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This question paper consists of 4 questions on 5 printed pages.

UNIVERSITI TUNKU ABDUL RAHMAN

ACADEMIC YEAR 2006/2007

SEPTEMBER EXAMINATION

UCCM1203 PROBABILITY AND STATISTICS I

WEDNESDAY, 13 SEPTEMBER 2006 TIME : 9.00AM 11.00AM (2 HOURS)

BACHELOR OF INFORMATION TECHNOLOGY (HONS)

COMMUNICATIONS AND NETWORKING

BACHELOR OF SCIENCE (HONS) ACTUARIAL SCIENCE

BACHELOR OF SCIENCE (HONS)

APPLIED MATHEMATICS WITH COMPUTING

YEAR ONE

Instructions to Candidates :

Answer ALL questions. All questions carry equal marks.

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UCCM1203 PROBABILITY AND STATISTICS I

This question paper consists of 4 questions on 5 printed pages.

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Q1. (a) A researcher in a particular city wishes to obtain information on the number of

children in households that receive welfare support. A random sample of 300

households was selected from the welfare rolls of the city.

(i) Identify the relevant population. (2 marks)

(ii) What is the sample? (2 marks)

(iii) What is the variable of interest? Is it qualitative or quantitative

(discrete or continuous)? (3 marks)

(b) Consider three events A, B and C. The probabilities of the various

intersections are shown in the table below. For example, P(ABC)= 0.10.

B B

C C C CA

A

0.05 0.10

0.13 0.15

0.12 0.17

0.18 0.10

(i) Determine the probabilities P(AC) and P(C). (3 marks)

(ii) Find the conditional probability of B given that both A and C occur.(4 marks)

(iii) Are events B and C independent? Explain. (4 marks)

(c) The body temperatures (in degrees Fahrenheit) of 100 healthy adults have a

mean of 98.40F, a median of 98.00F, a range of 1.50F and a standard

deviation of 0.62F.

(i) Describe how the range and standard deviation of the data are affected

if the same constant kis added to each value of the data set. (2 marks)

(ii) Find the values of new mean and new standard deviation after eachtemperature has been converted to the Celsius scale.

[Hint: ( )329

5= FC ] (3 marks)

(iii) If it is found that one of the data 99.10F was wrongly recorded as98.80F, could you determine the new value of median? If no, give areason. Otherwise, find this value. (2 marks)

[Total : 25 marks]

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UCCM1203 PROBABILITY AND STATISTICS I

This question paper consists of 4 questions on 5 printed pages.

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Q2. (a) The following SPSS output shows the final examination scores of 80 students

in a statistics course from the university A.

Statistics

Scores

N Valid 80Missing 0

Mean 58.00Variance 81.00Range 69Minimum 20Maximum 89Percentiles 25 47.50

50 60.0075 70.00

Extreme Values

CaseNumber

Value

Scores Highest 1 80 892 79 883 78 864 77 835 74 79

Lowest 1 1 202 2 233 3 244 4 315 5 31

(i) Draw a box-and-whisker plot for the data. Does this data contain anyoutliers? Justify your answer. (6 marks)

(ii) Compute the Pearsons coefficient of skewness and comment on theshape of the distribution. Hence, find the percentage of students who

scored between 40 and 76 marks. (6 marks)

(iii) Find the value of the 5th percentile. (2 marks)

(iv) Suppose two students are randomly selected from this data. What is theprobability that only one of them scored more than 80 marks?

(4 marks)

(b) A sample of 4 balls is to be selected at random from an urn containing 15 balls

numbered 1 to 15, where six balls are green, five are white and four are black.

(i) How many different samples can be selected? (3marks)

(ii) How many samples can be selected that contain at least one white ball?(4 marks)

[Total : 25 marks]

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UCCM1203 PROBABILITY AND STATISTICS I

This question paper consists of 4 questions on 5 printed pages.

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Q3. (a) A discrete random variableXhas a probability mass function given by

k(2-x) , x = -2, -1, 0

f(x) =x

k , x = 1, 2

0 , elsewhere

(i) Show that the value ofkis21

2. (3 marks)

(ii) Find the cumulative distribution of X. Then, use it to evaluate

P( X 1). (5 marks)

(b) LetXbe a geometric random variable with parameter p and probability mass

function

f(x) = qx-1p , x = 1, 2, 3, where q = 1-p.

(i) Show that the moment generating function is

t

t

Xqe

petM

=

1)( , t< - lnq. (5 marks)

(ii) Using the result from (i), find E(X2). (5 marks)

(c) (i) On the average, 30 calls per hours arrive at an office. Find the

probability that no calls arrive in a three-minute period. (4 marks)

(ii) The probability that a person will believe a rumor about thetransgressions of a certain politician is 0.70. Find the probability that

the tenth person to hear the rumor will be the seventh to believe it.

(3 marks)

[Total : 25 marks]

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UCCM1203 PROBABILITY AND STATISTICS I

This question paper consists of 4 questions on 5 printed pages.

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Q4. (a) A continuous random variableXis uniformly distributed on [1, a] with a mean

of 2.5.

(i) Find the value ofa. (4 marks)

(ii) Find the probability density function for Y = 2lnX.Hence, evaluate E(Y). (9 marks)

(b) Suppose the duration of double-free operation of a new vacuum cleaner is

normally distributed with a mean of 530 days and a standard deviation of 100

days.

(i) What is the probability that the vacuum cleaner will work for at least 2years without trouble? (3 marks)

(ii) The company wishes to set the warranty period so that no more than

10% of the vacuum cleaners would need repair services while under

warranty. How long a warranty period must be set? (5 marks)

(c) In a certain city, the daily consumption of electric power (in millions of

kilowatt-hours) can be treated as a random variable having a gamma

distribution with =2 and =3. If the power plant of this city has a dailycapacity of 10 million kilowatt-hours, what is the probability that this power

supply is inadequate on any given day? (4 marks)

[Hint: 0,

)(

1)(

1 >

=

xexxf

x

]

[Total : 25marks]