ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 42 - 46 of 165

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Question number: 42

» Probability » Tchebycheffs Inequality

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Appeared in Year: 2013

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Let X be a random variable with E [X] = 4 and E [X 2] = 20. Use Chebyshev’s inequality to determine a lower bound for the probability P [0 < x < 8].

Explanation

Let X be a random variable with mean µ and variance σ 2. Then any k > 0, the Chebyshev’s inequality is

or ‎

σ 2 = E [X 2] - (E [X] ) 2 =4

Then, a lower bound for the probability

Using Chebyshev’s inequality

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Question number: 43

» Probability » Probability Generating Functions

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Appeared in Year: 2014

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Let P (s) be the probability generating function associated with a non-negative, integer valued random variable X. Show that

Explanation

To show this we use the left hand side, express the summation term

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Question number: 44

» Probability » Bayes' Theorem

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Appeared in Year: 2014

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There are three identical bags U 1, U 2 and U 3. U 1 contains 3 red and 4 black balls; U 2 contains 4 red and 5 black balls; U 3 contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red. Find the probability that the first bag is chosen.

Explanation

Given that there are three identical bags U 1, U 2 and U 3. Then probabilities of selecting a bag are,

Let X be the event of selecting a red ball

Probability of selecting a red ball in U 1 is

Similarly, Probability of selecting a red ball in U 2 is

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Question number: 45

» Statistical Methods » Measures of Location

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Appeared in Year: 2010

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Show that for any discrete distribution, standard deviation is not less than mean deviation from mean.

Explanation

Here, we show that standard deviation is greater than mean deviation from mean.

S. D. ≥ mean deviation from mean

(S. D. ) 2 ≥ (mean deviation from mean) 2

Let x 1, x 2, …, x n are n observations in a set have frequency f 1, f 2, …, f n and the mean of observation is , then

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Question number: 46

» Statistical Methods » Non-Parametric Test » Mann-Whitney

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Appeared in Year: 2014

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Consider two samples as follows:

Sample 1: 1,4, 7,9, 16,17,22,24

Sample 2: 2,6, 10,12,18,20,26,28,32

Test whether the examples have come from the same population by Wilcoxon-Mann-Whitney test. [Given value of Z for α = 0·05 = 1·645, where Z is N (0,1) ]

Explanation

Let the sample 1 consider the population X and sample 2 consider the population Y. The hypothesis whether two sample comes from same identical population. We test the following null and alternative hypothesis is

In Wilcoxon-Mann-Whitney test, first consider the combined ordered sequence o

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