### Probability Sampling

#### Introduction

#### Simple Random Samples

#### Stratified Samples

#### Cluster Samples

#### Sample Planning

#### Appendix

### Probability Sampling

#### Introduction

#### Simple Random Samples

#### Stratified Samples

#### Cluster Samples

#### Sample Planning

#### Appendix

# Survey Sampling Methods

**Sampling method** refers to the way that observations
are selected from a
population
to be in the
sample for a
sample survey.

## Population Parameter vs. Sample Statistic

The reason for conducting a sample survey is to estimate the value of some attribute of a population.

**Population parameter**. A population parameter is the true value of a population attribute.**Sample statistic**. A sample statistic is an estimate, based on sample data, of a population parameter.

Consider this example. A public opinion pollster wants to know the
percentage of voters that favor a flat-rate income tax. The
*actual* percentage of all the voters is a population
parameter. The *estimate* of that percentage, based on
sample data, is a sample statistic.

The quality of a sample statistic (i.e., accuracy, precision, representativeness) is strongly affected by the way that sample observations are chosen; that is., by the sampling method.

## Probability vs. Non-Probability Samples

As a group, sampling methods fall into one of two categories.

**Probability samples**. With probability sampling methods, each population element has a known (non-zero) chance of being chosen for the sample.**Non-probability samples**. With non-probability sampling methods, we do not know the probability that each population element will be chosen, and/or we cannot be sure that each population element has a non-zero chance of being chosen.

Non-probability sampling methods offer two potential advantages - convenience and cost. The main disadvantage is that non-probability sampling methods do not allow you to estimate the extent to which sample statistics are likely to differ from population parameters. Only probability sampling methods permit that kind of analysis.

## Non-Probability Sampling Methods

Two of the main types of non-probability sampling methods are voluntary samples and convenience samples.

**Voluntary sample**. A voluntary sample is made up of people who self-select into the survey. Often, these folks have a strong interest in the main topic of the survey.Suppose, for example, that a news show asks viewers to participate in an on-line poll. This would be a volunteer sample. The sample is chosen by the viewers, not by the survey administrator.

**Convenience sample**. A convenience sample is made up of people who are easy to reach.Consider the following example. A pollster interviews shoppers at a local mall. If the mall was chosen because it was a convenient site from which to solicit survey participants and/or because it was close to the pollster's home or business, this would be a convenience sample.

## Probability Sampling Methods

The main types of probability sampling methods are simple random sampling, stratified sampling, cluster sampling, multistage sampling, and systematic random sampling. The key benefit of probability sampling methods is that they guarantee that the sample chosen is representative of the population. This ensures that the statistical conclusions will be valid.

**Simple random sampling**. Simple random sampling refers to any sampling method that has the following properties.- The population consists of N objects.
- The sample consists of n objects.
- If all possible samples of n objects are equally likely to occur, the sampling method is called simple random sampling.

There are many ways to obtain a simple random sample. One way would be the lottery method. Each of the N population members is assigned a unique number. The numbers are placed in a bowl and thoroughly mixed. Then, a blind-folded researcher selects n numbers. Population members having the selected numbers are included in the sample.

**Stratified sampling**. With stratified sampling, the population is divided into groups, based on some characteristic. Then, within each group, a probability sample (often a simple random sample) is selected. In stratified sampling, the groups are called**strata**.As a example, suppose we conduct a national survey. We might divide the population into groups or strata, based on geography - north, east, south, and west. Then, within each stratum, we might randomly select survey respondents.

**Cluster sampling**. With cluster sampling, every member of the population is assigned to one, and only one, group. Each group is called a cluster. A sample of clusters is chosen, using a probability method (often simple random sampling). Only individuals within sampled clusters are surveyed.Note the difference between cluster sampling and stratified sampling. With stratified sampling, the sample includes elements from each stratum. With cluster sampling, in contrast, the sample includes elements only from sampled clusters.

**Multistage sampling**. With multistage sampling, we select a sample by using combinations of different sampling methods.For example, in Stage 1, we might use cluster sampling to choose clusters from a population. Then, in Stage 2, we might use simple random sampling to select a subset of elements from each chosen cluster for the final sample.

**Systematic random sampling**. With systematic random sampling, we create a list of every member of the population. From the list, we randomly select the first sample element from the first*k*elements on the population list. Thereafter, we select every*kth*element on the list.This method is different from simple random sampling since every possible sample of

*n*elements is not equally likely.

## Test Your Understanding

**Problem**

An auto analyst is conducting a satisfaction survey, sampling from a list of 10,000 new car buyers. The list includes 2,500 Ford buyers, 2,500 GM buyers, 2,500 Honda buyers, and 2,500 Toyota buyers. The analyst selects a sample of 400 car buyers, by randomly sampling 100 buyers of each brand.

Is this an example of a simple random sample?

(A) Yes, because each buyer in the sample was randomly sampled.

(B) Yes, because each buyer in the sample had an equal chance of
being sampled.

(C) Yes, because car buyers of every brand were equally represented
in the sample.

(D) No, because every possible 400-buyer sample did not have an
equal chance of being chosen.

(E) No, because the population consisted of purchasers of
four different brands of car.

**Solution**

The correct answer is (D). A
simple random sample requires that
every
sample
of size *n* (in this problem, *n*
is equal to 400) has an equal chance of being selected. In this
problem, there was a 100 percent chance that the sample would
include 100 purchasers of each brand of car. There was
zero percent chance that the sample would include, for example,
99 Ford buyers, 101 Honda buyers, 100 Toyota buyers, and 100
GM buyers. Thus, all possible samples of size 400 did not have
an equal chance of being selected; so this cannot be a simple
random sample.

The fact that each buyer in the sample was randomly sampled is a necessary condition for a simple random sample, but it is not sufficient. Similarly, the fact that each buyer in the sample had an equal chance of being selected is characteristic of a simple random sample, but it is not sufficient. The sampling method in this problem used random sampling and gave each buyer an equal chance of being selected; but the sampling method was actually stratified random sampling.

The fact that car buyers of every brand were equally represented in the sample is irrelevant to whether the sampling method was simple random sampling. Similarly, the fact that population consisted of buyers of different car brands is irrelevant.

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