# Sampling Techniques (ISS (Statistical Services) Statistics Paper III): Questions 22 - 27 of 30

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## Question number: 22

» Sampling Techniques » Stratified Random Sampling

Appeared in Year: 2012

### Describe in Detail

Discuss the allocation of a sample to different strata in stratified sampling. Explain proportional allocation and hence obtain the variance of mean of a stratified sample under proportional allocation.

### Explanation

For a population consisting of N sampling units it can be divided into k relatively homogeneous sub-groups of sizes, , …, (called as strata) such that N=.

Then a simple random sample of size (i = 1, 2, …, k) is drawn from

## Question number: 23

» Sampling Techniques » Simple Random Sampling With and Without Replacement

Appeared in Year: 2012

### Write in Short

Show that for a simple random sample unbiased estimate of

## Question number: 24

» Sampling Techniques » Probability Sampling or Random Sampling

Appeared in Year: 2012

### Write in Short

Let be the *y*-value of the i^{th} unit and be corresponding selection probability,

i = 1, 2, …, N. Then, based on a sample of size n drawn with replacement, show that an unbiased estimator for population total is

## Question number: 25

» Sampling Techniques » Simple Random Sampling and Systematic Sampling

Appeared in Year: 2013

### Describe in Detail

Explain circular systematic sampling. Show that minimum value of ρ is** , **when **, **where ρis intraclass correlation and** **is the sample mean.

### Explanation

**Circular systematic sampling**

This method consists of choosing a random start from 1 to N and selecting the unit corresponding to the random start and thereafter every k^{th} unit in a cyclic manner till a sample of size n units is obtained, k being the integer nearest to

## Question number: 26

» Sampling Techniques » Probability Sampling or Random Sampling

Appeared in Year: 2012

### Write in Short

Obtain an unbiased estimator of the gain due to PPSWR sampling as compared to SRSWR.

## Question number: 27

» Sampling Techniques » Simple Random Sampling and Sampling With Probability Proportional to Size

Appeared in Year: 2013

### Describe in Detail

In simple random sampling without replacement (SRSWOR), the sample mean** **is an unbiased estimator of population mean** . **Obtain the variance of the estimator of the population total Y.

### Explanation

Sample mean is an unbiased estimator of population mean i. e.

Sample mean is given by

It can also be written as

Where

Hence

**Variance of the estimator of the population total Y: ****: **

Let