Anna University, Chennai

**SRINIVASAN ENGINEERING COLLEGE, PERAMBALUR**

**MA1254 -STATISTICS AND NUMERICAL METHODS**

**QUESTION BANK**

**PART –A**

**UNIT I**

- Define standard error.

The S.D of a sampling distribution of a statistic is know as its standard error.

- Define null hypothesis.

For applying the test of significance, we first set up of a hypothesis a define statement about the population parameter. Such a hypothesis is usually a hypothesis of no difference .

- What is type I , type II error?
- Type I error : Reject null hypothesis , when it is true.
- Type II error : Accept null hypothesis, when it is wrong.

i.e) accept null hypothesis, when alternative hypothesis is true.

4. Define test statistics.

Z= t – E(t) / s E(t).

- Write the application of “ F” test and chi- square test.

F test - To test if two samples have come from the same population.

Chi square test – To test the significance of discrepancy between experimental values and the theoretical values.

- Write two applications of Chi square test.

Chi square is used to test whether differences between observed and expected frequencies are significances.

- Define a F – variance.

F = Greater variance / smaller variance.

8. When we use the large sample and what is the test we use for that?

When n > 30 we use z test.

- What are the tests we use in small samples?

t- test, F test and Chi square test.

10. What is the degrees of freedom of Chi square test?

Degrees of freedom = (r- 1) ( s -1).

**UNIT – II**

- Write the basic design of experiments.

Randomised blocks, replication and plan of an experiment.

- Define ANOVA.

Analysis of variance (ANOVA) is a technique that willenable us to test for the significance of the difference among more than two sample means.

According to R.A. Fisher, “ The separation of variance ascribable to one group of causes from the variance ascribable to other group.

- What are the basic steps in analysis of variance.

i) One estimate of the population variance from the variance among the sample means.

ii) Determine a second estimate of the population variance from the variance within the sample.

iii) Compare these two estimates if they are approximately equal in value, accept the null hypothesis.

- What are the technique of ANOVA ?

i) one way classficition

ii) two way classification separately

iii) Randomized block design

iv) Latin square.

- What are the advantages of C.R.D?

i) Easy to lay out.

ii) Allows flexibility

iii) Simple statistical analysis.

iv) The lots of information due to missing data is smaller than with any other design.

- State the advantages of Latin square over other design.

i) Latin square controls more of the variation than the completely randomized block design with a two way stratification.

ii) The analysis is simple.

iii) Even with missing data the analysis remains relatively simple.

- Write the differences between RBD and LSD.

RBD

i) Available for wide there is diagonal trend range of treatments’

ii) no restriction on the number of replications.

iii) Flexible and easier to mange.

iv) Can be accommodated in a field of any shape.

LSD.

i) More efficient when there is diagonal ternd of fertility.

ii) Suitable only in the special cases where the land exhibits marked trends in fertility.

iii) Suitable mainly for 5 or 10 treatments.

iv) Shape of the field should be approximately square or rectanglar.

- Define replication.

To estimate the magnitude of an effect in an experiment the principle of randomization and replication are applied. Randomisation by itself is not necessarily sufficient to yield a valid experiment. The replication or repetition of experiment or experimental unit is also necessary. Randomisation must be invariably accompanied by sufficient replication so as to ensure validity in an experiment.

- What is the degrees of freedom of LSD.

DOF = (n-1)(n -2)

- How can we find the F ratio.

F = MSC/ MSE.

**UNIT – 3**

1. Compare Gauss – Jacobi and Gauss –Seidal methods for solving linear systems of the form

AX =B.

Gauss elimination is direct method.

Gauss Seidal method is iterative method.

2. For solving a linear system, compare Gauss elimination method and Gauss – Jordan method.

In Gauss elimination method the given system is transformed system with upper – triangular coefficient matrix which can be solved by back substitution. In Gauss – Jordan method the coefficient matrix is reduced to an unit matrix and then directly we can find the unknowns.

3. State the two differences between direct and iterative methods for solving system of equations.

Direct Method | Iterative Method |

It gives exact value | It gives only approximate solution |

Simple, take less time | Time consuming and labourious |

This method determine all the roots at the same time. | This method determine only one root at a time. |

4. What is the condition for convergence of Gauss – Jacobi method of iteration?

The coefficient matrix should diagonally dominant.

5. What are the elementary transforms?

Elementary Transforms:

i) Interchange the i th and j th row ( or column)

ii) Multiply all the elements in the ith row by a number K.

iii) Adding the elements i n the i th row to the corresponding elements in the j th row multiply by a constant K.

6. Give the name of any 2 iteration method in numerical methods.

Gauss – Seidel method , Jacobi ‘s method.

7. In Gauss elimination method , the coefficient matrix is transformed to what form.

Upper triangular form.

8. What type of eigen value can be obtained using power method?

We can obtain dominant egien value of the given matrix.

9. State a sufficient condition for Gauss – Jacobi method to converge.

Let the given equations be

The sufficient condition is

10. State Newton’s Raphson iterative formula.

UNIT 4

1. Construct a linear interpolating polynomial given the points ((

2.Form the divided difference table for the following data:

X | 2 | 5 | 10 |

Y | 5 | 29 | 109 |

Solution:

Divided difference table:

X | Y | I divided difference | II divided difference |

2 | 5 | 8 16 | 1 |

5 | 29 | ||

10 | 109 |

3.Taking h to be the interval of differencing, find .

4.When Newton’s backward interpolation formula is used?

To find the unknown values of ‘y’ for some ‘x’ which lies at the end of table of values, we use Newton’s backward formula.

5.Why is Trapezoidal rule is so called?

Since we approximate the given integral by the sum of ‘n’ Trapezoids we call it as Trapezoidal rule.

6.When does Simpson’s rule given exact result?

The Simpson’s rule will give exact result when we are decreasing the interval h.

7.Why Simpson’s 1/3 rule is called a closed formula?

Since the end point ordinates and are included in the Simpson’s 1/3 rule, it is called closed formula.

8.State Newton’s formula to find f’(x) using formula differenc es.

9.What is the order of error in Trapezoidal rule?

The order of error in Trapezoidal rule

10.What is the order of the error in Simpson’s formula?

Two.

**UNIT – 5**

1. Where the Taylor series method of solving differential equation is powerful?

If it is possible to find the successive derivatives in a very easy manner , then only Taylor series method is powerful.

2. Write down the third order Taylor’s algorithm.

3. Write the merits and demerits of the Taylor method of solution.

Merits:

1. Taylor formula is easily derived for any order according to our interest.

2. The values of y(x) for any x are easily obtained.

Demerits:

This method suffers from the time consumed in calculating the higher derivatives.

4. Using Euler’s method , find y(0.1) if , y(0) = 0.

= 0 + 0.1 f(0,0)

= 0.1 (1) = 0.1.

5. In the derivation of fourth order Runge – kutta formula, why it is called fourth order.

The fourth order Runge – Kutta method agree with Taylor series solution upto the terms of . Hence it is called fourth order R.K method.

6. Compare Taylor series and R.K method.

a) The use of R.K ‘s method gives quick convergence to the solutions of the differential equations than Taylor’s series.

b) The labour involved in R.K’ s method is comparatively lesser.

c) In R.K’ s method , the derivatives of higher order are not required for calculation as in Taylor’s series method.

7. What are the advantages of R.K method over Taylor’s method?

The R.K method is desgined to give greater accuracy and they possess the advantage of requiring only the function values at some selected points on the sub intervals.

8. What do we mean by saying that a method is self starting? Not self – starting?

Iteration method is self starting since we can take value which lies in the given interval [a,b] in which the root lies. But Milne’s method is not self . Since we should know any 4 values prior to the value which we need.

9. State the Milne’s correction formula.

10. Define a difference quotient.

A difference quotient, is the quotient obtained by dividing the difference between two values of a function , by the difference between the two corresponding values of the independent variable.