R09 - December, 2011 - Regular Examinations - Set - 3.

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
****
1. (a) Among 100 sh caught in a large lake, 18 were inedible due to the pollution
of the environment. With what con dence can we assert that the error of this
estimate is at most .065?
(b) A machine puts out 16 imperfect articles in a sample of 500 articles. After
the Machine is overhauled it puts out 3 imperfect articles in a sample of 100
articles. Has the Machine improved? [15]
2. Given n=10, x = 4.5, y = 3.6 and sum of the product of deviation from the
mean of X and Y is 64 nd the correlation co-e cient. [15]
3. (a) 2Check whether the following marker chain is regular and ergodic?
664
1 1=2 1=2 0
1=2 0 0 1=2
1=2 0 0 1=2
0 1=2 1=2 1=2
3
775
(b) The transition probability matrix of a marker chain is given by
2
4
0:3 0:7 0
0:1 0:4 0:5
0 0:2 0:8
3
5
verify whether the matrix is irreducible or not? [15]
4. Consider a self servie store with one cashier. Assume poisson arrivals and exponen-
tial service time. Suppose that a customers arrive on average of every 5 minutes
and the cashier can serve in 5 minutes. Find:
(a) The average number of customers queuing for service.
(b) The probability of having more than 10 customers in the system.
(c) The probability that the customer has to queue for more than 2 minutes. [15]
5. (a) If the Mean of a Poisson variable is 1.8, then nd
i. p(x > 1)
ii. p(x=5)
iii. p(0<x<5)
(b) If the masses of 300 students are normally distributed with mean 68 kgs and
standaard deviation 3kgs how many students have masses.
i. Greater than 72 kgs
ii. Less than or equal to 64 kg

iii. Between 65 and 71 kg inclusive [7 8]
6. (a) The average losses of workers, before and after certain pro are given. Use 0.05
level of signi cance to test whether the program is e ective (paired sample
t-test).
40 and 35, 70 and 65, 45, -42, 120 and 116, 35 and 55 and 50, 77 and 73.
(b) The table gives the biological values of protein from 6 cow's milk and 6 bu alo's
milk. Examine whether the di erences are signi cant. [15]
Cow's milk 1.8 2.0 1.9 1.6 1.8 1.5
Bu alo's milk 2 1.8 1.8 2.0 2.1 1.9
7. (a) Prove that for a random sample of size n, n, X1, X2........Xn taken from a nite
population s2=1
n
Pn
i??1
??
Xi ?? X
2
is not unbiased estimator of the parameter 2
but 1
n
Pn
i??1
??
Xi ?? X
2
is unbiased.
(b) A sample of 100 iron bars is said to be drawn from a large number of bars.
Whose lengths are normally distributed with mean 4 feet and S.D..6 ft. If the
sample mean is 4.2 feet can the sample be regarded as a truly random sample?
[8 7]
8. (a) There are six boxes, of which two are round and four are square. Each round
box contains two green marbles and three blue marbles. Each square box
contains one green marble and three blue marbles. A box is chosen at random
and a marble is chosen at random from it. If the marble is blue, what is the
probability that it is from
i. Round box
ii. Square box.
(b) Find the mean and the variable of uniform probability distribution given by
f(x) = 1
n for x=1,2,3,4,..........n [7 8]
*****