ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 42 - 46 of 165
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Question number: 42
» Probability » Tchebycheffs Inequality
Appeared in Year: 2013
Describe in Detail
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
Appeared in Year: 2014
Describe in Detail
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
Appeared in Year: 2014
Describe in Detail
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
Appeared in Year: 2010
Describe in Detail
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
Appeared in Year: 2014
Describe in Detail
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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