# Statistical Methods (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 55 - 60 of 72

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## Question number: 55

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2010

### Describe in Detail

Let (X, Y) be jointly distributed with density function

Obtain correlation coefficient between X and Y

### Explanation

First find the marginal distribution of X

find the marginal distribution of Y

The mean of X and Y is

The second moment of X and Y is

The variance of X and Y

## Question number: 56

» Statistical Methods » Non-Parametric Test » Sign

Appeared in Year: 2011

### Describe in Detail

Let the temperature before and after administration of aspirin be

Patient | Before | After |

1 | 100.0 | 98.1 |

2 | 102.1 | 97.2 |

3 | 100.6 | 98.6 |

4 | 100.1 | 99.1 |

5 | 101.5 | 97.6 |

6 | 102 | 98.6 |

7 | 99.9 | 98.2 |

8 | 102.7 | 98.1 |

9 | 100.40 | 98.2 |

10 | 100.8 | 97.1 |

Test by the sign test; whether aspirin is effective in reducing temperature. What is the p-value of the calculated statistic?

### Explanation

Let M _{1} and M _{2} is the median temperature of the patients. The hypothesis for test is

Fir this first find the difference of median

Also find the difference of temperature of before and after

d _{i} 2.1 4.9 2.0 1.0 3.9 3.4 1.7 4.6 2.4 3.7

## Question number: 57

» 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

## Question number: 58

» 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

## Question number: 59

» 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

## Question number: 60

» 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