ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 72 - 78 of 164

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Question 72

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Appeared in Year: 2012

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Essay▾

In a ticket counter, at some point of time the sequence of males (M) and females (F) was found as

MMFMFFFMFMMFFFFMFMMMM

Use runs test to examine if the sequence is random (5 % critical value of the number r of runs with n1 = 11, n2 = 10 is 6) .

Explanation

Suppose a sample size n contains n1 symbols of one type and n2 symbols of the other type. The null hypothesis is the symbols occur in random order. The alternative is the symbols occur in a set pattern. The lower and upper critical value is obtained from tables. If the observed number of runs in a sample lies in between these critical values, the n…

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Question 73

Appeared in Year: 2015

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Essay▾

Let X have the probability density function

Find the mean, geometric mean and harmonic mean.

Explanation

Then mean is

The geometric mean is

The geometric mean is

The harmonic mean is

Question 74

Appeared in Year: 2011

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Essay▾

Let X be a positive valued random variable. Prove that

Hence deduce the Chebychev՚s inequality.

Explanation

The expectation of X is define as

For any x ⩾ r

This implies that

Hence, any function of X, assume g (x) the Chebychev՚s inequality is

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Question 75

Standard Probability Distributions

Appeared in Year: 2015

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Essay▾

Let Y1 denote the first order statistic in a random sample of size n from a distribution that has the pdf

Obtain the distribution of Zn = n (Y1 -θ) .

Explanation

To find the distribution of order statistic, the pdf is

Here denote the first order statistic that is

The cdf of x is

The pdf of Y1 is

Then, the distribution of Zn = n (Y1 -θ) is

The pdf of Zn is

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Question 76

Newton (Dividend Difference)

Appeared in Year: 2010

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Essay▾

If , find the divided differences f (a, b) and f (a, b, c) .

Explanation

we can write the Newton՚s divided difference formulas as

First find f (b, c)

Then, from (1) and (2)

(remember a2-b2 = (a + b) (a-b) )

[cancelling (c-a) from both numerator and denominator]

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Question 77

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Appeared in Year: 2015

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Essay▾

If X follows binomial b (n1 , p1) distribution and Y follows binomial b (n2 , p2) , provide an appropriate test at level α for H0: p1 = p2 against H1: p1 > p2 .

Explanation

The hypothesis is equivalent to testing the null hypothesis that p1 − p2 = 0 against the alternative is p1 -p2 > 0. The statistic on which we base our decision is the random variable . Suppose the samples of sizes n1 and n2 are selected from binomial populations are independent.

For large sample size, the estimator was approximately normally distri…

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Question 78

Appeared in Year: 2014

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Essay▾

Given the following correlation matrix of order 3 × 3

Calculate (i) r12.3 (ii) r1.23

Explanation

In general, the correlation matrix is in these notations

(i)

(ii)