# 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

Appeared in Year: 2010

### 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

… (55 more words) …

## Question number: 133

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2010

### 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

… (11 more words) …

## Question number: 134

» Statistical Methods » Measures of Location

Appeared in Year: 2013

### 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

… (10 more words) …

## Question number: 135

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2015

### 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 |

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

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2011

### 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

» Statistical Methods » Tests of Significance » T-Test

Appeared in Year: 2015

### 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

… (168 more words) …