# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 116 - 121 of 164

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## Question 116

Appeared in Year: *2013*

### Describe in Detail

Essay▾Describe the run test for randomness. For the sequence of outcomes of 14 tosses of a coin,

HTTHHHTHTTHHTH, .

test whether the outcomes are in random order. (Given the lower and upper critical values R_{L} = 3,

R_{u} = 12 at 0.05 significance level.)

### 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 117

Appeared in Year: *2011*

### Describe in Detail

Essay▾Use Simpson՚s rule with five ordinates to compute an approximation to π with the help of the integration of the function (1 + x ^{2}) ^{-1} from 0 to 1.

### Explanation

T he rule of the integrand using Simpson՚s rule for five ordinates is

where , n = 5

We divide the range 0 to 1 in five equal part i.e..

Use Simpson՚s rule

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## Question 118

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let X follow log-normal with parameters µ and σ ^{2} . Find the distribution of Y = aX ^{b} , a > 0, - ∞ < b < ∞

### Explanation

If X follow log-normal with parameters µ and σ ^{2} , then Z = logX follow normal distribution.

First find the cdf of Y

Let

The upper limit is c = , lower limit is

Putting the value of c

The distribution of Y is

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## Question 119

Appeared in Year: *2015*

### Describe in Detail

Essay▾With 3 variables X_{1} , X_{2} and X_{3} , it is given that r_{13} = 0.71, R_{1.23} = 0.78. Find r_{12.3} .

### Explanation

We know that

The relationship between the partial and multiple correlations is

## Question 120

Appeared in Year: *2012*

### Describe in Detail

Essay▾For a set of 10 pairs of observations (x_{i} , y_{j}) , i = 1 (1) 10, the following calculations are available

Examine at 5 % level of significance if the two variables arc uncorrelated in the population.

### Explanation

First, we calculate the sample correlation coefficient r

for i = 1 to 10

Putting the value in the formula for n = 10

The test hypothesis is

The test statistic is the t-test

The tabulated value is t_{0.05,8} = 2.306

Since the calculated value is less than the tabulated value of t, we accept H_{o} . It means that there is no correlation between x and y variable.

## Question 121

Appeared in Year: *2009*

### Describe in Detail

Essay▾Find the density, if its characteristic function is

### Explanation

Rewrite the characteristic function Φ (t)

The density function of X is given by Fourier inversion theorem using the characteristic function,

The general form is

Putting the characteristic function

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