# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 43 - 47 of 165

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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

and probability of selec

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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

Assume

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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 of the sample values is

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## Question number: 47

» Statistical Methods » Correlation Ratio

Appeared in Year: 2012

### Describe in Detail

Scores in a UPSC examination in Statistics Paper-I and Paper-II awarded to10 candidates were as under:

Candidate | A | B | C | D | E | F | G | H | I | J |

Score in Paper- I | 91 | 65 | 80 | 120 | 80 | 140 | 105 | 80 | 42 | 72 |

Score in Paper-II | 86 | 38 | 75 | 135 | 85 | 135 | 94 | 101 | 45 | 65 |

Calculate Spearman’s rank correlation coefficient between the scores in the two papers.

### Explanation

The method of calculating the association between the ranks is known as Spearman’s rank correlation. The candidates are rank by the UPSC examination taken for two papers which is independent. The Spearman’s rank correlation is

where n is the number of candidates and d _{i} is the difference between the rank of the i ^{th } individu

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