This document contains an exam for a Probability and Statistics course, with 8 multiple choice questions covering various probability and statistics concepts. The questions assess students' understanding of topics like: probability calculations for single and multiple events; probability density functions; Poisson distributions; normal distributions; confidence intervals; hypothesis testing; and correlation. Students are instructed to answer any 5 of the 8 questions, with each question worth equal marks towards the exam's total of 80 marks.

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Code No: ECM2221
RA
III B.Tech I Semester Regular Examinations, November 2008
PROBABILITY AND STATISTICS
(Electronics & Computer Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
1. (a) For any three arbitrary events A, B, C , prove that
P (A B C) = P (A) + P (B) + P (C) - P (A n B) - P (B n C) - P (C n
A) + P (A n B n C )
(b) In a certain town 40% have brown hair, 25% have brown eyes and 15% have
both brown hair and brown eyes. A person is select at random from the town
i. If he has brown hair, what is the probability that he has brown eyes also
ii. If he has brown eyes, determine the probability that he does not have
brown hair [8+8]
2. (a) Probability density function of a random variable X = 1/2 sin x in 0 = x = p
= 0 else where.
Find the mean, mode and median for the distribution and also nd the prob-
ability between 0 and p
2
(b) Two dice one thrown 5 times. If getting a double of is a success. Find the
probability that getting the success
i. atleast once
ii. Two times. [8+8]
3. (a) If a Poisson distribution is such that P(x=1). 3 2 = P(x=3).
Find
i. p(x =1)
ii. p(x = 3)
iii. p(2= x = 5)
(b) A sales tax o cer has reported that the average sales of the 500 business that
he has to deal with during a year is Rs.36,000 with a standard deviation of
10,000. Assuming that the sales in these business are normally distributed,
nd
i. the number of business as the sales of while are Rs.40,000/-
ii. the percentage of business the sales of while are likely to range between
Rs. 30,000/- and Rs.40,000/- [8+8]
4. (a) What is the probability that X will be between 75 and 78 if a random sample
of size 100 taken from an in nite population has mean 76 and variance 256.
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(b) According to the norms established for a mechanical aptitude test persons
who are 18 years old should average 73.2 with a standard deviation 8.6. If
4 randomly selected persons of that age averaged 76.7, test the null hypoth-
esis µ = 73.2 against the alternate hypothesis µ > 73.2 at the 0.01 level of
signi cance? [8+8]
5. (a) In a study of an automobile insurance a random sample of 80 body repair
costs had a mean of Rs. 472.36 and a standard deviation of Rs 62.35. If x is
used as a point estimate to the true average repair costs, with what con dence
we can assert that the maximum error doesn’t exceed Rs. 10?
(b) A manufacturer of electric bulbs claims that the percentage defectives in his
product doesn’t exceed 6. A sample of 40 bulbs is found to contain 5 defectives.
Would you consider the claim justi ed? [8+8]
6. A pair of dice are thrown 360 times and the frequency of each sum is indicated
below:
Sum 2 3 4 5 6 7 8 9 10 11 12
Frequency 8 24 35 37 44 65 51 42 26 14 14
Would you say that the dice are fair on the basis of the chi-square test at .05 level
of signi cance. [16]
7. Fit an equation of the form Y=abx to the following data:
x: 2 3 4 5 6
y: 144 172.8 248.8 298.5 298.5 [16]
8. (a) If there are no ties in the ranks then show that the rank correlation is given
be = 1 - 6Sd2 i
n(n2 -1) where di = rank of xi rank of yi
(b) For 20 army personnel, the regression of weight of kidneys (y) on weight of
heart (x), both measured in oz, is y = 0.399 x + 6.394 and the regression of
weight of heart on weight of kidneys is x = 1.212y - 2.461. Find the correlation
coe cient between the two variables and also their means. [8+8]
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