R09 - December, 2011 - Regular Examinations - Set - 1.

II B.Tech I Semester Examinations,December 2011
PROBABILITY AND STATISTICS
Common to ME, MECT, MEP, AME, CSE
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
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1. (a) A can hit a target 4 in 5 shots. B can hit 3 among 4. C can hit 2 among 3.
What is the probability that at least two shots hit the target.
(b) A continous Random variable has the p.d.f f(x) = Kxe?? x If x 0; 0
Determine:
i. K
ii. The mean
iii. variance. [7 8]
2. (a) Explain student t-distribution, its properties and applications.
(b) A random sample of size 25 from a normal population has the mean
...
X =47.5
and the standard deviation S=8.4. Does this information tend to support or
refute the claim that the mean of the population is =42.5. [15]
3. (a) If 2% of light bulbs are defective. Find
i. At least one is defective.
ii. Exactly 7 are defective
iii. P(1<x<8) in a sample of 100
(b) A random sample of size 64 is taken from an in nite population having the
mean 45 and the Standard deviation 8. What is the probability that x will be
between 46 and 47.5. [7 8]
4. De ne the stochastic matrix which of the following stochastic matrices are regular.
[15]
(a)
2
4
1=2 1=4 1=4
0 1 0
1=2 0 1=2
3
5
(b)
2
4
2 1=2 0
1=2 1=2 0
1=4 1=4 1=2
3
5
5. Calculate coe cient of correlation between age of cars and annual maintenance and
comment. [15]
Age of cars ( years)                    2      4      6      7     8      10    12
Annual maintenance cost (rupees) 1600 1500 1800 1900 1700 2100 2000

6. (a) A sample of 400 items is taken from a population whose standard deviation
is 10. The mean of the sample is 40. Test whether the sample has come from
a population with mean 38. Also calculate4 95% con dence interval.
(b) Studying the 
ow of tra c at two busy intersections between 4 p.m and 6 p.m
to determine the possible need for turn signals. It was found that on 40 week
days there were on the average 247.3 cars approaching the rst intersection
from the south which made left turn, while on 30 week days there were on the
average 254.1 cars approaching the rst intersection from the south made left
turns.
The corresponding sample standard deviations are 15.2 and 12. Test the
signi cance between the di erence of two means at 5% level. [7 8]
7. At a one man barber shop, customers arrive according to poisson distribution with
a mean arrival rate of 5 per hour and the hour cutting time is exponentially dis-
tributed, with an average hair cut taking 10 minutes. It is assumed that because
of his excellent reputation customers are always willing to wait. Calculate.
(a) Average number of customers in the shop.
(b) Average number of customers waiting for hair cut.
(c) The percent of time on arrival can walk right in without waiting.
(d) The percent of customers who have to wait prior to getting into the barber's
chair. [15]
8. (a) A manufacturer claims that at least 95% of the equipment which he supplies to
a factory conforms to speci cations. An examination of a sample of 200 pieces
of the equipment revealed that 18 were faulty. Test the claim at a signi cant
level of 0.05.
(b) Random samples of 400 men and 200 women in a locality were asked whether
they would like to have a bus stop near their residence. 200 men and 40 women
were in favour of the proposal. Test the signi cance between the di erence of
two proportions at 5% level. [15]
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