MBOSE Class 12 Question Paper 2023 for Statistics

Total No. of Printed Pages—11
HS/XII/A. Sc./S/23

2023

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 : 1×10=10
(a) If a random variable X takes the values 1, 2, 3 with
r
probability P (X = r ) = ; r = 1, 2, 3. Then E (X ) is equal
6
to
7
(i)
6
7
(ii)
3
3
(iii)
7
2
(iv)
7

/53 [ P.T.O.

( 2 )

(b) If X is a random variable, then

(i) E (X 2 ) ³ { E (X )} 2

(ii) E (X 2 ) = { E (X )} 2

(iii) E (X 2 ) £ { E (X )} 2

(iv) None of the above

(c) If X follows binomial distribution with parameters n
and p, then

æX ö
(i) E ç ÷ < p
èn ø

æX ö
(ii) E ç ÷ = p
èn ø

æX ö
(iii) E ç ÷ > p
èn ø

(iv) None of the above

HS/XII/A. Sc./S/23/53 [ Contd.

( 3 )

(d) If X follows Poisson distribution and
P (X = 0) = P (X = 1) = a , then

(i) a = e -1

(ii) a = e 2

(iii) a = e -2

(iv) a = e

(e) In Normal distribution

(i) Mean > Median

(ii) Mean > Mode

(iii) Mean < Median

(iv) Mean = Median = Mode

(f) Fisher’s Ideal index number is

(i) the product of Laspeyres’ and Paasche’s index
number

(ii) the GM of Laspeyres’ and Paasche’s index
number

(iii) the AM of Laspeyres’ and Paasche’s index
number

(iv) the addition of Laspeyres’ and Paasche’s index
number

HS/XII/A. Sc./S/23/53 [ P.T.O.

( 4 )

(g) Time Reversal Test is satisfied by

(i) Laspeyres’ index number

(ii) Paasche’s index number

(iii) Fisher’s index number

(iv) None of the above

(h) Time series consists of

(i) one component

(ii) two components

(iii) three components

(iv) four components

(i) If s 2 is the population variance, then which of the
following is correct in case of SRSWR?

s2
(i) V (x ) >
n

s2
(ii) V (x ) <
n

s2
(iii) V (x ) =
n

(iv) None of the above

HS/XII/A. Sc./S/23/53 [ Contd.

( 5 )

(j) Changes that take place as a result of natural
calamities are classified under the head

(i) secular movement

(ii) seasonal variations

(iii) cyclical variations

(iv) irregular variations

2. Fill in the blanks : 1×5=5

(a) If X is a random variable and a is any constant, then
V (aX ) = _____.

(b) The recurrence relation for the probabilities of
Poisson distribution is p (x ) = _____.

(c) Cost of living index number is used in adjustment of
_____ allowances to compensate the rise in the price
level.

(d) Natural cause is associated with _____ variation of
time series.

(e) In case of SRSWOR, E (x ) = _____.

HS/XII/A. Sc./S/23/53 [ P.T.O.

( 6 )

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

(a) If X, Y, Z are random variables, then
E (XYZ ) = E (X )E (Y )E (Z ).

(b) Normal distribution is unimodal.

(c) Fisher’s formula satisfies ‘Factor Reversal Test’.

(d) Fire in a factory is a seasonal component of time
series.

(e) Sample survey is superior than census.

SECTION—II
( Marks : 30 )

4. Answer the following questions : 3×10=30

1 æ1 ö
(a) If f (x ) = , x = 2, 4, 8, 16, find Eç ÷ where
4 èX ø
f (x ) = P (X = x ).

(b) Prove that

E[(X - c )2 ] = var(X ) + [E (X ) - c ]2

HS/XII/A. Sc./S/23/53 [ Contd.

( 7 )

(c) “The mean of a binomial distribution is 5 and
standard deviation is 3.” State whether the statement
is true or false. Give your answer with justification.

(d) From a population of defective and non-defective
items, 100 items are drawn at random. Find the
probability of getting 2 defective items. Given that the
population consists of 1% defective items.

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

(f) What is time reversal test? Show that Paasche’s index
number formula does not satisfy this test.

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

(h) What is a time series? State its uses.

(i) “Index numbers are economic barometers.” Explain.

(j) From a large population, a random sample of 49
observations is taken. It is given that population SD
is 3·5. Find the SE of the sample mean.

HS/XII/A. Sc./S/23/53 [ P.T.O.

( 8 )

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

Answer four questions, taking at least one from each Group

GROUP—A

5. (a) The random variable X is normally distributed
with mean 2 and standard deviation 5. Find
æX - 2 ö
P (-8 < X < 1.
) Given that P ç < 0 × 2÷ = 0 × 5793
è 5 ø
æX - 2 ö
and P ç < 2÷ = 0 × 9772.
è 5 ø 5

(b) If X follows the binomial distribution
x 18 - x
æ1 ö æ 2 ö
f (x ) = 18C x ç ÷ ç ÷ ; x = 0, 1, 2, L, 18
è3ø è3ø

Find the following :

æ X - 6ö
(i) E ç ÷
è 10 ø

(ii) E (X - 6)2
3+3=6

(c) Write down the standard deviation of the following
distribution : 1½

e -4 4 x
P (x ) = ; x = 0, 1, 2, 3, L
x!

HS/XII/A. Sc./S/23/53 [ Contd.

( 9 )

6. (a) Prove that V (aX + b ) = a 2V (X ), where a and b are
constants. 4

(b) If X and Y are two random variables, then prove that

V (X + Y ) = V (X ) + V (Y ) + 2 cov(X , Y ) 4½

(c) If for a random variable X, E (X ) = 12 and V (X ) = 9, find
the value of E (2X + 5) and V (2X + 5). 4

GROUP—B

7. (a) Calculate the cost of living index number from the
following data : 6½

Price

Items Base year Current year Weight

Food 39 47 4

Fuel 8 12 1

Clothing 14 18 3

House Rent 12 15 2

Misc. 25 30 1

(b) Show that Laspeyres’ price index number and
Paasche’s index number do not satisfy Factor
Reversal Test. 3+3=6

HS/XII/A. Sc./S/23/53 [ P.T.O.

( 10 )

8. (a) (i) What are the components of time series?

(ii) State the various methods of measuring trend.
Give two merits and one demerit of the least
squares method. 2+2+2+1½=7½

(b) State the components of a time series with which the
following can be associated : 1×5=5

(i) An era of prosperity

(ii) Puja sales in a cloth store

(iii) Recession

(iv) Decrease in death rate due to advancement of
medical services

(v) Fire in a factory

GROUP—C

9. (a) Write any three demerits of Simple Random Sample
(SRS). 2×3=6

(b) A simple random sample of 10 households was drawn
from a town of 1200 households. The number of
persons in the selected houses were 2, 2, 3, 3, 4, 5, 6,
3, 3, 4. Estimate the number of persons in the town. 4

(c) What do you mean by stratified random sampling? 2½

HS/XII/A. Sc./S/23/53 [ Contd.

( 11 )

10. (a) What are the differences between sampling error and
non-sampling error? 6

(b) In case of SRSWOR, prove that E (x ) = X , i.e., the
sample mean is an unbiased estimator of population
mean. 6½

HHH

HS/XII/A. Sc./S/23/53 K23—90