# Statistical Methods (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 24 - 28 of 72

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## Question number: 24

» 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: 25

» 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: 26

» 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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## Question number: 27

» Statistical Methods » Order Statistics » Maximum

Appeared in Year: 2014

### Describe in Detail

Let X _{1}, X _{2}, …, X _{n} be n independently identically distributed random variables following an exponential distribution with mean parameter θ. Obtain the distributions of (i) Maximum (X _{1}, X _{2}, …, X _{n}); (ii) Minimum (X _{1}, X _{2}, …, X _{n}) and (iii) Median for n = 2m + 1, (m > 0)

### Explanation

Let X _{1}, X _{2}, …, X _{n}, be n independently identically distributed random variables following an exponential distribution with mean parameter θ. The probability density and distribution function is

Let X _{ (1) }, X _{ (2) }, …, X _{ (n) } denote the order statistic of a random sample, X _{1}, X _{2}, …, X _{n} from ex

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## Question number: 28

» Statistical Methods » Non-Parametric Test » Wilcoxon

Appeared in Year: 2010

### Describe in Detail

The following data represent lifetimes (hours) of batteries for two different brands:

Brand A | 40 | 30 | 40 | 45 | 55 | 30 |

Brand B: | 50 | 50 | 45 | 55 | 60 | 40 |

Test whether two brands are the same.

### Explanation

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.

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