Part-A

1. Given n=12, s=0.0086, σ =0.010. Calculate χ 2 value.

2. Given n= 10, s=650, x=6000 , μ=5500 calculate t statistic.

3. When do you apply analysis of variance technique?

4. Define mean square.

5. Derive the Newton’s iterative formula to find √

3
N where N is a positive real number.

6. When Gauss – Elimination method fails?

7. Write the inverse interpolation formula for Lagrange’s method.

2
dy d y
8. Specify the Newton’s forward and backward difference formulae for ,
dx d x 2

dy 2
9. Using Eulers method find y(0.2),given =x +y, y(0)=1
dx

10. State the formula of Euler modified method and Runge Kutta method.

11. State Type I and Type II error

12. Mention various steps involved in testing hypothesis?

13. Define a treatment and a yield in an experimental design.

14. What is contrast and orthogonal contrast in a 22 factorial design?

15. Derive the Newton’s iterative formula to find √ N where N is a positive real number.

16. Compare Gauss –Jacobi and Gauss-Seidel methods.

17. Find the divided difference table for the following data:

x 2 5 10
y 5 29 109
1
18. Apply Trapezoidal method to evaluate I =∫ e dx ,taking h=0.2 .
2
x

19. Use Euler’s formula to find y ( 0.2 ) and y ( 0.4 ) given y ' =x + y ,

y ( 0 )=1.

20. What is the condition to apply Adams-Bash forth predictor corrector method?

Part-B
1. The mean life time of a sample of 100 light bulbs produced by a company is computed to be 1570
hours with a standard deviation of 120 hours. If µ is the mean life time of all the bulbs produced by the
company, test the hypothesis µ=1600 hours, against the alternative hypothesis µ≠ 1600 hours with
α =0.05 and 0.01?
2.Test made on the breaking strength of 10 pieces of metal gave the following results
578,572,570,568,572,570,570,572,596 and 584 kg. Test if the mean breaking strength of the wire can be
assumed as 577 kg.

3. A sample of heights of 6400 Englishmen has a mean of 67.85 inches and a S.D. of 2.56 inches, while a
sample of heights of 1600 Australians has a mean of 68.55 inches and a S.D. of 2.52 inches. Do the data
indicate that Australians are on the average taller than Englishmen?

4. The independent samples from normal populations with equal variance gave the following:

Sample Size Mean S.D

1 16 23.4 2.5
2 12 24.9 2.8
Is the difference between the means significant?

5. A group of 10 rats fed on diet A and another group of 8 rats fed on diet B, recorded the following
increase in weight.

Diet 5 6 8 1 12 4 3 9 6 10
A:
Diet 2 3 6 8 10 1 2 8 - -
B:
6.Using the data given in the following table to test at the 0.01 level of significance whether a person’s
ability in mathematics is independent of his/her interest in Statistics.

Ability in Mathematics

Low Average High

Low 63 42 15

Interest in Statistics Average 58 61 31

High 14 47 29

7. Two independent samples of sizes 8 and 7 contained the following values:

Sample I : 19 17 15 21 16 18 16 14

Sample II : 15 14 15 19 15 18 16

Is the difference between the sample means significance? Use 5 % level of significance.
8. The table shows the yield of paddy in arbitrary units obtained from four different varieties planted in
five blocks where each block is divided into four plots. Test at 5% level whether the yields vary
significantly with (i) soil differences (ii) differences in the type of paddy.

Blocks Types of paddy

I II III IV
A 12 15 10 14
B 15 19 12 11
C 14 18 15 12
D 11 16 12 16
E 16 17 11 14

9. The table shows the seeds of four different types of corns planted in three blocks. Test at 5% level
whether the yields in kilograms per unit area vary significantly with different types of corns.
Blocks Types of corns
I II III IV
A 4.5 6.4 7.2 6.7
B 8.8 7.8 9.6 7.0
C 5.9 6.8 5.7 5.2
10. The following table provides the data collected for a yield for five different cultivation in a Latin
square experiment. Perform analysis of variance.

A 48 E 66 D 56 C 52 B 61
D 64 B 62 A 50 E 64 C 63
B 69 A 53 C 60 D 61 E 67
C 57 D 58 E 67 B 65 A 55
E 67 C 57 B 66 A 60 D 57

11. The following is the Latin square layout of a design, when 4 varieties of seeds are being tested. Set up
the analysis of variance table and state your conclusion.

A 18 C 21 D 25 B 11
D 22 B 12 A 15 C 19
B 15 A 20 C 23 D 24
C 22 D 21 B 10 A 17
12. Obtain Newton-Raphson formula for finding √ N, where N is a positive real number and hence
evaluate √ 142

13. Solve the following system of equations using Gauss-Jacobi method:

4 x+ y+ z =6 , x+ 4 y + z=6 , x + y +4 z=6

14. Find by Newton’s method, the real root of the equation

3 x=cosx+1 , correct to 4 decimal places.

15. Solve the following system of equations by Gauss-Seidel method

27 x +6 y−z=85 , x + y +54 z=110 , 6 x+15 y +2 z=72
 16. Find the numerically largest Eigen value and the corresponding Eigen vector of a matrix A =

( )
1 3 −1
3 2 4 with the initial vector ( 1 1 1 )T .
−1 4 10

17.Solve by using Gauss-Seidal method, the equations

8 x− y + z=18 , 2 x+5 y −2 z=3 , x + y−3 z=−6.

18. Find the numerically largest Eigen value and the corresponding Eigen vector of a matrix A =

( )
25 1 2
1 3 0
2 0 −4

19.Solve by using Gauss-Jordan method, the equations

2 x+ y+ 4 z=12 , 8 x−3 y +2 z =20 , 4 x +11 y−z=33.
20. Find the polynomial f(x) by using Lagrange’s formula and hence find f(3) .
x 0 1 2 5
f(x) 2 3 12 147
21. Using Newton’s divided difference formula, find the value of f(8) from the given table:

x : 4 5 7 10 11 13
f(x) : 48 100 294 900 1210 2028
22. Using Newton’s divided difference, find u(3) given
u(1) = -26, u(2) = 12, u(4) = 256, u (6) = 844.

23. Find f ' ( 3 ) ∧f ' ' (3) for the following data:

x 3.0 3.2 3.4 3.6 3.8 4.0

f(x) -14 -10.032 -5.296 -0.256 6.672 14

24. Find the first , second and third derivatives of f ( x ) at x =1.5 if

X 1.5 2.0 2.5 3.0 3.5 4.0

Y 3.375 7.000 13.625 24.000 38.875 59.000

6
dx
∫ 2
25.Evaluate 0 1+x using Trapezoidal rule
and Simpson’s 1/3 rule.
26. Using Lagrange’s interpolation , calculate the profit in the year 2000 from the following data:
 Year 1997 1999 2001 2002

Profit in lakhs of Rs. 43 65 159 248

1.2 1.4
1
27. Evaluate ∫ ∫ dxdy with h=k =0.1 by using Trapezoidal rule
1 1 1+ x

dy 2
28. Consider the I.V.P = y−x +1 , y ( 0 ) =0.5
dx
i) Using modified Euler method find y(0.2)
ii) Using 4th order RK method find y(0.4) and y(0.6)
Using Adams- Bashforth predictor-corrector method find y(0.8)

dy 2 2
29. Using modified Euler method, find y ( 0.1 )given =x + y ; y ( 0 )=1
dx

30. Solve y ' = x+y ; y(0) = 1 by Taylor’s series method.Find the values of y at x = 0.1 and x=0.2.

31. Using Milne’s method find y(2) if y(x) is the solution of

¿ dy 1
= (x+y) given y(0) =2 , y(0.5) = 2.636, y(1) = 3.595 and y(1.5) = 4.968.
dx 2
2 2
dy y −x
32.Using R.K method of fourth order, solve = 2 2 , with y(0) =1 at x =0.2.
dx y +x

33. Given y ' =1− y∧ y ( 0 )=0 , find

i) y(0.1) by Euler’s method
ii) y(0.2) by Modified Euler’s method
iii) y(0.3) by Taylor’s method
iv) y(0.4) by Milne’s method