MBOSE Class 12 Question Paper 2022 for Statistics

Total No. of Printed Pages—8
HS/XII/A. Sc/S/22

2022

STATISTICS

Full Marks : 100
Time : 3 hours

The figures in the margin indicate full marks for the questions

General Instructions :
(i) Write all the answers in the Answer Script.
(ii) Attempt Part—A (Objective Questions) serially.
(iii) Attempt all parts of a question together at one place.

( PART : A—OBJECTIVE )
( Marks : 50 )

SECTION—I
( Marks : 20 )

1. Choose and write the correct answer (any ten) : 1×10=10

(a) The variance of the binomial distribution is

(i) np

(ii) npq

(iii) npq

(iv) nq

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( 2 )

(b) In sampling distribution, a finite population of
12 units, samples of size 7 can be selected in
(i) 792 ways
(ii) 868 ways
(iii) 84 ways
(iv) 704 ways

(c) The expectation of the number of heads in 15 tosses
of a coin is
(i) 7·5
(ii) 15
(iii) 3·75
(iv) None of the above

(d) If the mean of a binomial distribution having n = 8 is
4, then the value of P is
(i) 32
(ii) 1/3
(iii) 1/2
(iv) 16

1
(e) If X is a random variable and E (X )  , then the
2
value of E (7 X  2) is
(i) 11/2
(ii) 2/11
(iii) 14/2
(iv) 9/2

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(f) If Z  2  3X  5Y and E ( X )  1, E (Y )  2 , then the
value of E (Z ) is
(i) 10
(ii) 15
(iii) 13
(iv) 12

(g) In an index number formula, the time reversal
test is
(i) P01  P10  1

(ii) P01  P10  0

(iii) P01  P10  1

(iv) None of the above

(h) Time series consists of
(i) one component
(ii) two components
(iii) three components
(iv) four components

(i) Which of the following is not a method for measuring
trend?
(i) Graphic method
(ii) Moving average method
(iii) Harmonic analysis method
(iv) Least square method

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(j) Which of the following is not a method for random
sampling?
(i) Simple random sampling
(ii) Stratified sampling
(iii) Systematic sampling
(iv) Cluster sampling

(k) Any statistical measure computed from sample data
is known as
(i) parameter
(ii) statistic
(iii) census
(iv) None of the above

(l) Changes that take place as a result of natural
calamities are classified under the head
(i) secular movements
(ii) seasonal variations
(iii) cyclical variations
(iv) irregular variations

2. Fill in the blanks (any five) : 1×5=5
(a) If X is a random variable and a is constant,
then V (aX )  _____.

(b) The sum of probabilities in a Poisson distribution
is _____.

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(c) Simple random sampling is not suitable if the
population is _____.

(d) The expected value of a constant is the _____ itself.

(e) If X is a random variable and E ( X )  a , E (Y )  b ,
then the value of E (aX  bY )  _____.

1
(f) For a binomial distribution with P  and n  10 ,
4
the value of mean = _____.

(g) An overall tendency of rise or fall in a time series
is called _____.

3. State whether the following statements are
True or False (any five) : 1×5=5

(a) In normal distribution, mean < median < mode.

(b) Fisher’s index number does not satisfy ‘factor
reversal test’.

(c) In time series analysis, the free-hand method can
represent both linear and non-linear trends.

(d) A time series is affected by economic factors alone.

(e) The number of parameters of a binomial distri-
bution is one.

(f) Index numbers are called barometers of economic
change.

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SECTION—II
( Marks : 30 )

4. Answer the following questions (any ten) : 3×10=30

(a) In a binomial distribution, prove that mean
> variance.

(b) A die is thrown at random. What is the expectation
of number on it?

(c) If a random variable X follows Poisson distribution
such that P (X  1)  P (X  2) , find the mean of the
distribution.

(d) Write any three uses of cost of living index number.

(e) Enumerate the objective of analysis of time series.

(f) Write any three advantages of sample survey over
census.

(g) What do you understand by seasonal variations in
time series? Explain with two examples.

(h) Write down the properties of normal distribution.

(i) Explain briefly how Fisher’s ideal index number is
constructed. Justify its being called ‘ideal’.

(j) Obtain the binomial distribution whose mean is 10
and standard deviation is 2 2 .

(k) Name the components of a time series.

(l) If X and Y are independent random variables, then
show that
E [{ X  E ( X )} {Y  E (Y )}]  0

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( PART : B—DESCRIPTIVE )
( Marks : 50 )

Answer four questions, taking at least one from each Group

GROUP—A
5. (a) Find the mean and variance of a Poisson
distribution. 4

(b) (i) Define random variable and probability
distribution. 1+1=2
(ii) A coin is tossed 7 times. What is the probability
that head appears an odd number of times? 3½

(c) The mean and variance of a binomial distribution
4
are 4 and . Find P ( X  1). 3
3

6. (a) In binomial distribution, prove that
n x p
p ( X  x  1)    P (X  x ) 5
x 1 q

(b) The mean of a binomial distribution is 20 and the
standard deviation is 4. Calculate n, p and q. 3

(c) Write the characteristics of Poisson distribution. 4½

GROUP—B
7. (a) The following table gives the group index numbers
and the weights of different heads of expenditure in

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the calculations of a cost of living index except, the
index for the group fuel and lighting :
Group Index Weight
Food 221 35
Clothing 198 14
Fuel and lighting ? 15
Rent 183 8
Miscellaneous 161 20
If the cost of living index is 193, find the index
number of fuel and lighting group. 5½
(b) Define a time series. What are its components?
Explain any one of them. 2+2+3=7

8. (a) Discuss the differences between seasonal and
cyclical variations. 6½
(b) What are the different types of index numbers?
Discuss the main steps followed in the construction
of cost of living index number. 6
GROUP—C
9. (a) In SRSWOR, show that E (x )  X , i.e., the sample
mean is an unbiased estimator of population, where
x and X have usual meanings. 6½
(b) Write short notes on the following : 2×3=6
(i) Random sampling
(ii) Stratified random sampling
(iii) Sampling error

10. (a) Write a note on the difference between census and
sample survey. 6
(b) What is sampling? Give its objects and name the
laws which form the basis of sampling. 6½
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