ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 55 - 61 of 165

we use Wilcoxon two-sample test to the following data

Let n ₁ be the number of breakdown times in the Brand A with median time m ₁, and n ₂ the number of breakdown times in the Brand B with median time m ₂. Then the hypothesis is that the median times of the two brands are same against the alternative that they are not same.

To find the mean and variance, we use the conditional expectation and conditional variance

By the given definition of conditional expectation

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.

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The variance is

Let X ~ N (0, σ ²). The density function is

First we find the distribution of Y = X ²

Let

The limit is also change

Differentiate with respect to y,

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So

Rewrite the characteristic function Φ (t)

The density function of X is given by Fourier inversion theorem using the characteristic function,

The general form is

Putting the characteristic function

Let the contingent table is

Class	A	α	Total
B	a	b	a + b
β	c	d	c + d
Total	a + c	b + d	a + b + c + d =N

Here A denotes the presence of any attributes and α denotes the absence of attributes A

Since the marginal frequencies is fixed, therefore

Probability for a individual belong

Let U = F (x), then the distribution function of G of U is given by

The inverse exists. Since F is non-increasing and its continuous

G (u) =F (F ^-1 (u) ) {F is a distribution of X}

G (u) =u

Then the p. d. f of U = F (x) is given by

Since F is a distribution function takes value in range [

We choose k as constant that gives the integral over the range x for the density function is equal to one.

The expression of the distribution function is

Assume

The limit is also change.

U = , L= and put