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

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## Question 72

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 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, the n…

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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

Appeared in Year: *2015*

### Describe in Detail

Essay▾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 distribution of Z_{n} = n (Y_{1} -θ) is

The pdf of Z_{n} is

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## Question 76

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 a^{2}-b^{2} = (a + b) (a-b) )

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

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## Question 77

Appeared in Year: *2015*

### Describe in Detail

Essay▾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 normally distri…

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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) r_{12.3} (ii) r_{1.23}

### Explanation

In general, the correlation matrix is in these notations

(i)

(ii)