Solve your IGNOU Doubts

Question:

Every clinical thermometer is classified into one of the four catergories A, B, C and D on the basis of inspection and test. From the past experience it is known that thermometers produced by a certain manufacturer are distributed among the four categories in the following proportions:

Category:	A	B	C	D
Proportion:	0.87	0.009	0.003	0.01

A new lot of 1336 thermometers is submitted by the manufacture for inspection and test and the following distribution of categories obtained:

Category:	A	B	C	D
No. of thermometers Reported:	1188	91	47	10

At 5% level of significance test whether this new lot of thermometers differ from the previous experience.

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Question:

b) The position $f(x)$ of a particle moving in a line at various times $x_k$ is given in the following table. Estimate the velocity and acceleration of the particle at $x=1.2.$

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

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Question:

a) Derive a suitable numerical differentiation formula of $0(h^2)$ to find $f''(2.4)$ with $h=0.1$ given the table

x	0.1	1.2	2.4	3.9
f(x)	3.41	2.68	1.37	-1.48

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Question:

The joint probability distribution of x and y is given below:

$x\rightarrow$	$y\downarrow$	0	1	2
0		$\frac{1}{12}$	$\frac{1}{6}$	$\frac{1}{24}$
1		$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{40}$
2		$\frac{1}{8}$	$\frac{1}{20}$	__
3		$\frac{1}{120}$	__	$\frac{1}{120}$

Find (i) ) $P(x=1,=2)$
(ii) ) $P(x=0,1\leq y<3)$
(iii) ) $P(x+y\leq1)$
(iv) ) $P(x>y)$

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Question:

c) Using finite differences, show that the data

x	-3	-2	-1	0	1	2	3
f(x)	13	7	3	1	1	3	7

represents a second degree polynomial. Obtain this polynomial using interpolation and find f (2.5).

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Question:

b) The function $f(x)=1n(1+x)$ is to be tabulated at equispaced points in the interval [2, 3] using linear interpolation. Find the largest step size h that can be used so that the error $\leq 5\times 10^{-4}$ in magnitude.

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Question:

a) Determine the constants $a,\beta ,y$ in the differentiation formula $y'(x_0)=ay(x_0-h)+\beta y(x_0)+\gamma y(x_0+h)$ so that the method is of the highest possible order. Find the order and the error term of the method.

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Question:

Show that variance can be expressed in terms of the mutual differences $x_i-x_j$ of the observations i.e.

$S^2=\frac{1}{2n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_i-x_j)^2.$

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Question:

b) Solve the system of equations

$\bg_white 3x+2y+4z=7$

$2x+y+z=7$

$x+3y+5z=2$

with partial pivoting. Store the multipliers and also write the pivoting vectors.

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Question:

a) Find the dominant eigenvalue and the corresponding eigenvector for the matrix

$A=\begin{bmatrix} -4 &14 & 0\\-5 &13 &0 \\-1 &0 &2 \end{bmatrix}$

using five iterations of the power method and taking $y^0=[111]^T$ as the initial vector.

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Question:

Let $f(x,y)=x+y;0<x<1,0<y<1.$ Find (i) the correlation coefficient between x and y, and (ii) E( y / x) .

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Question:

An unbiased die is rolled twice. Let $A_1$ denote the event: odd face roll on the first die, $A_2$ denote the event that total score is Odd. Check the independence of $A_1$ and . $A_2$

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Question:

b) For the linear system of equations $\begin{bmatrix} 1 &2 &-2 \\1 &1 &1 \\2 &2 &1 \end{bmatrix}\begin{bmatrix} x_1\\x_2 \\x_3 \end{bmatrix}=\begin{bmatrix} 1\\3 \\5 \end{bmatrix}$

set up the Gauss-Jacobi and Gauss-Seidal iteration schemes in matrix form. Also check the convergence of the two schemes.

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Question:

Let p be the probability that a coin will fall head in a single toss in order to test $H_0:p=\frac{1}{2}$ against $H_1:p=\frac{3}{4},$ The coin is tossed 3 times and $H_0$ is rejected if more than 2 heads are obtained. Find the probability of type I and type II errors. Also obtain the power of the test.