# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 72 - 78 of 164

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## Question 72

Edit

Appeared in Year: 2012

### Describe in Detail

Essay▾

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 n1 = 11, n2 = 10 is 6) .

### Explanation

Suppose a sample size n contains n1 symbols of one type and n2 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, the n&#8230;

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## Question 73

Appeared in Year: 2015

### Describe in Detail

Essay▾

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

## Question 74

Appeared in Year: 2011

### Describe in Detail

Essay▾

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 75

Standard Probability Distributions

Appeared in Year: 2015

### Describe in Detail

Essay▾

Let Y1 denote the first order statistic in a random sample of size n from a distribution that has the pdf

Obtain the distribution of Zn = n (Y1 -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 Y1 is

Then, the distribution of Zn = n (Y1 -θ) is

The pdf of Zn is

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## Question 76

Newton (Dividend Difference)

Appeared in Year: 2010

### Describe in Detail

Essay▾

If , 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)

(remember a2-b2 = (a + b) (a-b) )

[cancelling (c-a) from both numerator and denominator]

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## Question 77

Edit

Appeared in Year: 2015

### Describe in Detail

Essay▾

If X follows binomial b (n1 , p1) distribution and Y follows binomial b (n2 , p2) , provide an appropriate test at level α for H0: p1 = p2 against H1: p1 > p2 .

### Explanation

The hypothesis is equivalent to testing the null hypothesis that p1 − p2 = 0 against the alternative is p1 -p2 > 0. The statistic on which we base our decision is the random variable . Suppose the samples of sizes n1 and n2 are selected from binomial populations are independent.

For large sample size, the estimator was approximately normally distri&#8230;

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## Question 78

Appeared in Year: 2014

### Describe in Detail

Essay▾

Given the following correlation matrix of order 3 × 3

Calculate (i) r12.3 (ii) r1.23

### Explanation

In general, the correlation matrix is in these notations

(i)

(ii)