DOEACC Society 2006 DOEACC B Level B4.1 Computer Based Statistical & Numerical Techniques ( ) - Question Paper

B4.1-R3: COMPUTER BASED STATISTICAL AND NUMERICAL TECHNIQUES
NOTE:
Time: three Hours Total Marks: 100
1.
a) An analog signal received at a detector (measured in microvolts) may be modeled as a
Normal random variable N(200, 256) at a fixed point time. What is the probability that the
signal is larger than 240 microvolts?
b) If 2 random variables have the joint density provided by
f( x, y) = 2e-2y e-x if 0 < x < µ , 0 < y < µ
=0, elsewhere.
Are X and Y independent?
c) A random sample of 225 interactive response time measured at user terminals yields
x = seven units with a sample standard deviation of three units. obtain a 95% confidence time
interval for the interactive response time.
d) Inquiries to an on-line computer system arrive on 5 communication lines. The
percentage of messages received from lines 1, 2, 3, 4, five are 20, 30, 10, 15 and 25
respectively. The probabilities that the length of an enquiry will exceed 100 characters
are 0.4, 0.6, 0.2, 0.8 and 0.9. What is the probability that a randomly opted inquiry
will be longer than 100 characters?
e) Construct a divided-difference table for the function f provided beneath and write down the
Newton form of the interpolating polynomial.
x one 3/2 0 2
f(x) three 13/4 three 5/3
f) Evaluate = ò +
1
0
I x /( x2 1)dx
using Simpson’s 1 3rd rule with the number of subintervals=6. Hence deduce an
approximate value of log 2.
g) Show that sample variance
å=
-
-
=
n
i 1
( xi x )2
n 1
S two 1
is an unbiased estimator of the population variance.
(7x4)
B4.1-R3 Page one of three July, 2006
1. ans ques. one and any 4 ques. from two to 7.
2. Parts of the identical ques. should be answered together and in the identical
sequence.
2.
a) obtain the cubic polynomial, which takes the subsequent values:
y(0)=1, y(1)=0, y(2)=1 and y(3)=10
Hence find y(4).
b) Factorize the matrix
5 -2 1
A = seven one -5
3 seven 4
into LU form, where L is unit lower triangular and U is upper triangular and hence solve
the system of equations
5x – 2y + z= 4
7x + y - 5z= 8
3x + 7y + 4z= 10
(8+10)
3.
a) Consider the system consisting of 5 stages with independent components. Let the
reliability Ri of component i be described as the probability that the component is
functioning properly. provided that R1=0.95, R2=0.99, R3=0.70, R4=0.75, R5=0.9,
obtain the reliability of the system.
b) We are provided a box containing 5,000 IC chips of which 1,000 are manufactured by company
X and the rest by company Y. Ten percent of the chips made by company X and five percent
of the chips made by company Y are defective. If a randomly chosen chip is obtained to be
defective, obtain the probability that it came from company X.
(10+8)
4.
a) Suppose that the travel time from your home to your office is normally distributed with mean
40 minutes and standard deviation seven minutes. If you want to be 95 % certain that you will not
be late for an office appointment at one P.M., what is the recent time that you should leave
home?
b) Let X be uniformly distributed on (0, 1). Show that Y = -?-1 1n(1 – X) has an exponential
distribution with parameter ? >0.
(10+8)
5.
a) The servicing of a machine requires 2 separate steps, with the time needed for the
first step being an exponential random variable with mean 0.2 hour and the time for the
second step being an independent exponential random variable with mean 0.3 hour. If a
repair person has 20 machines to service, obtain approximate probability that all the work
can be completed in eight hours.
B4.1-R3 Page two of three July, 2006
R1 R2 R3
R3
R3
R4
R4
R5
b) If S1 and S2 are the standard deviations of independent random samples of size n1 =61
and n2 =31 from normal populations with s1
2
=12 and s2
2
=18, obtain
P(S1
2
/ S2
2
> 1.16).
(9+9)
6.
a) Among 6 measurements of the boiling point of a silsicon compound, the size of the
fault was 0.07, 0.03, 0.14, 0.08 and 0.03o C. Assuming that these data can be looked
upon as a random sample from the population provided by
f(x; ?) = two (? - x ) / ?2 for 0 < x < ?
= 0 elsewhere
obtain an estimator for ? by the method of moments.
b) presume that the time to failure, X, of a telephone switching system is exponentially
distributed with a failure rate l. Estimate the maximum- likelihood failure rate l from a
random sample of n times to failure.
(9+9)
7.
a) In a random sample of visitors to a famous tourist attraction, 84 of 250 men and 156 of
250 women bought souvenirs. Construct a 95 % confidence interval for the difference
ranging from the actual proportions of men and women who buy souvenirs at this tourist
attraction.
b) A Chemical Engineer is investigating the effect of process operating temperature on
product yield. The study outcomes in the following:
Temp(°C) (x): 100 110 120 130 140 150 160 170 180 190
Yield % (y) : 45 51 54 61 66 70 74 78 85 89
Estimate linear regression model of y on x.
(9+9)
B4.1-R3 Page three of three July, 2006

Earning:   Approval pending.