# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 72 - 77 of 165

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## Question number: 72

» Statistical Methods » Non-Parametric Test » Run

Appeared in Year: 2012

### Describe in Detail

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 n _{1 } = 11, n _{2 } = 10 is 6).

### Explanation

Suppose a sample size n contains n _{1} symbols of one type and n _{2} 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, t

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## Question number: 73

» Statistical Methods » Measures of Location

Appeared in Year: 2015

### Describe in Detail

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

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## Question number: 74

» Probability » Tchebycheffs Inequality

Appeared in Year: 2011

### Describe in Detail

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 number: 75

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

### Describe in Detail

Let Y _{1} denote the first order statistic in a random sample of size n from a distribution that has the pdf

Obtain the distribution of Z _{n} =n (Y _{1} -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 Y _{1} is

Then, the dis

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## Question number: 76

» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)

Appeared in Year: 2010

### Describe in Detail

If f (x) =1/x ^{2}, 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)

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## Question number: 77

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2015

### Describe in Detail

If X follows binomial b (n _{1}, p _{1}) distribution and Y follows binomial b (n _{2}, p _{2}), provide an appropriate test at level α for H _{0}: p _{1} =p _{2} against H _{1}: p _{1} > p _{2}.

### Explanation

The hypothesis is equivalent to testing the null hypothesis that p _{1} −p _{2} = 0 against the alternative is p _{1} -p _{2} > 0. The statistic on which we base our decision is the random variable . Suppose the samples of sizes n _{1} and n _{2} are selected from binomial populations are independent.

For large sample size, the estimator was approximately

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