# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 13 - 18 of 165

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## Question number: 13

» Statistical Methods » Dispersion and Skewness

Appeared in Year: 2010

### Describe in Detail

The first three moments of a distribution about the value 1 are 2, 25 and 80. Find its mean standard deviation and the moment-measure of skewness.

### Explanation

Given that a = 1,

The mean is

The standard deviation is

Measure of skewness

## Question number: 14

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2013

### Describe in Detail

Prove that for among the discrete distributions, the geometric distribution has the lack of memory property.

### Explanation

The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.

Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is

## Question number: 15

» Statistical Methods » Non-Parametric Test » Sign

Appeared in Year: 2010

### Describe in Detail

Describe clearly sign test. State its asymptotic relative efficiency with respect to t-test.

### Explanation

Let x _{ (1), } x _{ (2), …, } x _{ (n) } be the ordered sample values from a population F (X) and M be its median.

Here we test where M _{0} is the given value of the median and hence

To perform the sign test, first find the differences (X

## Question number: 16

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### Describe in Detail

Prove that the sum of two independent chi-squared random variables is also chi-squared.

### Explanation

Let X and Y are two independent chi-squared random variables with degree of freedom n and m respectively. We proof this by moment generating function. The moment generating function of chi-squared distribution is

Then moment generating function of sum of two random variable (X + Y) is

because

## Question number: 17

» Statistical Methods » Correlation Ratio

Appeared in Year: 2012

### Describe in Detail

Two persons Amal and Bimal come to the club at random points of time between 6 p. m. and 7 p. m. , and each stays for 10 minutes. What is the chance that they will meet?

### Explanation

Let *x* denote the time Amal arrives at the bar and *y* denote the time Bimal arrives at the bar from 6 p. m. We will draw the lines *y* = *x* – 10 and *y* = *x* + 10 and shade the area in between these two lines to

## Question number: 18

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

Appeared in Year: 2009

### Describe in Detail

wo drugs were given to two batches of 6 students. The numbers of days to get a complete cure are given below:

Durg A | 6 | 7 | 8 | 9 | 12 | 16 |

Durg B | 10 | 11 | 13 | 14 | 15 | 17 |

Using Mann-Whitney test decide whether the median days for cure by the two drugs are equal. (Table values of U - Statistic at 0.05 level are: U _{5, 5} = 2, U _{5, 6} = 3, and U _{6, 6} = 5. )

### Explanation

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

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

Sample: