UNIVERSITY OF COLOMBO

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UNIVERSITY OF COLOMBO SCHOOL OF
COMPUTING
IT3201 Mathematics for Computing-II
DEGREE OF BACHELOR OF INFORMATION TECHNOLOGY
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Semester 3 Mathematics for Computing II Model
Paper I MCQ Basic Statistics Part (10 questions)
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1) The number of data-entry mistakes per hour
made by an operator is a random variable X that
has the following probability distribution

• What is the mean number of mistakes of one-hour
  sessions?
• 0.85 (b) 0.95 (c) 0.65
• (d) 0.80 (e) 0.90

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Answer The mean number of mistakes of one-hour
sessions E(X) 0(0.40)1(0.30)2(0.25)3(0
.05) 0.95
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2) Consider the following statements about a
normal distribution (i) The total
area under the normal curve is equal to
one. (ii) The standard normal distribution has
a mean of zero and standard deviation of
one. (iii) The normal curve is skewed to the
left.  Which of the following is (are) correct?
(a) (i) and (ii) only. (b) (i) only. (c) (ii)
only. (d) (i) and (iii) only. (e) (iii) only.  
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(No Transcript)
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3) The random variable X has the following
probability distribution function    Which of
the following is (are) correct?
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(a) X is a continuous random variable. (b) X has
a binomial distribution with n5 and p0.5.
(c) The mean value of X is 2.5. (d) The mean of
X is equal to its standard deviation. (e) X can
be thought of as a sum of 5 independent and
identically distributed random variables.
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Answer
If X Bin (n, p) then, probability distribution
function of X is
So, E(X) n p and V(X) n p q Also, Binomial
distribution consist of sum of n independently
and identically distributed Bernoulli trials. In
this case, n 5 and p 0.5 Therefore E(X)
5(0.5) 2.5 and V(X) 5(0.5)(0.5) 1.25
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4) A box contains ten items of which three are
defective. Two items are selected one at a time
without replacement and X is the number of
defectives in the sample of two. The following
are four statements regarding X.
(i)    X is a binomial random variable with n2
and p0.3. (ii)    X is not a binomial random
variable since the trials are not independent.
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(iii)   X is not a binomial random variable
since n is not fixed. (iv)  X is a binomial
random variable with n10 and p0.3.  Which
of the above is(are) true?
(a) (i) only. (b) (ii) only. (c)
(ii) and (iii) only. (d) (iii) only. (e) (iv)
only.
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Answer Binomial experiment - Properties
There are n trials (n is finite and fixed). Each
trial can result in a success or a failure. The
probability p of success is the same for all the
trials. All the trials of the experiment are
independent.
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D-Defective item
2/9
D

3/10
D
7/10
D
3/9
Selection of 1st item
Selection of 2nd item
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5) Consider the following density curve of a
normal distribution
Which of the following statement(s) is (are) true
about the shaded area?
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• It is the Pr ( X gt 120 ).
• It is the Pr ( Z gt 20 / ? ).
• (c) It is the Pr ( Z lt - 20 / ? ).
• (d) It is 1 - Pr ( Z lt - 20 / ? ).
• (e) It is 0.5 - Pr ( Z gt 20 / ? ).
•  

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Answer
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0.5 P(Zgt 20/6)
Pr (Z gt 20/?) Pr (Z lt - 20/?)
1- Pr (Z lt - 20/?)
Z

0
20/?
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• Suppose X has a binomial distribution with n
  150, p 0.4 and Z has the standard normal
  distribution. Then

Pr ( X gt 35 ) is approximated by
(a) . (b) . (c) .
(d). (e) .
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Answer Since X Bin (150, 0.4) E(X) 150(0.4)
60 and V(X) 150 (0.4)(0.6) 36 Also np
60gt5 and nq 90gt5 Using Normal approximation to
Binomial distribution Pr (X gt 35) ? Pr (Y gt 35
0.5) where Y N(60, 36)
Continuity correction factor
Pr (Z gt (35.5 60)/6)
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7) Suppose Y has a distribution given by the
density function    Which of the following is
(are) false?
(a) (b) E ( Y ) 50 and Var ( Y ) 50
2. (c) E ( Y ) 0.02 and Var ( Y ) 0.02
2. (d) Y has a Poisson distribution. (e)
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Answer If Y exp(?) then, E(Y) 1/ ? and V(X)
(1/ ?)2 Also f(y) ? e -?y for y 0
Here ? 0.02 Pr (Y gt 25)
E(Y) 1/0.02 50 and V(Y) (1/0.02)2 (50)2
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8) The useful life of a computer terminal at a
university computer center is represented by a
random variable U. U is normally distributed with
mean 3.25 years and standard deviation 0.5 years.
What is the probability that a computer terminal
will have a useful lifetime of at least 3 but
less than 4 years?
(c) Pr ( - 0.5 lt Z lt 1.5 ). (d) Pr ( 3 lt Z lt 4
). (e) None of the above.
(a) (b) Pr (U lt 4).
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Answer Given that U N (3.25, 0.52) Have to
find Pr (3 lt U lt4) Pr ( lt Z
lt ) Pr (- 0.5 lt Z lt 1.5)
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9) During rush hours, accidents occur in a
particular metropolitan area at an average of
three per hour. Let Y be a Poisson random
variable representing the number of accidents per
hour with the following probability distribution
function
Which of the following is (are) correct?
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(a) E ( Y ) 3. (b) Pr ( Y lt 1 ) e 3. (c) Y
can only take values 0, 1, 2, and 3. (d) Y has a
continuous probability distribution. (e)
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Answer If YPois (?) then, Pr (Y y)
where y 0, 1, 2,
And E (Y) V (Y) ?
Here ? 3 Therefore E (Y) 3 Pr (Ylt1) Pr (Y
0) e 3 Pr (Y 1) 1- Pr (Ylt1) 1-e 3
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10) Which of the following is (are) true? (a) A
random variable that assumes any value between 5
and 6 is a discrete random variable. (b) The area
under the curve to the left of the mean for a
uniform distribution is 0.5. (c) As the
standard deviation increases, the height of the
normal curve increases. (d) In a binomial
experiment, the probability of success 1 -
the probability of failure. (e) A continuity
correction is needed whenever normal
probabilities are computed.
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Answer A random variable X is said to be
uniformly distributed in the interval a,b if
its density function is
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f(x)
mean
1/b-a
x
a
b
(ab)/2
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Semester 3 Mathematics for Computing II Model
Paper II Structured Questions Basic Statistics
Part (1 question)
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Records show that 30 of the customers in a shoe
store make their payments using a credit
card. This morning 20 customers purchased shoes.
(a)
(i) What is the probability that at least 12
customers used a credit card?
Answer Let X be a random variable, number of
customers who used a credit card. This is a
binomial experiment with n20 and p.30
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i.e. X Bin (20, 0.30) and Use Binomial
probability table with n 20 and p 0.30
Here Pr (X ? k) is given. i.e. Cumulative
Probability
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P(At least 12 used credit card)
P(X?12)1-P(X?11) 1-.995 .005
(ii) What is the probability that at least 3 but
not more than 6 customers used a credit card?
Answer P(3 ? X ? 6) P(X3 or 4 or 5 or 6) P(X
? 6) -P(X ? 2) .608 - .035 .573
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• What is the expected number of customers who used
  a credit card?
• Answer
• E(X) np 20(.30) 6

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• Find the probability that exactly 14 customers
  did not use a credit card.
• Answer
• Let Y be the number of customers who did not use
  a credit card.P(Y14) P(X6) P(X ? 6) - P(X
  ? 5)
• .608 - .416 .192

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• The time it takes to write a standard entrance
• exam is normally distributed, with a mean of 60
  minutes and a standard deviation of 8 minutes.
• (i) What is the probability that a student will
  finish the exam in between 60 and 70 minutes?

Answer If X denotes the time taken to write the
exam, we seek the probability P(60ltXlt70). This
probability can be calculated by creating a new
normal variable the standard normal variable.
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Every normal variable with some m and s, can be
transformed into this Z.
Therefore, once probabilities for Z are
calculated, probabilities of any normal variable
can found.
E(Z) 0
V(Z) 1
To complete the calculation we need to compute
the probability under the standard normal
distribution
The tabulated probabilities correspond to the
area between Z0 and some Z z0 gt0
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P(0ltZlt1.25)
0.3944
Z
1.25
0
0.04
0.4495
1.6
0.4505
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• If only 5 of the students takes more than x
  minutes to finish the exam, estimate the value
• of x.

Answer
Pr (X gt x) 0.05 Pr (
) Pr (Z gt )
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0.5 - Pr (0 lt Z lt ) 0.05 Pr ( 0 lt
Z lt ) 0.45 By standard normal tables
Pr (0 lt Z lt 1.645) 0.45 Therefore
1.645 x 73.16 minutes
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Thank you