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Posted By: lalitha rajappa Member Level: Diamond Posted Date: 25 Aug 2008
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2007 Gujarat University B.Com Advance Statistics (Subsidiary) Old Course Question paper
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Paper-I
(Old Course)
Time : 3 Hours] [Max. Marks : 70
Instructions : (i) All questions carry equal marks.
(ii) Figure to the right hand side indicates marks.
(iii) Use of simple calculator is allowed.
1. (a) Define the continuity of a function at point x = a. (4)
(b) Answer the following questions : (6)
(i) If ƒ(x) = 2x2 +
1x
, find the value of ƒ(x) + ƒ(– x).
(ii) The cost function of producing x units of an item is C = 3x + 240 and the
selling price per unit is Rs. 6. Find break-even point. Find the number of
units to be produced to get profit of Rs. 600.
(c) Find the limits of following (any two) : (4)
(i) lim
x ? 1
x2 + x – 2
2x2 – x – 1
(ii) lim
x ? 0
7x2 + 5x + 6
x2 + 2x + 3
(iii) lim
x ? 1
x – 1
3 x – 1
(iv) lim
x ? 8
(x + 2) (x2 + 1)
2x3 + 1
OR
(a) State the rules of limit. (4)
(b) Answer the following questions : (6)
FC-13 14
(i) Discuss the continuity of ƒ(x) =
x2 – 4x + 3
x2 – 1 at point x = 1.
(ii) Verify the continuity of the following function at point x = 1.
ƒ(x) = 3x , where 0 = x < 1.
= 3 , where x = 1.
= 6 – 3x , where 1 < x = 2.
(c) Differentiate the following functions with respect to x (any two) : (4)
(i) y =
x2 – 6x + 5
x2 – 3x + 2
(ii) y = x2·log x
(iii) y = (x2 + 2x + 5)3
(iv) y = 11 + x + x2 +
3x
2. (a) Define nPr. Prove that nPr =
n!
(n – r) ! (4)
(b) Answer the following questions : (6)
(i) Find the value of
10
S
i = 1
i (i + 5).
(ii) Find sum of n-terms of the series 2.5 + 5.8 + 8.11 + …..
(c) Find the fourth term in the expansion of (x – 3y)7. (4)
OR
(a) Write the formula of Binomial expansion and state its characteristics. (4)
(b) Answer the following questions : (6)
(i) If nC5 : nC2 = 28 : 5, find the value of n.
(ii) Find the value of ( 6 + 2 )4 + ( 6 – 2 )4
(c) Using principle of mathematical induction prove that : (4)
1.3 + 2.5 + 3.7 + 4.9 + ….. + n (2n + 1) =
n (n + 1) (4n + 5)
6
FC-13 15 P.T.O.
3. (a) Define central moments and state the formula to find coefficients of skewness and
kurtosis of a frequency distribution. (4)
(b) Answer the following questions. (6)
(i) If P(A) =
13
, P (B) =
14
and P (A n B) =
16
, find the value of P (A ? B),
P (A' n B') and P (A'/B').
(ii) Three machines A, B and C produces 15%, 55% and 30% of items daily in a
factory. The percentage of defective items. of these machines are
respectively 4%, 5% and 6%. An item is taken at random from the
production and is found to be defective. Find the probability that it is
produced by machine A.
(c) There are 5 tickets in a box and numbers 1, 1, 2, 2 and 3 are written on tickets.
Two tickets are taken at random from the box. Find the expected value of the sum
of the numbers on the tickets. (4)
OR
(a) Define mathematical expectation. State the characteristics of the mathematical
expectation. (4)
(b) Answer the following questions : (6)
(i) Probability distribution P(x) of a random variable X is as follows :
P(X) = K.x3, where X = 1, 2, 3.
Find the constant K. Find the expected value of X.
(ii) The first four moments about 3 of a frequency distribution are 1, 3, 7 and
20. Find central moments.
(c) The opinion in favour of a book given by three critics is in the ratio of 3 : 5, 2 : 5
and 3 : 4. Find the probability that at least two will give good opinion about a
book. (4)
4. (a) Write the probability mass function of Poisson distribution. State its properties. (4)
(b) Answer the following questions : (6)
(i) The probability of a defective screw in a manufacturing process is
1
10 . Find
mean and variance of defective screws in a sample of 400 screws.
(ii) The mean of a Poisson variate x is 0.64. Find its standard deviation. Find
P (x = 1). [e–0.64 = 0.5273)
FC-13 16
(c) There are 30 screws in a packet of which 5 are defective. If 10 screws are taken at
random from the packet, find the probability that none of them is defective. Also
find mean and variance of the defective screws. (4)
OR
(a) Obtain the probability mass function of Binomial distribution and state its
properties : (4)
(b) Answer the following questions : (6)
(i) Quartiles of a normal distribution are 20.7 and 39.3. Find its mean and
variance.
(ii) There are 40 bulbs in a lot and 5 of them are defective bulbs. If 10 bulbs are
taken one after the other from it, find the probability that at the most one
bulb is defective.
(c) The average daily profit of a hotel is Rs. 200 per customer and its standard
deviation is Rs. 50. Assuming the distribution of daily profit normal, how many
days will have the profit less than Rs. 150 in a year of 365 days ? (4)
[When z = 1, Area = 0.3413 and when z = 2, Area = 0.4772]
5. (a) Explain the difference between population survey and sample survey. (5)
(b) For studying some characteristic of a population, the observations are 8, 12, 15, 18
and 22. Taking all possible samples of size 2 without replacement, verify the
following results : (9)
(i) E–y =
–
Y
(ii) V(–y) =
N – n
N ×
S2
n
(iii) E(s2) = S2.
OR
(a) Explain systematic sampling method. State its merits and demerits. (4)
(b) A population is divided into two strata. The information regarding them is as
follows : (6)
Stratum Number of
Observations
Mean of
Stratum
Variance of
Stratum
1 30 52 15
2 70 48 35
Find the population mean. If 10% random sample is taken from each stratum, then
find variance of stratified sample mean.
FC-13 17 P.T.O.
(c) Ten observations of a population are 18, 20, 23, 25, 28, 29, 21, 26, 28 and 35
respectively. Taking all possible systematic samples of size 2 from it verify
E(–ysy) =–
Y.
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