Anna University, Chennai

**SRINIVASAN ENGINEERING COLLEGE, PERAMBALUR**

**MA1254 -STATISTICS AND NUMERICAL METHODS**

**QUESTION BANK**

**PART – B**

**UNIT I**

1.Car stereo manufacturer of A have mean life time of 1400 hrs with S.D of 200 hrs while those

of manufacturer B have mean lifetime of 1200 hrs with S.D of 100 hrs. If a random

sample of 120 stereos of each manufacturer are tested. i) What is the probability that the

manufacturer of A’s stereo’s will have a mean life time of at least 160 hrs more than the

manufacturer B’s stereo’s ii) and 250 hrs more than the manufacturer B stereo’s.

2. In a sample of 1000 people in Mumbai, 540 are rice eaters and the rest are wheat eaters. Can

we assume that both rice and wheat are equally popular in this state at 1% LOS?

3. In a test given of two groups of students the marks obtained were as follows.

I Group: 18 20 36 50 49 36 34 49 41

II Group: 29 28 26 35 30 44 46

Examine the significant difference between the means of marks secured by students of the above 2 groups.

4. The weight gains in pounds under two systems of feeding of calves of 10 pairs of identical

twins is given below.

Twin pair: 1 2 3 4 5 6 7 8 9 10

System A: 43 39 39 42 46 43 38 44 51 43

System B: 37 35 34 41 39 37 37 40 48 36.

Discuss whether the difference between the 2 systems of feeding is significant.

5.Two random samples gave the following results;

Sample | Size | Sample mean | Sum of squares of deviations from the mean |

1 | 10 | 15 | 90 |

2 | 12 | 14 | 108 |

Test whether the samples come from the same normal population at 5% LOS.

6. 4 coins were tossed 160 times and the following results were obtained.

No of heads: 0 1 2 3 4

Observed Fr:17 52 54 31 6

Under the assumption that the coins are balanced, find the expected frequencies of getting

0, 1,2, 3,4 heads and test the goodness of fit.

7.A sample of 200 persons with a particular disease was selected. Out of these , 100 were given a

drug and the others were not given any drug. The result are as follows:

No of persons | Drug | No Drug | Total |

Cured | 65 | 55 | 120 |

Not cured | 35 | 45 | 80 |

Total | 100 | 100 | 200 |

Test whether the drug is effective or not.

- The following data relate to a random sample of government employees in 2 states of the Indian Union.

Sample | Size | Mean monthly income of the employees | Variance |

1 | 16 | 440 | 40 |

2 | 25 | 460 | 42 |

Test whether the samples come from the same normal population

- A normal population has a mean of 6.48 and S.D of 1.5 . In a sample of 400 members mean is 6.75 .I s the difference significant?

- A simple sample of heights of 6400 English men has a mean of 170 cm and S. D of 6.4 cm, while a simple sample of heights of 1600 Americans has a mean of 172 cm and a S.D of 6.3 cm. Do the data indicate that Americans are, on the average , taller than the Englishmen?

- 1000 students at college level were graded according to their I.Q and their economic conditions. What conditions can U draw from the following data:

Economic conditions | I.Q level | |

High | Low | |

Rich | 460 | 140 |

Poor | 240 | 160 |

**UNIT – II**

1. The following are the number of mistakes made in 5 successive days by 4 technicians working for a photographic laboratory. Test whether the difference among the 4 sample means can be attributed to chance. LOS at 1%.

Technician

I | II | III | IV |

6 | 14 | 10 | 9 |

14 | 9 | 12 | 12 |

10 | 12 | 7 | 8 |

8 | 10 | 15 | 10 |

11 | 14 | 11 | 11 |

2. Set up the analysis of variance for the following results of a Latin square design at 1% LOS.

A 12 | C 19 | B 10 | D 8 |

C 18 | B 12 | D 6 | A 7 |

B 22 | D 10 | A 5 | C 21 |

D 12 | A 7 | C 27 | B 17 |

3. To study the performance of 3 detergents and 3 different water temperatures, the whiteness readings were obtained with specially designed equipment.

Detergent

Water temp | A | B | C |

Cold water | 57 | 55 | 67 |

Warn water | 49 | 52 | 68 |

Hot water | 54 | 46 | 58 |

Perform a 2 way analysis of variance using 5 % LOS.

4. The following data are obtained from a factorial experiment blocked 3 times. Evaluate the sum of squares for all factorial effect by the contrast method. Draw conclusion.

Blocks

Treatment combination | I | II | III |

(1) | 12 | 19 | 10 |

k | 15 | 20 | 16 |

p | 24 | 16 | 17 |

kp | 24 | 17 | 29 |

5.Perform two way ANOVA for the given below:

Treatments | ||||

Plots of land | A | B | C | D |

I | 38 | 40 | 41 | 39 |

II | 45 | 42 | 49 | 36 |

III | 40 | 38 | 42 | 42 |

6.Analyze the variance in LSD .

S122 | P121 | P123 | Q122 |

Q124 | R123 | P122 | S125 |

P120 | Q199 | S120 | R121 |

R122 | S123 | Q121 | P122 |

7. The following data are obtained from a factorial experiment blocked 3 times. Evaluate the sum of squares for all factorial effect by the contrast method. Draw conclusion.

Blocks

Treatment combination | I | II | III | IV |

(1) | 12 | 12.3 | 11.8 | 11.6 |

k | 12.8 | 12.6 | 13.7 | 14 |

p | 11.5 | 11.9 | 12.6 | 11.8 |

kp | 14.2 | 14.5 | 14.4 | 15 |

8. Find ANOVA using CRD.

Samples | ||

8 | 7 | 12 |

10 | 5 | 19 |

7 | 10 | 13 |

14 | 9 | 12 |

11 | 9 | 14 |

9. Perform two way ANOVA for the given below:

Water temp | Detergent A | Detergent B | Detergent C |

Cold | 57 | 55 | 67 |

Warm | 49 | 52 | 68 |

Hot | 54 | 46 | 58 |

10.Perform ANOVA for the given below using CBD method.

A6 | C5 | A8 | B9 |

C8 | A4 | B6 | C9 |

B7 | B6 | C10 | A6 |

Unit 3

1. Solve the system of equations by Gauss Elimination method. 10 x- 2y +3z =23;

2 x+10 y -5z = - 33; 3x -4y +10 z =41.

2. Find the iterative formula for finding the value of where N is a real number , using Newton –

Raphson method also find 1/26.

3.Solve the following system of equations by Gauss – Jacobi method

27 x +6y –z =85; x + y +54 z =110; 6x +15 y +2z =72.

4.Using power method , find the eigen value of A =

5. Solve the system of equations by Gauss Jordan method

3x+ y – z = 3; 2x – 8y + z = -5; x -2y + 9z = 8

6.Solve the following system of equations by Gauss – Seidel method

4x + 2y + z =14; x + 5y – z =10; x + y +8z =20.

7.Find the inverse of the matrix

8.Find the dominant of eigen value and eigen vector of A =

9.Compute the real root of x log x =1.2 using Newton’s Raphson method.

Unit 4

1.The following table gives the population of a town during the last 6 census. Estimate using Newton’s

interpolation formula, the increase in the population during the period 1946 to 1948.

Year | 1911 | 1921 | 1931 | 1941 | 1951 | 1961 |

Population(in 1000’s) | 12 | 13 | 20 | 27 | 39 | 52 |

2.By dividing the range into 10 equal parts, evaluate by using Simpson’s 1/3 rule.

3.Using the data given below find the value of by Trapezoidal rule.

X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Y | 2.061 | 2.312 | 2.891 | 3.106 | 3.670 | 4.721 | 6.103 | 7.950 | 9.942 |

4.A rod is rotating in a plane about one of its ends. If the following table gives the angle radians through which the rod has turned for different values of time t seconds. Find it angular velocity and angular acceleration when t= 0.7 seconds.

t seconds | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |

0.0 | 0.12 | 0.48 | 1.10 | 2.0 | 3.20 |

5.Using Lagrange’s formula to calculate f(3):

x | 0 | 1 | 2 | 4 | 5 | 6 |

F(x) | 1 | 14 | 15 | 5 | 6 | 19 |

6..Find the cubic function from the following table:

x | 0 | 1 | 3 | 4 |

F(x) | 1 | 4 | 40 | 85 |

7.From the following table find at x = 43.

x | 40 | 50 | 60 | 70 | 80 | 90 |

184 | 204 | 226 | 250 | 276 | 304 |

8.Find the value of x = 2.5

x | 0 | 1 | 2 | 3 | 4 |

y | 7 | 10 | 13 | 22 | 43 |

9.Find the polynomial using Newton’s forward method.

x | 4 | 6 | 8 | 10 |

y | 1 | 3 | 8 | 16 |

10.Find the value of y’’ at the point x =0.96

X | 0.96 | 0.98 | 1 | 1.02 | 1.04 |

F(x) | 0.7825 | 0.7739 | 0.7651 | 0.7563 | 0.7473 |

Unit 5

1.Evaluate the values of y(0.1) given y’’ –+ =0 , y(0) =1, y’(0) =0 by using Taylor series

method.

2. Using Euler’s method solve y’ = x + y + xy, y(0) =1 . Compute y at x =0.1 by taking h =0.05

3. Using R.K method of 4^{th} order solve with y(0) = 1 at x=0.2.

4. Consider the second order initial values problem y’’ -2y’ = sint with y(0)= -0.4 and

y’(0)= -0.6 using fourth order R.K method , find y(0.2)

5. Using Milne’s method find y(4.4) given 5xy’ + -2 = 0 given y(4) =1, y(4.1)=1.0049,

y(4.2) =1.0097 & y(4.3) = 1.0143

6. Solve the system equations with y(0) = 1, z (0) = 1 by taking h= 0.1 to get

y(0.1) and z(0.1) by Taylor series method.

7.Solve = 1- y , y(0) =0 by modified Euler’s method.

8.Using R.K method solve with y(0) = 1 at x= 0.2

Milne’s method to find y(0.8) and y(1).

10. Using improved Euler’s method find y(0.2) and y(0.4) from y’ = x +y , y(0) = 1 with h =0.2.