ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 157 - 162 of 165

Let X ₁, X ₂, …, X _n and Y ₁, Y ₂, …Y _m are the samples taken from independent N (µ ₁, σ ₁²) and N (µ ₂, σ ₂²).

In this question the hypothesis for testing is

Let s ₁² and s ₂² be the estimates variances of σ ₁² and σ ₂² based on sample sizes n and m.

and

Then

Let the a polynomial is P (X) = kx ² +lx + m, where k, l, m are constants which is determine by using Lagrange’s interpolation polynomial because the x values is not equal interval.

where

x ₀ =0, x ₁ =1 _, , x ₂ =3 and f (x ₀) =1, f (x ₁) =3, f

Let X is a random variable follows normal distribution with mean µ and variance σ ². The testing of null hypothesis is that the population variance σ ² equals a specified value against one of the usual alternatives σ ² < , σ ² > , or σ ². The appropriate statistic on which to base our decision is the chi-squared statistic.

In chi

Let A and B are two possible events in the sample space.

(i) Additive law of probability is

{Commutative property }

We know that

This shows that

(ii) This also show by additive law of probability

Let assume BUC = D, then

To find the missing value, we use binomial expansion method. Here 4 values are known, we would take fourth order finite difference zero. Thus,

Here for x = 1, U ₀ =7, U ₁ =? , U ₂ =13, U ₃ =21, U ₄ =37

To find the value of a, we known that the probability density function under the range of x is equal to one.

To find order which is exist for this p. d. f. is

Upto moments is exist is k-2 < 0 = k < 2

That is only we find the first moment for k