IT MAth Questions

5/21/2018 IT MAth Questions

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Computer Oriented Numerical and Statistical Methods 060010306

Mr. Divyang Gor & Mr. Nikhil Choksi 1 | P a g e

Unit-1

Introduction and solution of algebraic and Transcendental Equation

Short Answer Questions:

1. Define the following terms:-a) Round-off errorb) Absolute errorc) Relative errord) Significant digitse) Root of an equationf) Rate of convergenceg) Algebraic Equationh) Transcendental Equation

2. State Rolles theorem.3. State Intermediate value theorem.4. State Mean-value theorem for derivatives.5. State Taylors series for a function of one variables.6. State Maclaurins expansion.7. State Taylors series for a function of two variables.8. State Taylors series for a function of several variables.9. Round off the following number up to four significant figures:

a) 1.6583b) 30.0567c) 0.859378d) 3.14159

10.An approximate value of is given by= = 3.1428571 and its true value is X = 3.1415926. Find theabsolute and relative errors.

11. Evaluate the sum S = 3 + 5 + 7to 4 significant digits and find its absolute and relative errors.12. Round-off the following numbers to two decimal places:

a)48.21416

b)2.3742

c)52.275

d)2.375

e)2.385

f)81.255

13. Calculate the value of 102 101correct to four significant figures.14. Explain the rule to round off a number into n significant digits.

15. Write down the formula of Bisection Method.

16. Write down the formula of Newton-Raphson Method.

17. When Newton-Raphson method fails to find root of an algebraic equation?

18. Define error also discuss different types of errors.

19. Explain the use of Lagranges interpolation formula.20. Obtain the Lagranges interpolating polynomial for the observed data of points (1,1),(2,1) and (3,-2).

Long Answer Questions:

1. Use Bisection Method to find the smallest positive root, correct up to four significant digits, of theequation : 10 = 0.

2. Use Bisection Method to find the smallest positive root, correct up to three significant digits, of theequation : x= 1.


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3. Use Bisection Method to find the smallest positive root, correct up to three decimal places, of theequation : + + + 7 = 0.

4. Use Bisection Method to find the smallest positive root, correct up to three decimal places, of theequation : 18 = 0.

5. Use Bisection Method to find the smallest positive root, correct up to three decimal places, of theequation : 1 = 0.

6.

Use Bisection Method to find the smallest positive root, correct up to three decimal places, of theequation : 1 = 0.7. Use Bisection Method to find the smallest positive root, correct up to three decimal places, of the

equation : 5 + 3 = 0. 8. Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation :

3 = 1.9. Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation :

3 9+ 8 = 0.10.Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation

:sin = 1 .11.Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation :

5 + 3 = 0. 12.Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation :

+

80 = 013.Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation : cos = 0.

14.Use Newton-Raphson Method to find the a root, correct up to three decimal places, of the equation :x + log = 2.

15.Find the real root of the equation x3+ x 2= 1 by iteration method.16.Use Bisection Method to find the smallest positive root, correct up to f ive significant digits, of the

equation : sin 1.17.Use Bisection Method to find the smallest positive root, correct up to f ive significant digits, of the

equation : 3 1 + sin .18.Use Newton-Raphsonmethod to find the smallest positive root, correct up to five significant digits, of the

equation : sin 1.19.Use Newton-Raphson method to find the smallest positive root, correct up to five significant digits, of the

equation : 3 1 + sin .20.Use Newton-Raphson method to find the smallest positive root, correct up to five significant digits, of theequation : 3 1 +tan .

Multiple choice question:

1. The number 0.032040 has ____ significant figures.(a) 3 (b) 4 (c) 5 (d) 6

2. How many significant figures are there in the number 0.02040?(a) 3 (b) 4 (c) 5 (d) 6

3. Round the following measurement to three significant figures: 0.90985 cm2(a) 0.91 cm

2(b) 0.9099 cm

2(c) 0.910 cm

2(d) 0.909 cm

2

4. How many significant figures are in the measurement 1.3000 meters?(a) 4 (b) 3 (c) 5 (d) 2

5. How many significant figures are in the measurement 0.00130 cm?(a) 5 (b) 3 (c) 2 (d) 4

6. How many significant figures are in the measurement 102.400 meters?(a) 5 (b) 3 (c) 4 (d) 6

7. The measurement, 20600 molecules, has how many significant digits?(a) 2 (b) 3 (c) 4 (d) 5

8. The measurement, 2.060 x 10 -3 coulombs, has how many significant digits?


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(a) 2 (b) 4 (c) 5 (d) 6

9. The measurement, 0.0020600 mole, has how many significant digits?(a) 5 (b) 6 (c) 7 (d) 8

10. How many solutions are there of the equation sin(2x5)2 =(3 4)in R ?(a) none (b) 1 (c) 2 (d) infinitely many

11. How many solutions are there to the equation ln( + 1)= - 2x - 3 in R?(a) none (b) 1 (c) 2 (d) infinitely many12. How many solutions are there to the equation + x + 1 = cosin R?(a) none (b) 1 (c) 2 (d) infinitely many

13. The next iterative value of the root of -4 = 0 using the Newton-Raphson method, if the initial guess is 3,(a) 1.5 (b) 2.067 (c) 2.167 (d) 3.000

14. The order of convergence of Newton Raphson method is

(a) 2 (b) 3 (c) 0 (d) none

15. The Newton-Raphson method fails when

(a) f'(x) is negative (b) f'(x) is too large (c) f'(x) is zero (d) never fails

16. Using the successive Bisection method the approximate value of a root of the equation - x- 4 = 0 lyingbetween 1 and 2 at the end of the third iteration is

(a) 1.875 (b) 1.796 (c) 1.8125 (d) none of these

17. Root of the equation -9x + 1 = 0 lying between 2 and 3 using the successive bisection method is (after 2iteration)(a) 2.875 (b) 2.75 (c) 2.987 (d) none of these

18. The value of 12correct to 3 decimal places by Newton-Raphson method is given by(a) 3.464 (b) 3.462 (c) 3.467 (d) none of these

19. A root of equation + 3x +1 =0 by using Newton-Raphson method which is near to 0 is given by (useonly 2 iteration)

(a) 0.303 (b) 0.313 (c)0.301 (d) none of these

20. If the root of the equation + - 1 = 0 is near to 1.0 then by Newton-Raphson method the firstcalculated approximate value of this roots is

(a) 0.9 (b) 0.6 (c) 1.2 (d) 0.80

True or False:

1. Round-off the number 1.5055 correct to four significant figures is 1.506.


3. Round-off the number 0.012348 correct to four decimal figures is 0.0123.

4. Round-off the number 0.00024145 correct to two decimal figures is 0.00.


6. Absolute error for 5/6 approximating 0.8333 is 0.0001.

7. Percentage error for 5/6 approximating 0.8333 is 0.2%.

8. Relative error in the computation of x-y for x = 12.05 and y = 8.02 having absolute error = 0.005 and = 0.001is 0.003.

9. Relative error in the computation of x-y for x = 1.135 and y = 1.075 having absolute error = 0.011and = 0.12is 0.38.

10. If f(x) is continuous in a x band if f(a) and f(b) are of same sign, than f(c) = 0 for at least one number c

such that a < c < b.11. To round off number n-significant digit if nth digit less than half a unit than increase nth digit by unity.

12. To round off number n-significant digit if nth digit greater than half a unit than decrease nth digit by unity.

13. To round off number n-significant digit if nth digit exactly half a unit than increase nth digit by unity if it is

odd.

14. To round off number n-significant digit if nth digit exactly half a unit than increase nth digit by unity if it is

even.

15. If f(a) and f(xr) < 0, the root lies in interval (xr, b).

16. If f(a) and f(xr)> 0, the root lies in interval (xr, a).

17. If f(a) and f(xr) = 0, the root lies in interval (xr, b).

18. Newton rapshon process has a third-order convergence.

19. Newton rapshon process has quadratic convergence.


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20. The order of convergence is one of the primary ways to estimate the actual rate of convergence.

21. The order simpally estimates the rate in terms of polynomial behavior.

Fill In The Blanks:

1. Round-off the number 48.2146 four significant place is __________.2. Round-off the number 0.0005549 four significant place is __________.

3. Round-off the number 53.1487 four significant place is __________.

4. Round-off the number 48.2146 two decimal place is __________.

5. Round-off the number 1.657898 five signif icant place is __________.

6. Using the successive bisection method the approximation value of a root of the equation 4 = 0lying between 1 and 2 at the end of third iteration is ________.

7. Root of the equation 9 + 1 = 0lying between 2 and 3 using the successive bisection method(after 2iterations) is ____________.

8. The newton rapshon method for finding the root of an equation f(x) = 0, is ________________.

9. The value of 12correct to 3 decimal place by newton rapshon method is given by ________.10. If 9 + 1 = 0root lies between ____ and ____.11. If

10 + 5 = 0root lies between ____ and ____.

12. The notation that function and sequence approach a limit under certain condition is ___________.

13. Newton rapshon process has a __________ order convergence.

14. For bisection method, if __________ the root lies in interval (a, xr).

15. For bisection method, if __________ the root lies in interval (xr, b).

16. For bisection method, if __________ xrit self is the root of equation f(x) = 0.

17. If f(x) is continuous in a x b, f(x) exists in a < x < b and f(a) = f(b) = 0. then, there exists at least one

value of x, say c, such that ________, a < x < b.

18. To round off number n-significant digit if nth digit less than half a unit than _________ nth digit by unity.

19. To round off number n-significant digit if nth digit greater than half a unit than _______ nth digit by unity.

20. To round off number n-significant digit if nth digit exactly half a unit than _______ nth digit by unity if it is

odd.

21.To round off number n-significant digit if nth digit exactly half a unit than _______ nth digit by unity if it is

even.


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Unit-2

Interpolation


1. Define symbolic operators E and and find a relation between them.2. Define symbolic operators and and find a relation between them.3. Prove the following operators relations

a) (1 + )(1 )= 1b) = c) = d) + =

4. What is the degree of the first order forward difference of a polynomial of degree n?

5. What is the degree of the nth order forward difference of a polynomial of degree n?

6. Construct the difference table for()= ( + 2)for x = 1, 2, 3, 4 and find (3).7. Find the value of (5)for the following table of f(x) given below

X 0 5 10 15

F(x) 1.0 1.6 3.8 6.3

8. Compute(0.2)and(0.6)from the following table.

X 0.2 0.4 0.6

F(x) 1.6596 1.6698 1.6804

9. Compute(1.0)and(1.6)from the following table.

X 1.0 1.2 1.4 1.6

F(x) 2.7183 3.3201 4.0552 4.9530

10. Construct a difference table for ()= + 2 + 1for x = 0, 1, 2, 3, 4, 5 and comment on the differenceof third order.11. Construct a linear interpolation formula for f(x) with given values f(1) = 3 and f(2) = -5.

12. Determine the polynomial which interpolates the following data: y(0) = 1, y(1) = -1, y(3) = 1, y(4) = 5,

comment on the degree of the polynomial.

13. Evaluate ()14. Given = 1, = 11find .15. Determine the polynomial representation of f(x), given f(0) = 8, f(1) = 12, f(1) = 19, f(3) = 29

16. Find the linear interpolation for the function f(x), if f(3) = 5 and f(5) = 3.

17. Determine the Newton divided difference interpolation for the function for the function for whichf(2) = 5,

f(5) = 29, f(10) = 109.

18. Certain corresponding value of x and log10 x are (300, 2.4771), (304, 2.4829), (305, 2.4843) and (307,

2.4871). Find log10301.19. If y1 = 4, y3= 12, y4= 19 and yx= 7 find x.

20. The function y = sin x is tabulated below, Using Lagranges interpolation formula, find the value of sin ().

X Y = sin x

0 04

0.70711

2

1.0


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21. State Lagranges Interpolation formula.

22. State Newtons Forward difference interpolation formula.

23. State Newtons Backward difference interpolation formula.


1. Explain forward differences.

2. Explain backward differences.

3. Given the table values of = (), from the diagonal difference table find the values of (8), (7),(6).X 6 7 8 9 10

F(x) 8 12 21 52 85

4. Given the table values of = (), from the diagonal difference table find the values of (5), (8),(7).X 6 7 8 9 10

F(x) 8 12 21 52 85

5. Explain Central differences.6. Derive newtons formula for forward differences.

7. Derive newtons formula for backward differences.

8. Using Newtons forward difference formula, find the sum = 1+ 2+ 3+ + .9. Find the missing term in the following table:

x Y

0

1

2

3

4

1

3

9

-

81

Explain why result differs from 3= 27?10. Show that 1 + and (1 ). Also deduce that 1+ ( 1).11. Value of x (in degree) and sin x are given in the following table:

x Y

35

40

45

50

55

60

0.573576

0.642788

0.707107

0.766044

0.819152

0.866025

12. The table below gives the values of tan x for 0.10 0.30:x Y

0.10

0.15

0.20

0.25

0.30

0.1003

0.1511

0.2027

0.2553

0.3093

Find tan (0.40) and tan (0.50).


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13. Use lagranges interpolation formula, find the form of the function y(x) from the following table:

X Y

0

1

3

4

5

-10

-3

0

15

2814. Usinglagranges formula, express the rational function,

5 + 2 + 2( 1)( 2)( 3)

as a sum of partial fractions.

[Hint: Let ()= 5+ 2 + 2. From a table of values of () for x = 1, 2, 3. Obtain the seond orderLagrange polynomial ()]

15. Given the data points (1, -3), (3, 9), (4, 30) and (6, 132), satisfying the function y = f(x), compute f(5) using

Lagrange polynomials of order 1 to 3.

16. Usinglagranges formula, express the rational function,

3+ + 1( 1)( 2)( 3)

as a sum of partial fractions.[Hint: Let ()= 3+ + 1. From a table of values of () for x = 1, 2, 3. Obtain the seond orderLagrangepolynomial ()].

17. Establish Newtons divided-differences formula and give an estimate of the reminder term in terms of the

appropriate derivative.

Deduce newtons forward and backward interpolation formulae as particular cases.

18. Determine the interpolating polynomial of degree three for the table below.

X F(x)

-1

0

1

2

1

1

1

-3

19. Evaluate the values of f(2) and f(6.3) using Lagrangian interpolation formula for the table of value givenbelow.

X F(x)

1.2

2.5

4

5.1

6

6.5

6.84

14.25

27

39.21

51

58.25

20. Compute f(0.23) and f(0.29) by using suitable interpolation formula with the table of data given below.

X F(x)

0.20

0.22

0.24

0.26

0.28

0.30

1.6596

1.6698

1.6804

1.6912

1.7024

1.7139

21. What is interpolation?

22. Discuss interpolation and the lagrange polynomial.


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Multiple Choice Questions

1. What is the polynomial that can be fitted into these data by newtons forward difference formula?

a)

2

7 3 b)

3

5 + 2 c)

7

4 4 d) 3

2

5 3e) 5 2 3 + 52. What is the approximate value of y for x = 3.5 given by newtons interpolation?

a) -8 b) 45 c) 53 d) 40 e) 32

3. Which of the following statement/s is/are true?

1. The quality of newtons interpolation increases with the degree of the polynomial used.

2. The quality of newtons interpolation decreases with the degree of the polynomial used.

3. In newtons interpolation, the degree of the polynomial is independent of the number of given data

points.

a) 1 only b) 2 only c) 3 only d) 1 and 3 only e) none

4. An important application of finite differences is in______, especially in numerical differential equations,

which aim at the numerical solution of ordinary and partial differential equations respectively.

a) Numerical analysis b) calculus c) Mathematical analysis d) optimization e) none.5. Occurs frequently in the calculus of finite differences, where it plays a fole formally similar to that of the

_____, but used in discrete circumstances.

a) Derivative b) Integral c) Differential d) Differential Calculus e) none

6. If a finite difference is divided by b-a, one gets a ______.

a) Derivative b) Integral c) Differential d) Differential Calculus e) none

7. If we retain r+1 terms in Newtons forward difference formula we obtain a polynomial of degree ____

agreeing with at , , , . , a) r+2 b) r+1 c) R d) R-1 e) none

8. P in Newtons forward difference formula is defined as

a) = b) =

c) =

d) =

e) none9. Newtons divided difference interpolation formula is used when the values of the independent variable are

a) Equally spaced b) not equally spaced c) constant d) variable e) none10. Below are all the finite difference methods EXCEPT _________.

a) Jacobis method

b) Newtons backward difference method

c) Stirlling formula

d) Forward difference method

e) None


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Unit-3

Least Squares Approximation


1. The table below gives the temperature T (in C) and lengths l (in mm) of a heated rod. If = + , findthe best value for and .

T L

40 800.6

50 800.8

60 801.0

70 801.1

2. Certain experimental values of x and y are given below

x Y

0 -1

2 55 12

7 20

3. If = + find approximate values of and .4. Find the least square line for the data points ( -1, 10),(0, 9),(1 , 7),(2, 5),(3, 4),( 4, 3),( 5, 0) and (6, -1) .

5. Find the least square power function of the form y = axbfor the data.

xi 1 2 3 4

yi 3 12 21 35

Solve following Examples:

1. Find the second degree curve y = a +bx + cwhich will fit the following data.

x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

y 3.1950 3.2299 2.2532 3.2611 3.2516 3.2282 9.1807 3.1266 3.0594 2.9759

2. If the curve f(x)= + x+ fits the following data determine , and degree parabola for the followingdata using the method of least square error.

X 0 1 2 3 4

f(x) 3.07 12.85 31.47 37.38 91.29

3. Fit a second degree parabola for the following data using the method of least square error.

X 0 1 2 3 4

f(x) 0 1.8 1.3 2.5 6.3

4. The table below gives the temperatures T (in C) and lengths l (in mm) of a heated rod. If l = + , find thebest values for .

T (in C) 20 30 40 50 60 70


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l (in mm) 800.3 800.4 800.6 800.7 800.9 801.0

5. Certain experimental values of x and y are given below. If y = + , find approximate value of and.

x 0 2 5 7

y -1 5 12 20

6. Fit a straight line of the form y = + to the data.

x 1 2 3 4 6 8

y 2.4 3.1 3.5 4.2 5.0 6.0

7. Find the values of and so that y = + , fit the data given the table:

x 0 1 2 3 4

y 1.0 2.9 4.8 6.7 8.6

8. Use the method of least squares to fit the straight line y = a + bx to the data:

X 0 1 2 3

y 2 5 8 11

w 1 1 1 1

9. Find the values of a, b and c so that y = a + bx + cis the best fit to the data:

x 0 1 2 3 4

Y 1 0 3 10 21


x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

y 3.2980 3.3399 3.2532 4.2611 3.2516 3.2282 9.1807 5.1266 6.0594

11. If the curve f(x)= + x+ fits the following data determine , and degree parabola for thefollowing data using the method of least square error.

X 5 6 7 8 9

f(x) 3.07 12.85 31.47 37.38 91.29


X 5 6 7 8 9

f(x) 0 1.8 1.3 2.5 6.3


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13. The table below gives the temperatures T (in C) and lengths l (in mm) of a heated rod. If l = + ,find the best values for .

T (in C) 40 50 60 70 80 90

l (in mm) 800.3 800.4 800.6 800.7 800.9 801.0


x 2 4 7 9

y -1 5 12 20


x 4 5 6 7 8 9

y 2.4 3.1 3.5 4.2 5.0 6.0





X 0 1 2 3 4

f(x) 0 1.8 1.3 2.5 6.3

4. The table below gives the temperatures T (in C) and lengths l (in mm) of a heated rod. If l = + , findthe best values for .

x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

y 3.1950 3.2299 2.2532 3.2611 3.2516 3.2282 9.1807 3.1266 3.0594 2.9759

X 0 1 2 3 4

f(x) 3.07 12.85 31.47 37.38 91.29

T (in C) 20 30 40 50 60 70

l (in mm) 800.3 800.4 800.6 800.7 800.9 801.0


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5. Certain experimental values of x and y are given below. If y = + , find approximate value of and .

x 0 2 5 7

y -1 5 12 20


x 1 2 3 4 6 8

y 2.4 3.1 3.5 4.2 5.0 6.0

7. Find the values of and so that y = + , fit the data given the table:

x 0 1 2 3 4

y 1.0 2.9 4.8 6.7 8.6

8. Use the method of least squares to fit the straight line y = a + bx to the data:

x 0 1 2 3

y 2 5 8 11

w 1 1 1 1

9. Find the values of a, b and c so that y = a + bx + cis the best fit to the data:

x 0 1 2 3 4

y 1 0 3 10 21


x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

y 3.2980 3.3399 3.2532 4.2611 3.2516 3.2282 9.1807 5.1266 6.0594 7.9759


X 5 6 7 8 9

f(x) 3.07 12.85 31.47 37.38 91.29


X 5 6 7 8 9

f(x) 0 1.8 1.3 2.5 6.3

13. The table below gives the temperatures T (in C) and lengths l (in mm) of a heated rod. If l = + , findthe best values for .T (in C) 40 50 60 70 80 90

l (in mm) 800.3 800.4 800.6 800.7 800.9 801.0


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x 2 4 7 9

y -1 5 12 20


x 4 5 6 7 8 9

y 2.4 3.1 3.5 4.2 5.0 6.0

16. Find the straight line fitting the following data

4 6 8 10 12

13.72 12.90 12.01 11.14 10.31

17. Use method of least square approximation to fit a straight line to the following observed data.

60 61 62 63 64

40 40 48 52 55

18. Use method of least squares to fit a straight time for the following data points.

x -1 0 1 2 3 4 5 6

y 10 9 7 5 4 3 0 -1

19. Determine the straight line y = a + bx which fits the following data, by the method of least square error.

x -2 -1 0 1 2 3 4

y 1 2 3 3 4 5 6

20. Determine by the method of least squares the straight line y = a + bx, which will fit the following data.

x 2 4 6 8 10 12

y 7.32 8.24 9.20 10.19 11.01 12.05


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Unit-4

Introduction to Statistics


1. Define Elements related to statistical data with an example.

2. Define Variables related to statistical data with an example.

3. Define Observations related to statistical data with an example.

4. Define Nominal scale related to statistical data with an example.

5. Define Ordinal scale related to statistical data with an example.

6. Define Ratio scale related to statistical data with an example.

7. Define Elements related to statistical data with an example.

8. Define data.

9. Define data set.

10. Define Qualitative data.

11. Define Quantitative data.

12. Define cross-sectional data.

13. Define Time series data.

14. What is difference between Quantitative data and Qualitative data?

15. What is difference between cross-sectional data and Time series data?

16. Define Frequency.

17. Define Cumulative frequency.

18. Discuss Bar graph.

19. Discuss pie chart.

20. Discuss Dot plot.

21. Discuss Histogram.

22. Discuss Ogive.

23. What is secondary data? state its chief source.


1. Discuss briefly the possible applications of statistical methods in business pointing out the limitaions if any.

2. What are the limitations of statistics?

3. Define primary data. What are the methods of collecting primary data?4. The following are the weights in Kg. of a group of 55 students.

42 53 68 66 72 74 110 69 49 50

40 76 104 77 79 60 84 80 90 52

82 50 79 84 103 115 67 79 76 96

41 78 54 42 51 61 77 73 64 86

75 63 59 69 78 83 65 81 70 94

63 95 100 80 71

prepare a frequency table taking the magnitude of class interval as 10 kg. and the first class interval equal to

40 and less than 50.

5. Compute the range of the following frequency distribution


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Marks out of 10: 0 1 2 3 4 5 6 7 8 9

No. of Students: 2 7 10 11 22 28 10 5 3 1

6. Find the value of range of frequency distribution

Age in year : 14 15 16 17 18 19 20

No of students : 1 2 2 2 6 4 0

7. Find the range of the following distribution

Class interval :10-20 20-30 30-40 40-50 50-60

frequency :8 10 15 18 19

8. Find the missing frequency distribution shown that the mean of the distribution is 1.46.

X: 0 1 2 3 4 5 total

Y: 46 ? ? 25 10 5 200

9. Find the unknown entries (a, b, c, d, e, f and g) from the following frequency distribution of hights 50

students in a class.

Class intervals Frequency Cumulative

frequency

150-155 12 A

155-160 B 25

160-165 10 C

165-170 D 43

170-175 E 48

175-180 Z f

Multiple Choice Questions:

1. Skewness is measure of

(a)shape of data distribution (b)relative variability (c) variability (d) none

2. The relation between variance and standard deviation is

(a) variance is square of standard deviation (b) variance is square root of standard deviation

(c) variance is positive root of standard deviation (d) none

3. Which of the following is graphical methods for Qualitative data?

(a) Bar graph (b) Dot plot (c) Histrogram (d) none

4. Graphical representation of cumulative distribution is known as,

(a) Pie chart (b) Ogive (c) Scatter diagram (d) none

5. Cross-sectional data are data:

(a) collected over time (b) that are collected over a single period of time

(c) consisting of a single variable (d) consisting of two or more variables

6. Ordinal data are:

(a) sorted into categories according to specified characteristics

(b) ordered or ranked according to some relationship to one another

(c) ordered, have a specified measure of a distance between observations but have no nature zero(d)

data which have a natural zero

7. The statistical measures that characterize dispersion are:

(a) Range, variance, standard deviation

(b) Mean, median, mode

(c) Range, mean, standard deviation

(d) Correlation, frequencies, standard deviation.

8. An advantage that a histogram has over a pie chart is that:

(a) When two categories are close in size, the difference in the size of the categories is easier to see

with the histogram.

(b) The histogram is more attractive

(c) The histogram is easier to construct than the pie chart


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(d) The histogram is usually used more often to depict budgets

9. Which of the following graphical depictions of data represents cumulative frequencies?

(a) histogram

(b)frequency polygon

(c) ogive

(d)pie chart

10. Which of the following is most useful when the decision maker wants to see running totals?(a)histogram

(b)ogive

(c)frequency polygon

(d)Pareto chart

11. In a frequency distribution, which of the following is a running total of frequencies through the

classes?

(a)relative frequency

(b)cumulative frequency

(c)class midpoint

(d)frequency

12. Suppose the table below contains the four largest companies in a given industry and each company's

respective annual sales (in $millions). If a pie chart were constructed to depict the sales of thesecompanies, how many degrees would be in the "slice" of the pie for company C?

(a)90

(b)44

(c)56

(d)61

13. What is the relative frequency of the 10-under 15 class?

(a).307

(b).545

(c).182

(d).239

14. Which of the following is sometimes more difficult to interpret than other graphical depictions

because the relative sizes of angles are hard to judge?

(a)pie chart

(b)triangle chart

(c)histogram

(d)ogive

15. Which of the following is named after an Italian economist?

(a)Pareto chart

(b)Roman chart

(c)pizza pie chart

(d)Ogive graph

16. What is the relative frequency for the class 20-under 30?

(a).30

(b)12

(c).425

(d).40

17. In a frequency distribution, which of the following is sometimes referred to as the class mark?

(a)relative frequency

(b)cumulative frequency

(c)class midpoint

(d)frequency

18. The pie chart below shows the amount of money Derrick spent on various things.


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How much money did he spend on the bag?

(a)$ 37.50

(b)$ 50

(c)$ 75

(d)$ 125

19.The graph below shows the rate at which a candle burns. At the same rate, how long does a

similar candle, measuring 15 cm, burn?

(a) 2h

(b) 3h

(c)4h

(d)6h

20. The table shows the parking charges at a carpark.

How much must MrRamu pay for parking his car from 11 a.m. to 3.45 p.m. on the same day?

(a)$6.0

(b)$6.80

(c)$7.60

(d)$8.40


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Unit-5

Measure of central tendency and Measure of Dispersion


1. What is arithmetic mean?

2. What is meant by mode?

3. What is empirical formula linking mean, median and mode?

4. Why arithmetic mean is most affected by extreme observations?

5. When is mode useful over other average?

6. Calculate arithmetic mean for 10, 50, 80, 54, 68, 28.59, -29 and -56.

7. Define Dispersion and what state types of dispersion.

8. What is Range?

9. Find the range of coefficient of range from the following:

200, 210, 208, 160, 220, 150.

10. Define Inter-quartile Range.

11. What is mean deviation?12. What is Standard deviation?

13. What is variance?

14. Define Quartiles and Percentiles.

15. Following are the price of shares of a company from Monday to Saturday.find Coefficient of Range.

Day Price

1

2

3

4

5

6

26.84

24.25

27

39.21

31

38.25

16. The monthly income of 10 employees working in a firm is as follows:4487 4493 4502 4446 4475 4492 4572 4516 4468 4489

find the average monthly income.

17. Calculate average monthly income by the assumed method from data of example 16 taking deviation from

4460 as the arbitrary point.

18. The following are the figures of profits earned by 1400 companies during 2011-2012

Calculate average profit for all companies:

Profit No. of

companies

200-400

400-600

600-800

800-1000

1000-1200

1200-1400

1400-1600

500

300

280

120

100

80

20

19. Give the formula for combined standard deviation of two sets of data.

20. State the formula of standard deviation.

21. What are the various methods to measure variation?

22. Distinguish between mean deviation and standard deviation.


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1. What is the arithmetic mean of first n natural numbers, 1, 2, 3,..n?

2. What are the measures of central tendency? Why are they called measure of central tendency?

3. Give a brief description of central tendency. why is arithmetic mean so popular?

4. For the following frequency table calculate mean, median and mode:

Day Price

200-400

400-600

600-800

800-1000

1000-1200

1200-1400

1400-1600

1600-1800

1800-2000

6

9

11

14

20

15

10

8

7

5. Calculate arithmetic mean price per bag of 20 kg. coal purchased by an industry for the half year.Month Price Bag

Jan

Feb

Mat

Apr

May

June

42.05

51.25

50.00

52

44.25

54

25

30

40

52

10

45

6. Following distribution represents the number of minutes spent by a group of teens in going to movies. What

is median?

Day Price

0-99

100-199

200-299300-399

400-499

500-599

600 and more

27

32

6578

58

38

9

7. An investor buys Rs. 1200 worth of shares in a company each month. During the first five months he bought

the shares at a price of Rs. 10, 12, 15, 20, 24 per share. After 5 months what is the average price paid for

the shares by him?

8. An incomplete distribution is given below:

Variable: 10-20 20-30 30-40 40-50 50-60 60-70 70-80 total

Frequency: 12 30 ? 65 ? 25 18 229

9. Given below is the frequency distribution of the marks obtained by 90 students. Compute the arithmetic

mean, median and mode:Day Price

20-29

30-39

40-49

50-59

60-69

70-79

80-89

90-99

2

12

15

20

18

10

9

4


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10. Find the missing frequency in the following distribution if N is 100 and median is 30:

Marks 0-10 10-20 20-30 30-40 40-50 50-60

No. of students 10 15 ? 30 10 8

11. Calculate the median and quartiles for the following data:

Class interval Frequency

0-50

50-100100-150

150-200

200-250

250-300

20

6050

30

24

16

12. Calculate mean and median for the following data:

Price: 15 20 25 30 35 40 45

No. 3 25 19 16 4 5 6

13. Calculate the arithmetic mean and the median of the frequency distribution given below. Hence calculate

the mode using the empirical relation between the three:

Heights No. of students

130-134

135-139140-144

145-149

150-154

155-159

160-164

5

1528

24

17

10

1

14. Calculate the median and mode for the distribution of the weights of 150 students from the data given

below:

Weight: 30-40 40-50 50-60 60-70 70-80 80-90

Frequency: 18 37 45 27 15 8

15. In the examination of 675 candidates the examiner supplied the following information:

Marks No. of

Candidates

Less than 10%

Less than 20%

Less than 30%

Less than 40%

Less than 50%

Less than 60%

Less than 70%

Less than 80%

7

39

95

201

381

545

631

675

16. Calculate the mean deviation for the following frequency distribution:

No. of

colds

No of

persons

0

1

2

3

4

5

6

7

8

9

15

46

91

162

110

95

82

26

13

2

17. Calculate arithmetic mean and standard deviation for the following data.

Value 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0

No. 4 19 30 63 66 29 18 1


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18. A study of 241 authors revealed the following data on the distribution of age:

Age.(Years) No of

Authors

Up to 30

Up to 40

Up to 50

Up to 60Up to 70

Up to 80

20

73

80

4422

2

19. Calculate the standard deviation from the following data;

Temperature No of

Days

-40 to -30

-30 to -20

-20 to -10

-10 to 0

0 to 10

10 to 20

20 to 30

10

24

30

42

65

180

14

Multiple Choice Questions:

1. The arithmetic mean of a set of 10 numbers is 20. If each number is first multiplied by 2 and then

increased by 5, then what is the mean of new numbers?

a) 20 b) 25 c) 40 d) 45

2. The mean of 25 observations is 36. The mean of first 13 observations is 32 and that of last 13 observations is

39. What is the value of 13th

observation?

a) 20 b) 23 c) 40 d) 32

3. The average age of 06 persons living in a house is 23.5 years. Three of them are majors and their average

age is 42 years. The difference in ages of the three minor children is same. What is the mean of the ages

of minor children?

a) 3 years b) 4 years c) 5 years d) 6 years

4. What is the weighted mean of first 10 natural numbers whose weights are equal to the

corresponding number?

a) 7 b) 5.5 c) 5 d) 4.5

5. In a class of 45 students a boy is ranked 20th

. When two boys joined, his rank was dropped by one. What is

his new rank from end?

a) 25th

b) 26th

c) 27th

d) 28th

6. The mean age of combined group of men and women is 25 years. If the mean age of group of men is 26 and

that of group of women is 21, then percentage of men and women in the group respectively is:

a) 60, 40 b) 80, 20 c) 30, 70 d) 50, 50

7. Sum of mode and median of the data

12, 15, 11, 13, 18, 11, 13, 12, 13

a) 26 b) 31 c) 36 d) 25

8. The arithmetic mean (average) of the first ten whole numbers is

a) 5.5 b) 5 c) 4 d) 4.5

9. The mean of 9 observations is 16. One more observation is included and the new mean becomes 17. The

10th

observation is

a) 18 b) 26 c) 30 d) 7


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10. The following table shows the points scored by the sixth grade basketball team in their first seven games.

Calculate the mean, median, mode and range. which statistical measure makes the team should the best?

Basketball team point

Game Point

1 26

2 203 25

4 30

5 20

6 31

7 41

a)mean b) range c)mode d) median

11. The table shows elevations of some cities in Alabama.

City Altitude

Dothan 326 feet

Montgomery 160 feet

Mobile 220 feet

Huntsville 641 feetTuscaloosa 227 feet

what are the mean and the range of the data?

a)mean = 305; range = 461 b) mean = 325; range = 481

c)mean = 315; range = 481 d) mean = 325; range = 489

12. The chart below shows the number of As that Mrs. Kimuras studentsrecived on each of their six tests last

year. what is the mean number of As receives?

Test Number Point

1 6

2 9

3 4

4 9

5 96 8

a)7.5 b) 8 c)8.5 d) 9

13. The table below shows the number of car washes Emily and her friends gave four days last week.

Day No. of

cars

1 5

2 6

3 16

4 15

a)10 car washes b) 10.5 car washes c)11 car washes d) 11.5 car washes

14. Bozemans daily high temperature(F) for one week were 82, 87, 88, 90, 84, 84 and 80. for this data, setwhich of the following is 85?

a)mean b) range c)mode d) median

15. Which data set has a mean of 6?

a)1, 10, 6, 5, 9 b) 6, 6, 3, 10, 4 c)2, 2, 8, 4, 9 d) 4, 7, 7, 7, 5

16. A soccer team has 12 players. their age are 14, 15, 14, 14, 16, 15, 14, 15, 16, 16, 14, 15. What is the mode

of this data set?

a)12 b) 15 c)14 d) 16

17. Mode is:

a)Middle most value b) least frequent value c)most frequent value d) none

18. While computing mean of grouped data, we assume that the frequencies are :

(a) evenly distributed over all the classes (b) centred at the class-marks of the classes


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(c) centred at the upper limits of the classes (d) centred at the lower limits of the classes

19. If mode = 80 and mean = 110, then the median is

(a)110 (b)120 (c)100 (d)90

20. Which of the following is not a measure of central tendency ?

(a) Mean (b) Median (c) Range (d) Mode

Unit-6

Correlation Analysis


1. What are the properties of correlation coefficient?

2. What are the limitation of correlation analysis?

3. What is meant by correlation?

4. What are the limits of coefficient of correlation?

5. Write down the formula for measuring the degree of relationship by karlpearson.

6. Discuss positive correlation with example?

7. Discuss negative correlation with example?

8. Discuss no correlation with example?

9. For the observation {(1, 2), (2, 5), (3, 7), (4, 8), (5, 10)} calculate karlpearsons coefficient of correlation.

10. When a case consist of more than two variables and the correlation is measured between two variables

keeping the other variable as constant which type of correlation it is?

11. In a scatter diagram if the plotted points from a straight line running from the lower left to the upper right

corner then which type of correlation it is?

12. When variables are said to be correlated?

13. When are the two variables said to be in perfect correlation?

14. Define karl- Pearsons coefficient of correlation

15. Mention any two properties of karl Pearsons coefficient of correlation.

16. Can simple correlation coefficient measure any type of relationship?

17. What is the difference between liner and non-liner correlation?

18. Name various methods of studying correlation. Describe any one.

19. How coefficient of correlation interpreted?

20. Enumerate the various methods available for finding the correlation.

21. What are the advantages and limitations of karlpearsons correlation coefficient?


1. Define coefficient of correlation. what is it intended to measure? bring out its usefulness. How does the

coefficient differ from karl Pearsons coefficient of correlation?

2. Prove that the correlation coefficient is unaffected by the change of origin and scale.

3. Calculate karl Pearsons coefficient of correlation from the following data and interpret its value;

Roll No. : 1 2 3 4 5

Marks in Accountancy : 48 35 17 23 47

Marks in Statistics : 45 20 40 25 45


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4. Following figures give in the rainfall in inches for the year and the production in 00s kgs. for the Rabi crop

and Kharif Crop. Calculate Karl Pearsons coefficient of correlation between rain fall and total production:

Rain fall : 20 22 24 26 28 30 32

Rabi Crop : 15 18 20 32 40 39 40

Kharif Crop : 15 17 20 18 20 21 15

5. Find karl Pearsons coefficient of correlation from the following index number and interpret it:

Wages : 100 101 103 102 104 99 97 98 96 96Cost of living : 98 99 99 97 95 02 95 94 90 91

6. Find Karl Pearsons coefficient of correlation between capital employed and profit obtained from the

following data:

Employed : 10 20 30 40 50 60 70 80 90 100

Profits : 2 4 8 5 10 15 14 20 22 50

7. From the following data calculate coefficient of correlation

Year : 1997 1998 1999 2000 2001 2002 2003

% on securities : 5.0 5.1 5.2 4.9 4.8 5.3 5.4

Index No. : 140 138 126 132 140 135 132

8. Find correlation by Karl Pearsons method between the two kids of assessment of post guard students

Performance:

Roll No. : 1 2 3 4 5 6 7 8 9 10Internal : 45 62 67 32 12 38 48 67 42 85

External : 39 48 65 32 20 35 45 77 30 62

9. Calculate the Karl Pearsons Coefficient of Correlation between age and playing habits from the data given

below.

Age : 20 21 22 23 24 25

No. of students : 500 400 300 240 200 160

Regular Players : 400 300 180 96 60 24

10. Calculate Pearsons Coefficient of correlation from the following taking 100 and 50 as the assumed average

of X and y respectively:

X : 104 111 104 114 118 117 105 108 106 100 104 105

Y :57 55 47 45 45 50 64 63 66 62 69 61

11. State and discuss Properties of correlation coefficient.

12. Calculate Pearsons Coefficient of correlation from the data given below:

X : 39 65 62 90 82 75 25 98 36 78

Y :47 53 58 86 62 68 60 91 51 84

13. Calculate Pearsons Coefficient of correlation from the data given below:

N = 10, = 110, = 200, = 3000, = 1313, = 658014. Calculate Pearsons Coefficient of correlation from the data given below:

N = 10, = 140, = 150, ( 10)= 180, ( 10)= 215, ( 10)( 10)= 60.15. Calculate coefficient of correlation between X and Y from the following data:

X :38 41 30 42 47

Y :55 60 45 50 50

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