MCQ

1. If the A.M of 2,6,x,5,7 be 4, then the value of x is

(a) -20 (b) 5 (c) 4 (d) 0

Ans (d)

2. The standard deviation of the following observations is 5,7,1,2,6,3.

(a) 4.66 (b) 2.16 (c) 1.47 (d) none of these

Ans (b)

3. The mode of the following data is : 2,1,3,2,1,5,2,2,1,6,4,21,3.

(a) 5 (b) 2 (c) 3 (d) 1

Ans (b)

4. The median of the following distribution is: 7,9,5,3,`10,15,21,19,17

(a) 15 (b) 9 (c) 10 (d) 17

Ans (c)

5. The relation between mean, median and mode is

(a) Mode=3 Median-2 Mean
(b) Mode=3 Median+2 Mode
(c) Mode=2 Median-3 Mean
(d) Mode=2 Median+3 Mean

Ans (a)

6. A.M of the group of 5 observation is 240; that of a group of 2 observation is 100. If the
two group are merged into one, then the A.M of the merged group is
(a) 250 (b) 125 (c) 200 (d) none of these

Ans (c )

7. In a simple frequency distribution of 200 items one frequency against a data is

missing; but the mean of all data is known as 7740. The missing frequency is.
(a) 49 (b) 50 (c) 48 (d) none of these

Ans (a)

8. 2 xi + yi = 3 is the relation between two sets of data  xi  and  yi  . If  x = 3 then  y =

(a) 3 (b) 4 (c) 6 (d) none of these.

And ©
9. The maximum and minimum values for corelation coefficient between x and y is
(a) 1,0 (b) 2,1 (c) 0,-1 (d) 1,-1

Ans (d)

10. The regression coefficient of x on y is given by

y 
(a) rxy (b) rxy x (c) rxy (d) ryx
x y

Ans: (a)

x−x y− y
11. If u = ,v = , then rxy =
x y
(a) ruv (b) cov( x, y) (c) cov(u, v) (d) none of these

Ans: ©

12. The regression line of x on y is 2 x − 3 y + 5 = 0 then the regression coefficient of x on y

is
2 3
(a) (b) (c) 2 (d) 3
3 2

Ans: (b)

13. If x + 4 y + 3 = 0 and 4 x + 9 y + 5 = 0 be the two-regression lines, then the expectation of

y is.
(a) 1 (b) 2 (c) -1 (d) none of these

Ans ©

14. An observation may repeat in a

(a) SRSWR sample (b) SRSWOR sample
(c) any type of sample (d) none of these

Ans: (a)

15. Under SRSWR the possible samples from the population 2,6,7 are
(a) 2, 2 ,2,6 , 6,7
(b) 2,6 , 2,7 , 6,7
(c) 2, 2 ,6,6 ,7,7 , 2,6 , 6,7 , 6,7
(d) None of these

Ans: ©
16. Sample standard deviation is
(a) A fixed quantity (b) a variable quantity
(c) always zero (d) none of these

Ans: (b)

17. Standard error of a statistic depends on

(a) Population size
(b) observations of sample
(c) observation of population
(d) sample size

ans: (d)

18. The mean and s.d of a normal population are 3 and 0.2. Then the standard error of the
sample mean, with sample size 100 is
(a) 0.2 (b) 0.02 (c) 0.002 (d) 0.6

Ans: (b)

19. The sampling distribution of the sample mean for a large population is approximately
normal if the sample size is
(a) 2 (b) 5 (c) 10 (d) 100

Ans: (d)

20. If a sample of size n is drawn from an infinite population with s.d.  then, for the
sample s.d. S ,
 n 2
(a) E  S  =2 (b) E ( S 2 ) =  2
 n − 1 
 n 
(c) E  S =2 (d) none of these
 n −1 

Ans: (a)

21. A normal population has a mean 0.1 and s.d. 2.1. The mean of the sampling
distribution of the sample mean with sample size 900 is
(a) 1 (b) 0.1 (c) 0.001 (d) none of these

Ans: (b)
22. Bisection method fails when
(a) f(a)f(b)=0 (b) f(a)f(b)<0 (c) both (a) and (b) (d) None of These

Ans d
23. Gauss Seidel method is
(a) direct method (b) indirect method
(c) iterative method (d) None of These

Ans c

24. Gauss Elimination method is

(a) direct method (b) indirect method
(c) iterative method (d) None of These

Ans a
25. Newton Raphson method fails when
(a) f’(x)=1 (b) f’(x)= -1 (c) f’(x)=0 (d) None of These

Ans a
26. Another name of Regula- Falsi method is
(a) Tangent method (b) Root method
(c) Method of False position (d) None of These
Ans c
27. If f(a)f(b)>0, then the equation f(x)=0 has
(a) Exactly one root in (a, b) (b) At least one root in (a, b)
(c ) No root in (a, b) (d) No conclusion can be drawn
Ans d
28. Gauss Seidel method is applicable when the coefficient matrix is
(a) Rectangular (b) Diagonal
(c) Diagonally dominant (d) None of These
Ans c
29. LU method will fail if
(a) any uij>0 (b) any lii=0 (c) uii<0 for all i (d) None of These
Ans b
30. Gauss Elimination method is applicable when the coefficient matrix is
(a) Rectangular (b) Orthogonal (c) Singular (d) None of These
Ans d
31. The other name of LU factorization method is
(a) Gauss’s mehtod (b) Newton’s method
(c) Crout’s method (d) None of These
 Ans c
32. x2-9x+1=0 is
(a) Algebraic equation (b) Transcendental equation
(c ) Exponential equation (d) None of these
Ans a
33. tan x+x=0 is
(a) Algebraic equation (b) Transcendental equation
(c ) Exponential equation (d) None ofthese
Ans b
34. If f(a)f(b)<0, then the equation f(x)=0 has
(a) Exactly one root in (a, b) (b) At least one root in (a, b)
(c ) No root in (a, b) (d) No conclusion can be drawn
Ans b
35. tan x+ ex +2 =0 is
(a) Algebraic equation (b) Transcendental equation
(c ) Exponential equation (d) None of these
Ans b
36. For convergence of Regula-Falsi method the initial interval must be
(a) very large (b) closed (c)very small (d) None

Ans c

37. Using Newton Raphson method the value of 27 , correct to five significant figures, is
a) 5.137 (b) 5.167 (c) 5.196 (d) none of these
Ans c
38. One root of 3x − cos x −1 = 0, correct to five decimal places, computed by method of
bisection is
a) 0.60710 (b) 0.59802 (c) 0.540041 (d) none of these
ans b

39. The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the
observations is
(a) 50000 (b) 250000
(c ) 252500 (d) 255000
Ans c
40. In Gauss-elimination method, the given system of equation by AX = B is converted to
another system UX=y, where U is
(a) diagonal matrix (b) null matrix
(c) identity matrix (d) upper triangular matrix.

Ans d

41. The mean and median of the data 88,72,33,29,70,54,86,91,57,61 is

(a) 65.5,63.1 (b) 64.1,65.5 (c) 64.3,65.5 (d) none

Ans b
42. The mean deviation about the arithmetic mean of the numbers 31,35,29,63,55,72,37 is

(a) 13.9 (b) 14.5 (c) 15.2 (d) 14.9

Ans d
43. The maximum likelihood estimate of p for the population having Binomial distribution
with parameter n,p is
x̅
(a) x̅ (b) nx̅ (c) n (d) none

Ans c

44. One root of the equation x2 + cos x + 2 = 0 lies between

(a) 1 and 2 (b) 0 and 0.5 (c) 0.5 and 1 (d) none of these.

Ans d

45. The two regression lines involving the two variables x and y are x + 4y + 3 = 0 and .
4x + 9y + 5 = 0. The mean of x and y are
(a) 0,1 (b) 1,-1 (c) 1,2 (d) -1,1

Ans b

46. Regula-falsi method has a convergence rate of the order of

(a) 2 (b) 1.62 (c) 1 (d) none of these.

Ans b

47. The correlation coefficient lies between

(a) (0,1) (b) (1,2) (c) (−∞, ∞) (d) (-1,1)

Ans d

48.
A normal population has a mean 0.1 and s.d 2.1. The mean of the sampling distribution of the sample
mean with sample size 900 is
(a) 1 (b) 0.1 (c) 0.001 (d) none of these

Ans b

49. Which of the following methods is an iterative method?

a) Gauss Elimination method b) Gauss- Jordon method
c) Gauss- Seidal method d) Crout’s mathod.

Ans c

50. The standard deviation of the data 49,63,46,59,65,52,60,54 is

(a) 636 (b) 639 (c) 632 (d) 649

Ans a

 −5 2 
One of the eigen value of   is
 −9 6 

Short answer type

The A.M calculated from the following frequency distribution is known to be 72.5. Find the
value of x:
Classes 30-39 40-49 50-59 60-69 70-79 80-89 90-99
:
Frequency: 2 3 11 20 x 25 7
Following is a frequency distribution lacking two class frequency. Find them if the mean is
7.74.
Value 3-5 5-7 7-9 9-11 11-13 Total
:
Frequency: 32 - 57 - 25 200

Find the variance and standard deviation of the following frequency distribution:
Weight(kg): 36-40 41-45 46-50 51-55 56-60 61-65 66-70
No of 14 26 40 33 50 37 25
persons:

The mean and standard deviation of marks of 70 students were found to be 65 and 5.2
respectively. Later it was detected that the marks of one student was wrongly recorded as 85
instead of 58. Obtain the correct s.d.
Find mean and standard deviation of the following distribution:
Class 4-6 6-8 8-10 10-12 12-14 14-16
interval
Frequency 13 111 182 105 19 7

1. If u + 3 x = 5 and 2 y − v = 7 and correlation coefficient of x and y is 0.12 then find

the correlation coefficient of u and v.
2. If x = 4 y + 5 and y = kx + 4 be two regression equations of ‘x on y’ and ‘y on x’
respectively then find the interval in which k lies.
Calculate the correlation coefficient between the height and weight of six persons given as
follow:
Height 162 165 167 168 170 175
Weight 58 60 65 67 72 75

3. If 3 y + 2 x = 9 and 3x + 2 y = 7 are the two regression lines, then what is the value
of the correlation coefficient?
4. The mean weight of 500 ball bearings is 5.02 gm. Their s.d. is 0.30 gm. Find the
probability that a random sample of 100 ball bearing would have a combined
2.96
weight more that 510 gm [Given   ( z)dz = 0.4985 where  ( z) is standard
0

normal function]
5. Mean and standard deviation of 100 observations were found to be 40 and 10,
respectively. If at the time of calculation two observations were wrongly taken as
30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.
6.
 You are given that the variance of x is 9. The regression equations are
8x −10 y + 66 = 0 and 40x −18 y = 214 . Find

(i) Average values of x and y.

(ii) Correlation coefficient between the two variables.

Find the equation of the line of regression of x on y for the following data:
x 1.0 1.5 2.0 2.5 3.0 3.5 4.0
y 5.3 5.7 6.3 7.2 8.2 8.7 8.4

Calculate mean and median from the following data(showing the weights of 60 apples)
Weights 65-84 85-104 105-124 125-144 145-164 165-184 185-204
(grams)
Frequency 9 10 17 10 5 4 5

Compute one root of x + ln x − 2 = 0, correct to two decimal places by method of bisection.

Compute one root of 3x − cos x −1 = 0, correct to two decimal places by method of bisection.
Compute one root of sin x = 10( x − 1) , correct to two decimal places by Regula Falsi method.

Use Newton Raphson method to find 5

3 , correct to five significant figures.

Solve the system of equations by Gauss Elimination method:

3x1 + 9 x2 − 2 x3 = 11
4 x1 + 2 x2 + 13x3 = 24
4 x1 − 2 x2 + x3 = −8

Correct to four significant figures.

Solve the following equations by LU method.
x+2y+-3z=1
2x-y+z=4
x+3y =5
If T is an unbiased estimator of 𝜃, show that √𝑇 is biased estimate of √𝜃.
22.
From the random sample of size 49 drawn from a normal population of s.d 2, find the 99%
confidence interval of the population mean. Find the interval if the mean of such a sample is 3.
2.58
[Given ∫0 ∅(𝑧) 𝑑𝑧 = 0.495]
A sample of weight of 200 students each are drawn from a very normal population of the
weights of students with s.d 10 pound. Variance in each sample is computed. Find
(i) the mean
(ii) the standard deviation of the sampling distribution of sample variance.

Compute one root of x ln x = 1 , correct to two decimal places by Regula Falsi method.

Find the dominant eigenvalue and corresponding eigenvectors of the matrix.

 2 −12 
 1 −5 
 

Long answer type question

1.
Calculate mean, median and hence find the approximate value of the mode from the
following frequency distributions:
Height 60-63 64-67 68-71 72-75 76-79 80-83
:
No of 8 3 18 6 16 8
students :

2.
Compute the s.d. for the following frequency distribution on average daily sales (in
Rs.) of 80 salesmen of a department store:
Class 50-59 60-69 70-79 80-89 90-99 100-109 110-119
Frequency 6 9 15 25 13 7 5

3.
The mean life in days and standard deviation for two types of electric bulbs are given
below:
Mean life in days s.d. in days
Type I : 310 9
Type II : 260 14
Compare the relative variability of life of the type of bulbs.
4.
Let ( x, y ) and ( u, v ) represent two sets of bivariate data such that u = ax + b and
ac
v = cy + d then prove that ruv = rxy , where a, b, c, d are constants.
a c
5.
Prove that −1  rxy  1 where rxy is correlation coefficient of x and y.
6.
If the equations of two regression lines obtained in a correlation analysis are
3x + 12 y = 19 and 3x + 9 y = 46 ,
(a) Determine which one of there is regression equations of x on y.
(b) Find means, correlation coefficient.
(c) Ratio of standard deviation of x and y.
7.
The marks obtained by nine students in Mathematics and Statistics in an examination
are as follow:
Students A B C D E F G H I
Mathematics 70 72 80 45 60 35 50 94 55
(X)
Statistics 60 83 72 63 74 54 40 85 58
(Y)
Find the correlation coefficient between the marks in Mathematics and statistics.
8.
If S 2 be the sample variance of a sample of size n drawn from a population with mean
 and s.d.  then prove that E ( S 2 ) =
n
 2 (where population size is infinite or the
n −1
sample is drawn with replacement).
9.
A normal population has a mean 0.1 and s.d. 2.1. Find the probability that the mean of
a sample of size 900 will be negative. Given that P ( z  1.43) = 0.847 .
10.
For finding the square root of ‘ a ’( a  0 ), derive the Newton Raphson iteration
1 a
formula xn +1 = ( x n + ), n = 0,1, 2,3,....
2 xn

Hence find 2 , correct to five significant figures.

11.
Solve the system of equations by Gauss Seidel method:
10 x − 5 y − 2 z = 3
4 x − 10 y + 3 z = −3
x + 6 y + 10 z = −3

Correct to four significant figures.

12.
Solve the system of equations by LU factorization method:

8 x1 − 3x2 + 2 x3 = 20
4 x1 + 11x2 − x3 = 33
6 x1 + 3x2 + 12 x3 = 36
Correct to three significant figures.
13. Find the solution of the System of linear equations
3x – y + z = -1
-x + 3y – z = 7
x – y + 3z = -7
by using Successive Over Relaxation (SOR) method, with value w = 1.25
( Relaxation parameter)

14.
Find the solution of the System of linear equations Ax=b where
 4 −2 1  2
   
A =  −1 5 −2  , b =  4 
 1 1 3  6
  

by using Successive Over Relaxation (SOR) method, with value w = 0.5

= (1 2 1) .
(0 ) T
( Relaxation parameter) and initial point x

15.
A random sample of 100 articles taken from a batch of 2000 articles with s.d 0.048
shows that the average diameter of the articles is 0.354. Find 95% confidence interval
for the average diameter of this batch of 2000 articles. [ Given area under the normal
curve between 𝑧 = 0 𝑎𝑛𝑑 𝑧 = 1.96 𝑖𝑠 0.475]
16.
Compute one root of log e x = cos x , correct to two decimal places by Regula Falsi
method.
17.
Solve the system of equations by LU factorization method:
2 x1 − 3x2 + 4 x3 = 8
x1 + x2 + 4 x3 = 15
3x1 + 4 x2 − x3 = 8

18.
Calculate seven iterations of the power method with scaling to approximate a dominant
eigenvector and eigen value of the matrix.
 1 2 0
A =  −2 1 2  ; use x0 = 1 1 1 as the initial approximation (up to two decimal
 1 3 1 
places)