MBOSE Class 12 Question Paper 2021 for Statistics

Total No. of Printed Pages—8
HS/XII/A.Sc/S/21

2021

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

9
(a) If E (X )  , then the value of E (2X  1) is
2
19
(i)
2
11
(ii)
2

(iii) 10

9
(iv)
2
/115 [ P.T.O.

( 2 )

(b) The mean of a binomial distribution having n = 8 and
1
p q  is
2
(i) 2
(ii) 4
(iii) 8
(iv) 6

(c) A binomial distribution is
(i) uniparametric
(ii) biparametric
(iii) triparametric
(iv) None of the above

(d) In SWR, where n is sample size and N is the
population size, the sampling fraction is
(i) N / n
(ii) n / N
(iii) N n
(iv) n N

(e) The term parameter is used to denote the
characteristic of the
(i) population
(ii) size of population
(iii) sample size
(iv) None of the above

HS/XII/A.Sc/S/21/115 [ Contd.

( 3 )

(f) The following is not a method for measuring trend :
(i) Graphic method
(ii) Moving average method
(iii) Harmonic analysis method
(iv) Least squares method

(g) If X is a random variable, then

(i) E (X 2 )  { E (X )}2

(ii) E (X 2 )  { E (X )}2

(iii) E (X 2 )  { E (X )}2

(iv) None of the above

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

(i) In a normal distribution
(i) mean = median = mode
(ii) mean < median < mode
(iii) mean > median > mode
(iv) None of the above

HS/XII/A.Sc/S/21/115 [ P.T.O.

( 4 )

(j) Decrease in death rate due to advancement in
medical science can be classified under the head
(i) random
(ii) cyclical
(iii) seasonal
(iv) trend
(k) The bias which has its origin in the sampling only is
(i) bias due to substitution
(ii) response bias
(iii) observational bias
(iv) None of the above
(l) In a given business venture, a man can make a
profit of R 1,000 with probability 0·8 or take
a loss of R 400 with probability 0·2, then the
expectation is
(i) R 270
(ii) R 720
(iii) R 40,000
(iv) None of the above
(m) The most important factor(s) causing seasonal
variations is/are
(i) growth of population
(ii) depression in business
(iii) weather and social customs
(iv) None of the above

2. Fill in the blanks (any five) : 1×5=5
(a) If  be the expected value of x, then E (x   )  _____.
(b) Normal distribution is a _____ theoretical
distribution.

HS/XII/A.Sc/S/21/115 [ Contd.

( 5 )

(c) If X and Y are two random variables satisfying
E ( XY )  E ( X ) E (Y ) , then the variables are _____.
(d) If X follows binomial distribution with parameters
X 
n and p, then E    _____.
n 
(e) _____ is the difference between the parameter and its
estimate obtained from the random sample.
(f) The number of parameters of a normal distribution
is _____.
(g) If the trend is absent in the data, then the seasonal
indices are computed by the method of _____
averages.

3. State whether the following statements are
True or False (any five) : 1×5=5
(a) In sampling distribution, a finite population of
10 units, samples of size 6 can be selected in
60 ways.
(b) The expected value of a constant is the constant
itself.
(c) Simple random sampling requires smaller sample
size than stratified random sampling for a fixed level
of precision.
(d) Trend can be measured by moving average method.
(e) In time-series analysis, the free-hand method can
represent both linear and non-linear trends.
(f) Sample error is defined as the difference between the
results of a sample and that of a census.
(g) The variance of the binomial distribution is
np 2  np .

HS/XII/A.Sc/S/21/115 [ P.T.O.

( 6 )

SECTION—II
( Marks : 30 )

4. Answer the following questions (any ten) : 3×10=30
(a) If X denotes the number of heads appear when two
coins are tossed, then by using the laws of
expectations, evaluate E (2X  1)2 .
(b) State some of the important factors responsible
for non-sampling errors in any survey (census or
sample).
(c) Write down the properties of normal distribution.
(d) A die is tossed thrice. A success is 1 or 6 on a toss.
Find the mean and variance of successes.
(e) Describe the models of a time series.
(f) Distinguish between seasonal variation and cyclical
variation.
(g) Define stratified random sampling.
(h) Let X and Y are independent random variables, then
show that
E [{ X  E ( X )}{ Y  E (Y )}]  0
(i) Explain the term ‘principle of statistical regularity’ in
sample survey.
(j) Write any three advantages of sample survey over
census.
(k) State whether the following statement is True or
False and justify :
‘‘The mean of a binomial distribution is 3 and
variance is 4.’’
(l) What is the random variable?
(m) Find the mean of the binomial distribution.

HS/XII/A.Sc/S/21/115 [ Contd.

( 7 )

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

Answer four questions, taking at least one from each Group

GROUP—A
5. (a) Define expectation of a random variable X. State the
theorems on the expectations of sum and product of
two variables. 1½+1½+1½=4½

(b) Show that E (aX )  aE ( X ) . 3

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

6. (a) Given E ( X  4)  10 and E [(X  4)2 ]  116 . Determine
2
E ( X ) and E ( X ) . 1½+3=4½

(b) For a normal distribution, mean = 57·9765 and
3rd quartile = 60, then find the standard deviation. 3

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

GROUP—B
7. (a) Define time series. What are its components?
Explain any one of them. 2+2+3=7

(b) Describe the method of moving average for
measurement of trend. 5½

HS/XII/A.Sc/S/21/115 [ P.T.O.

( 8 )

8. (a) A study of demand ( di ) for last 12 years
( t  1, 2, 3,     , 12 ) has indicated the following :
di  100(t  1, 2,    5)
 20(t  6)
 100(t  7, 8,    12)
Compute 5-yearly moving average. 6½
(b) Mention the merits and demerits of moving average
method. 6

GROUP—C
9. (a) Explain the concept of standard error. Discuss the
role of standard error in large sample theory. 3+3½=6½
(b) Prove that in simple random sampling, sample mean
is an unbiased estimate of the population mean. 6

10. (a) Define simple random sampling (SRS) and stratified
random sampling ( St RS ). What are their merits and
demerits? 2+2+2+2=8
(b) A population of size 720 was divided into three
strata whose sizes were 360, 240 and 120
respectively. A sample size 60 is taken from this
population using proportional allocation. Find the
sizes of the samples selected from each stratum. 4½

  

HS/XII/A.Sc/S/21/115 11-21—160