# Statistical Methods (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 46 - 50 of 72

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## Question number: 46

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2009

### Describe in Detail

Given the joint density of (X _{1}, X _{2}),

Find the marginal densities of X _{1} and X _{2}. Also find E [X _{1}].

### Explanation

The joint density is written as

The marginal distribution of X _{1}

The marginal distribution of X _{2 } is

The first integral is

So, the marginal distribution of X _{2 } is

The mean of X _{1} is

## Question number: 47

» Statistical Methods » Non-Parametric Test » Wilcoxon

Appeared in Year: 2011

### Describe in Detail

Apply the Wilcoxon two-sample test to the following data on the first breakdown times of two brands of computers:

Brand A 98, 102, 47, 85, 99, 140, 130

Brand B 95, 125, 160, 155, 148.

Use 1.96 as the critical point for the appropriate test.

### Explanation

Let n _{1} be the number of breakdown times in the Brand A with median time m _{1}, and n _{2} the number of breakdown times in the Brand B with median time m _{2}. Then the hypothesis is that the median times of the two brands are

## Question number: 48

» Statistical Methods » Non-Parametric Test » Mann-Whitney

Appeared in Year: 2015

### Describe in Detail

Consider the two samples as follows:

Sample I = 6, 7, 8, 10, 12, 14, 16, 23

Sample II: 9, 11, 13, 15, 17, 18, 19, 20, 24

Test whether the examples have come from the same population by Wilcoxon-Mann-Whitney test at 10 % level of significance. [You can use normal approximation]

### Explanation

Let the sample I consider the population X and sample II consider the population Y. The hypothesis whether two sample comes from same identical population. We test the following null and alternative hypothesis is

In Wilcoxon-Mann-Whitney test, first consider the combined ordered sequence of the sample values is

Sample:

## Question number: 49

» Statistical Methods » Regression » Linear

Appeared in Year: 2015

### Describe in Detail

For 20 pairs of heights of father (X) and son (Y) measured in cm, the following data are obtained:

Test whether the cut on the X-axis can be assumed to be zero, at 5 % level of significance.

### Explanation

In the regression model, the cut on the X-axis can be assumed to be zero means the intercept zero. The test of the hypothesis is

Can be tested by the statistic is

where is the standard error of an estimator b _{0}.

where

## Question number: 50

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2012

### Describe in Detail

125out of 285 college-going male students in 1995 from a city were found to be smokers. Another sample of 325 such students from the same city in 2012 included 95 smokers. Examine at 5 % level of significance if smoking habit among college-going students is on the decrease in this city.

### Explanation

The given information can be written as in a table:

Year | Smoker | Non-smoker | Total |

1995 | 125 | 160 | 285 |

2012 | 95 | 230 | 325 |

Total | 220 | 390 | 610 |

Here, the null hypothesis is that the proportion of smoker in this two year is same and the alternative is there is decrease in