Comprehensive Guide to Sampling Methods and Simple Random Sampling

Principles of Sampling and Simple Random Samples

• Objective of Sampling:

  • Whether conducting an experiment or an observational study, the primary objective is to obtain a sample that is representative of the underlying population.
  • Achieving a representative sample requires ensuring that every member of the population has the exact same chance of being selected for the sample, a task that is more difficult to execute in practice than it might initially seem.

• Definition of a Simple Random Sample:

  • A simple random sample of some number of subjects $n$ is selected in such a way that every possible sample of the exact same size $n$ has the exact same chance of being chosen.
  • A simple random sample is frequently referred to simply as a random sample in general terminology.

• Distinction Between a Simple Random Sample and a Random Sample:

  • Strictly speaking, a random sample carries the specific requirement that all individual members of the population have the same chance of being chosen.
  • A simple random sample carries the stronger requirement that every possible sample group of size $n$ has an equal probability of selection.

Four Basic Sampling Methods

• Method 1: Systematic Sampling:

  • Definition: A method in which researchers select a starting point and subsequently select every $k\text{th}$ element of the population.
  • Example: Inspecting every $3\text{rd}$ car that comes down an assembly line.

• Method 2: Convenience Sampling:

  • Definition: A method that relies on using data that is very easy to get.
  • Example: Polling followers on social media or asking next-door neighbors to determine or predict the outcome of a state election.
  • Representativeness Concerns: Data gathered via convenience sampling is unlikely to produce a sample that is representative of the entire population (e.g., all eligible voters in a state).

• Method 3: Stratified Sampling:

  • Definition: A method where the population is first divided into at least two distinct subgroups (or strata), where all subjects within the same subgroup share the same characteristic. A random sample is then drawn from each individual subgroup.
  • Example Context: Testing the effects of a new sleep aid medication to evaluate whether the medication affects men and women differently. The population is stratified by gender into male and female subgroups, followed by drawing a random sample from each subgroup.

• Method 4: Cluster Sampling:

  • Definition: A method that divides the population area into sections or clusters based on location. Certain clusters are then randomly selected, and all members from the selected clusters are included in the final sample.
  • Example: A polling company predicting voter turnout for an upcoming election by randomly selecting specific voting districts and polling everyone residing within those selected districts.

Scenario Analyses and Method Identification

• Scenario 1: Computer-Selected Phone Survey:

  • Scenario Details: A total of $500$ people are called in a survey after their telephone numbers are randomly selected by a computer.
  • Method Classification: Simple Random Sample.
  • Justification: Because every possible group of $500$ people has the exact same chance of being selected by the computer program, this process fulfills the criteria for a simple random sample.

• Scenario 2: Social Media Homework Question:

  • Scenario Details: To complete a school assignment, a student sends out the question "How many hours per day do you spend online?" to his friends on Facebook.
  • Method Classification: Convenience Sample.
  • Justification: The data is extremely easy for the student to obtain, defining it as convenience sampling.

• Scenario 3: Department Store Customer Survey:

  • Scenario Details: A department store posts an employee at the entrance door and surveys every $10\text{th}$ customer who enters.
  • Method Classification: Systematic Sample.
  • Justification: Selecting subjects based on a fixed periodic interval (every $10\text{th}$ customer) represents systematic sampling.

Multistage Sampling and Applications

• Multistage Sampling Design:

  • Definition: A sampling design in which researchers collect data by using a combination of basic sampling methods across sequential stages.
  • Process: Pollsters select a sample in different stages, where each individual stage may utilize a different sampling method.

• Real-World Application:

  • National Census: The United States government utilizes a multistage sampling design combining both cluster sampling and stratified sampling to complete the national census every $10\,\text{years}$.