QMT161

Sample Measurements

Yfx

1.

Mean,

x = ±=>— n

2.

Median,

x = Lm +

3.

Mode,

x=Lmo +

4.

Standard Deviation, s =

5.

Coeficientof Variation,

6.

Pearson's Measure of Skewness =

- Y f m-1 . C

+A

. C

n-1

fx 2 -

n

CV = — x 100 x Mean-Mode S tan dard Deviation

3(Mean-Median) S tan dard Deviation

where total frequency lower median class boundary lower modal class boundary

n L

m-1

A2

C

cumulative frequencies for the classes before the median class frequency of median class (modal class frequency) - (frequency for the class before the modal class) (modal class frequency) - (frequency for the class after the modal class) size of the class

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APPENDIX 2

CS/OCT2003/QMT161

Areas in Upper Tail of the Normal Distribution The function tabulated is 1 -
1

03

r

Thus I - 0{z) = -y== I *"*

2

is the probability that a standardised Normal variatc sdected at random wiU be greater/than a

value of i =

*-,"

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0 0.1 OJ2 0.3 0.4

.5000 .4602 .4207 .3821 3446

.4960

.4920 .4522

;4880 .4483 .4090 3707 J336

.4840 .4443 .4052 .3669 .3300

.4801 .4404

.4721 .4325 .3936 3557 3192

0_5 0.6 0.7

3085

J2981 J2643 .2327 .2033

.2946

.4761 .4364 3974 3594 3222 .2877 JZ546 .2236 .1949 .1685

.4.681 .4286 .3897 3520 3156 .2810

.4641 .4247 3859 3483 3121 .2776 J2451 J2148 -.1867 .1611 .1379 ,1170

a

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

1.6 1.7 1.8 1.9 2.0 2.1

2.2 2.2 2.4 2.5 2.6 2.7 2.3 2.9 3.0 3.1 3J2 3.3 3.4

.2743 .2420 .2119 .1841

.1335

.1539 .1314

.1151

.1131

.0968 .0808

.1112

:0951

.0934 .0778

.0668 .0548

.0446 .0359 .0287 .02275 .01786 .01390 .01072 .00820 .00621 .00466 .00347 .00256 .00187

3.8 3.9 4.0

.000032

5.0-}i-

.1814 .1562-

.4129 3745 3372 3015 .2676 .2358 .2061 .1788

.1587 .1357

.00135 .00097 .00069 .00048 .00034 .00023 .00016 .000108 .000072 .000048

3.5 3.6 3.7

.4562 .4168 3783 .3409 3050 J2.709 .2389 .2090

.0793 .0655

.0537 .0436 .0351 .0281

.0643 .0526

.1762 .1515 .1292 .1093 .0918 .0764 .0630

.07777

.02169

.01743 .01355 .01044 .00798

.00181

.01700 .01321 .01017 .00776 .00587 .00440 .00326 .00240 .00175

.00131 .00094 .00066. .00047 .00032

.00126 .00090 .00064 .00045 .00031

.0516 .0418 .0336 .0268 .02118 .01659 .01287 .00990 .00753 .-00570 .00427 .00317 .00233 .00169 .00122 .00087 .00062 .00043 .00030

.00022 .00015 .000104 .000069

.00022 .00015 .000100 .000067 .000044

.00021 .00014 .000096 .000064 .000042

.00604 .00453 .00336 .00248

.000046

o.aoo QOO 2867

.0427 .0344 .0274

5:5'-*•

.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

0.000 000 0190

.4013 3632

3264 J2912 .2578 J2266

.1977 .1711 .1469 .1251 .1056 .0885 .0735

.0606 .0495 .0401 .0322 .0256 .02018 .01578 .01222 .00939 .00714 .00539 .00402 .00298 .00219 .00159-

.1446 .1230 .1038 .0869 .0721 .0594 .0485

.0392

.2843

.2514 J22Q6 .1922 .1660 .1423 .1210 .1020 .0853 .0708 .0582

.Q475 .0384

.0314

.0307

.0250

.0244 .01923.01500 .01160 .00889 .00676 .00508 .00379 .00280 .00205 .00149 .00107 .00076 .00054 .00038 .00026 .00018 .00012 .000082 .000054 .000036

.01970

.01539 .01191 .00914 .00695 .00523

.00391 .00289

.00114 .00082 .00058 .00040 .00028

.00212 .00154 .00111 .00079 .00056 .00039 .00027

.00019 .00013 .000088 .000059 .000039

.00019 .00013 .000085 .000057 .000037

.2483 .2177 .1894

.1635 .1401 .1190 .1003 .0838 .0694 .0571 .0465

.0985 .0823 .0681

.0559 .0455 .0367 .0294 .0233 ,01831 .01426 .01101 .00842 .00639 .00480

.0375 .0301 .0239 .01876 .01463 .01130 .00866 .00657 .U0494 .00368 .00272 .00199 .00144 .00104 .00074 .00052 .00036 .00025

.00264 .00193 .00139 .00100 -.00071 .00050 .00035 .00024

.00017 .00012 .000078 .000052 .000034

.00»17 .00011 .000075 000050 000033

.00357

6.0 -v 0.000:0000 01.0. CONFIDENTIAL