# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 132 - 137 of 165

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## Question number: 132

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

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Appeared in Year: 2010

Essay Question▾

### Describe in Detail

Show that for discrete distribution β 2 > 1

### Explanation

We have to prove that

By definition of Kurtosis

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: 133

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Appeared in Year: 2010

Essay Question▾

### Describe in Detail

How large a sample must be taken in order that the probability will be at least 0·95 that X n will be within 0·5 of µ (µ is unknown and σ = l).

### Explanation

Here, you would know the standard deviation of a population but not know the mean of that population. So, you want to estimate the mean μ to within a given margin of error in a given confidence interval. The required sample size is

Margin of error E = 0.5,

1-α=0.95 →α=0.05

σ=1 and z α/2 =1.96

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## Question number: 134

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Appeared in Year: 2013

Essay Question▾

### Describe in Detail

Find the weighted arithmetic mean of the first ‘n’ natural numbers, the weights being the corresponding numbers.

### Explanation

If x 1, x 2, …, x n are the values having weights w 1, w 2, …, w n respectively, the weight mean is

Here X is first ‘n’ natural numbers that is 1,2, 3, …, n

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## Question number: 135

» Numerical Analysis » Interpolation Formulae » Lagrange

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Fit the exponential curve y = a + bx to the following data

x: 0 2 4

y: 5.01 10 31.62

### Explanation

his is not exponential equation, it is a straight line y = a + bx

 S. no x y Xy x 2 1 0 5.01 0 0 2 2 10 20 4 3 4 31.62 126.48 16 Sum 6 46.63 146.48 20

The normal equations is

Using the table, putting these values in normal equations. We get

Solving these two equations, we get the value of

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## Question number: 136

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Appeared in Year: 2011

Essay Question▾

### Describe in Detail

Let X have the continuous c. d. f. F (x). Define U = F (x). Show that both - log U arid -log (1 - U) are exponential random variables:

### Explanation

Let U = F (x), then the distribution function of G of U is given by

Since F is non-increasing and its continuous.

G (u) =F (F -1 (u) ) implies G (u) =u

Then the p. d. f is

Since F is a distribution function takes value in range [0,1]. Hence

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## Question number: 137

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let X 1, X 2, …, X 12 be a random sample from a normal N (µ 1, σ 1 2) and Y 1, Y 2, …Y 10 be another random sample from normal N (µ 2, σ 2 2), independently to each other. Carry out an appropriate test for testing

at 5 % level of significance. It is given that .

### Explanation

Let X 1, X 2, …, X 12 be a random sample from a normal N (µ 1, σ 1 2) and Y 1, Y 2, …Y 10 be another random sample from normal N (µ 2, σ 2 2), independently to each other. The test of hypothesis is

The sample size of X is n = 12 and Y is m = 10

The test is depen

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