MBOSE Class 12 Question Paper 2026 for Statistics

FOR MBOSE CLASS 12 EXAM PREPARATION

MBOSE Class 12 2026
Question Paper ·
Statistics
EXAM YEAR TYPE SUBJECT

MBOSE Class 12 2026 Question Paper Statistics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Total No. of Printed Pages—12
HS/XII/A.Sc./S/26
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2026
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STATISTICS
( FOR CANDIDATES WITH INTERNAL ASSESSMENT—NEW COURSE )

Full Marks : 80

( FOR CANDIDATES WITHOUT INTERNAL ASSESSMENT—OLD COURSE )

Full Marks : 100
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Time : 3 hours

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The figures in the margin indicate full marks for the questions.

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General Instructions :
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The question paper consists of three parts.
(i) Part—A : Consists of two Sections.
Section–I : Consists of multiple-choice and very short
answer-type questions of 1 mark each.
Section–II : Consists of short answer-type questions of
2 marks each.
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(ii) Part—B : Consists of long answer and very long
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answer-type questions. This part is divided into three

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a (iii) Part—C : This part is only for Repeaters/Improvement.
Internal choices are provided in the questions.
(iv) Write all the answers in the Answer Script.
(v) Attempt all parts of a question together at one place.

/67 [ P.T.O.

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( 2 )

( PART : A—OBJECTIVE )
( Marks : 32 )

SECTION—I
( Marks : 16 )

1. Choose and write the correct answer : 1×8=8

(a) If X is a random variable, then the standard
deviation is given by

(i) s = E 2 (X ) - E (X 2 )

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

(iii) s = [E (X 2 ) - { E (X )} 2 ]1/2

(iv) s = { E (X 2 )} 2 - E (X 2 )

(b) If for a random variable X , E (X ) = 12, then the value
of E (2X + 3) is

(i) 27

(ii) 36

(iii) 30

(iv) 45

HS/XII/A.Sc./S/26/67 [ Contd.

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( 3 )

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(c)
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If X is a random variable, then which of the following
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is correct?
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(i) { E (X )} 2 = E (X 2 )

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(ii) E (X 2 ) £ { E (X )} 2

(iii) E 2 (X ) = E (X 2 )

(iv) E (X 2 ) ³ { E (X )} 2

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(d) A discrete distribution having the same mean and
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variance is
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(i) binomial distribution la
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(ii) normal distribution

(iii) Poisson distribution

(iv) hypergeometric distribution

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(e) If X follows Poisson distribution such that

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P (X = 1) = 2 × P (X = 2), then the value of l is

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(i) 0

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a (ii) 1 / 2

(iii) 1

(iv) 2

HS/XII/A.Sc./S/26/67 [ P.T.O.

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( 4 )

(f) If n = 36 and p = q = 1, then the standard deviation
2
of binomial distribution is

(i) 2

(ii) 2

(iii) 3

(iv) 3

(g) Time reversal test is satisfied if

(i) I on ´ I no = 0

(ii) I on = I no

1
(iii) I on =
I no

(iv) I on ¹ I no

(h) The possible outcomes of a random experiment are
called

(i) event

(ii) sample space

(iii) sample point

(iv) None of the above

HS/XII/A.Sc./S/26/67 [ Contd.

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( 5 )

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2. Fill in the blanks :
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If X is a random variable, then V (a X + b ) = _____
a
(a)

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where a and b are constants.
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(b) Binomial distribution is symmetrical if _____.

(c) In SRSWR, the sample mean x is an unbiased
estimator of _____.

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c
. _____.
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(d) The term ‘statistic’ is used to denote

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3. State whether the following statements are True or False :
1×4=4

(a) If X and Y are two random variables, then covariance
between them is given by cov (X , Y ) = - cov (Y , X ).

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(b) In a sampling distribution, a finite population of N

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unit samples of size n can be selected as N ways.

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(c) Moving average method can be used to determine
only trend.

(d) Cyclical variations refer to the movements which
occur after a time interval of more than one year.

HS/XII/A.Sc./S/26/67 [ P.T.O.

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( 6 )

SECTION—II
( Marks : 16 )

4. Answer the following questions (any eight ) : 2×8=16

(a) A binomial variate X has mean 6 and variance 4.
Find n and p.

(b) Find the fallacy if any in the following statement,
by giving proper justification :
“The mean of a binomial distribution is 20 and
its standard deviation is 6.”

(c) The random variable X has the following
distribution :

X : 1 2 3
P (X ) : 1 1 1
6 3 2

Find the variance of X .

(d) If P (X = x ) = 1, x = 1, 2, 3, 4, then find E (X ).
4

(e) Mention two applications of index numbers in
Economics.

(f) A random sample of size 16 is drawn from
a population with standard deviation 8. Find the
standard error of the sample mean.

(g) Define the term reversal test for index numbers.

HS/XII/A.Sc./S/26/67 [ Contd.

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( 7 )

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(h)
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What are the objectives of time series analysis?
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“Index numbers
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(j)
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Define
analysis.

( PART : B—DESCRIPTIVE )

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( Marks : 48 )
Answer four questions, taking at least one from each Group .co
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GROUP—A
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5. (a) For a binomial distribution with
x 8- x
æ1 ö æ 2 ö
f (x ) = 8C x ç ÷ ç ÷ , x = 0, 1, 2, K, 8
è3ø è3ø
Find E (X + 1) and V (X + 1.
) 4

Or
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If E (X ) = 6 and V (X ) = 4 for a random variable X ,
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then find E (2X + 3) and V (2X + 3).
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s em (b) Show that V (a X ) = a 2V (X ), where a is a constant and
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a X is a random variable. 4

(c) If the sum of the mean and variance of a binomial
distribution for 5 trials is 1·8, then find the
distribution. 4

HS/XII/A.Sc./S/26/67 [ P.T.O.

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( 8 )

6. (a) Define Poisson distribution and binomial distribution.
When does a binomial distribution or a Poisson
distribution tend to a normal distribution? 5

(b) A random variable X is such that P (X = 1) = 2P (X = 2).
Find P (X = 0). 2

(c) A random variable X is normally distributed with
mean 10 and standard deviation 4. Find P (6 < X < 14).
Given P (z < 1) = 0 × 843, P (z < -1) = 0 × 1587. 5

GROUP—B

7. (a) Calculate the cost of living index number from the
following data : 5

Base year Current year
Items Weight
Price Price

Food 40 48 4

Clothing 15 18 2

Fuel 10 12 1

Rent 20 25 2

Misc. 12 15 1

(b) Show that Fisher’s ideal index number satisfies the
factor reversal test. 5

(c) Define sample and sampling. 2

HS/XII/A.Sc./S/26/67 [ Contd.

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( 9 )

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8. (a) Discuss the difference between seasonal and cyclical
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variations.
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(b) List the
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methods to measure the trend in a time ag
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series. one advantage and one disadvantage
of the moving average method. 2+2+2=6

GROUP—C

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9. (a) State two advantages of simple random sampling.

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2+2=4

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(b) A simple random sample of 4 shops from a market of
400 shops recorded sales as 12, 18, 24, 30 thousand
rupees. Estimate the total sale of the market. 4

(c) Write short notes on any two of the following : 2×2=4

(i) Sampling error

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(ii) SRSWOR
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a is an unbiased estimate of the population. 6

(b) Distinguish between sampling and non-sampling
errors. 6

HS/XII/A.Sc./S/26/67 [ P.T.O.

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( 10 )

( PART : C )
( Marks : 20 )
( These questions are for Repeaters/Improvement Candidates only )

11. Choose and write the correct answer (any four) : 1×4=4

(a) Which of the following is not correct?

(i) E (aX ) = a E (X )

(ii) E (aX ) = X E (a )

(iii) E (a ) = a

(iv) E (aX + bY ) = a E (X ) + b E (Y )

(b) Normal distribution is

(i) uniparametric

(ii) biparametric

(iii) triparametric

(iv) None of the above

(c) Least square method is one of the best ways of
obtaining

(i) trend values

(ii) data components

(iii) population size

(iv) None of the above

HS/XII/A.Sc./S/26/67 [ Contd.

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( 11 )

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(d)
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A player rolls one fair die. If the die shows an odd
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number, the player wins the value that appears on
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the die, else loses half the value that appears on it.
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The expected gain of the player is
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(i)

(ii) 0
1
(iii)
2
(iv) 1

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(e) In case of SRSWR, V (x ) equals to
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(i)
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n

(ii)
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n
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(iii)
n
s2
(iv)
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c. o(f) Sampling which provides for a known non-zero equal
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chance of selection is
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a
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(iii) quota sampling

(iv) purposive sampling

HS/XII/A.Sc./S/26/67 [ P.T.O.

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( 12 )

12. Answer any two of the following questions : 2×2=4
(a) In a binomial distribution, prove that
mean > variance.
(b) If X follows binomial distribution with mean 4 and
variance 2, then find P (X = 5).
(c) If f (x ) = 1, x = 2, 4, 8, 16, then find E (X ), where
4
f (x ) = P (X = x ).
(d) Given E (X + 4) = 10 and E [(X + 4)2 ] = 116. Determine
E (X 2 ).
13. (a) Find the mean and variance of a Poisson distribution. 7
(b) Explain briefly how Fisher’s ideal index number is
constructed and justify it being called ‘ideal’. 3+2=5
Or
14. (a) Assuming a four-yearly cycle, calculate the trend by
method of moving average for the following data : 6
Year Sales (in crore)
1961 50
1962 52
1963 55
1964 47
1965 51
1966 54
1967 56
1968 57
1969 59
1970 61
(b) Show that Laspeyre’s index number and Paasche’s
index number do not satisfy time reversal test. 6
HHH

HS/XII/A.Sc./S/26/67 26K—90

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