Chapter 4: Designing Studies - Section 4.1: Samples and Surveys

The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE

Chapter 4: Designing Studies

  • Sections:
    • 4.1 Samples and Surveys
    • 4.2 Experiments
    • 4.3 Using Studies Wisely

Section 4.1 Samples and Surveys

  • Learning Objectives: After this section, students should be able to:
    • IDENTIFY the population and sample in a sample survey
    • IDENTIFY voluntary response samples and convenience samples
    • DESCRIBE how to use a table of random digits to select a simple random sample (SRS)
    • DESCRIBE simple random samples, stratified random samples, and cluster samples
    • EXPLAIN how undercoverage, nonresponse, and question wording can lead to bias in a sample survey

Population and Sample

  • Definition of Population: The population in a statistical study is the entire group of individuals about which we want information.
  • Definition of Sample: A sample is the part of the population from which we actually collect information.
    • We use information from a sample to draw conclusions about the entire population.
  • Process of Sampling:
    • Collect data from a representative sample and make an inference about the population.

The Idea of a Sample Survey

  • Conclusions about a whole population are often drawn from a sample.
  • Steps to Conduct a Sample Survey:
    • Step 1: Define the population to describe.
    • Step 2: Specify what to measure.
    • Step 3: Decide how to choose a sample from the population.
  • A sample survey is defined as a study that utilizes an organized plan to choose a sample that represents a specific population.

Sampling Methods

  • Methods to select trustworthy samples include:
    • Convenience Sampling: Selecting individuals who are easiest to reach.
    • Voluntary Response Samples: Individuals decide whether to join the sample in response to an open invitation.

How to Sample Well: Random Sampling

  • A sample chosen by chance/rules out:
    • Favoritism by the sampler.
    • Self-selection by respondents.
  • Definition of Simple Random Sample (SRS): An SRS of size n consists of n individuals from the population chosen so that every set of n individuals has an equal chance to be the sample selected.
    • In practice, random numbers generated by computers or calculators are often used, but when technology is unavailable—tables of random digits may be utilized.
    • TI-84 Random Number Generator:
    • MATH → Prob → randInt(low, high, #)

How to Choose an SRS

  1. Label: Assign a numerical label to each member of the population of the same length.
  2. Table: Read consecutive groups of digits of the appropriate length from a Table of Random Digits.
    • The sample contains individuals corresponding to the labels found.
  • Definition of a Table of Random Digits: A long string of digits (0-9) where each entry is equally likely and independent of each other.

Example of Selecting an SRS

  • Problem: Use Table D at line 130 to choose an SRS of 4 hotels:
    • Digits: 69051, 64817, 87174, 09517,…
    • SRS of 4 hotels: 05 Beach Castle, 16 Radisson, 17 Ramada, and 20 Sea Club.
  • Sample Hotel List:
    1. 01 Aloha Kai
    2. 02 Anchor Down
    3. 03 Banana Bay
    4. 04 Banyan Tree
    5. 05 Beach Castle
    6. 06 Best Western
    7. 07 Cabana
    8. 08 Captiva
    9. 09 Casa del Mar
    10. 10 Coconuts
    11. 11 Diplomat
    12. 12 Holiday Inn
    13. 13 Lime Tree
    14. 14 Outrigger
    15. 15 Palm Tree
    16. 16 Radisson
    17. 17 Ramada
    18. 18 Sandpiper
    19. 19 Sea Castle
    20. 20 Sea Club
    21. 21 Sea Grape
    22. 22 Sea Shell
    23. 23 Silver Beach
    24. 24 Sunset Beach
    25. 25 Tradewinds
    26. 26 Tropical Breeze
    27. 27 Tropical Shores
    28. 28 Veranda

Other Sampling Methods: Stratified

  • Advantages of Stratified Sampling:
    • Provides statistical advantages by sampling important groups (called strata) separately.
  • Definition of Stratified Random Sample: First classify the population into strata (groups of similar individuals), then choose a separate SRS in each stratum and combine these to form the full sample.

Example: Sampling Sunflowers

  • Instructions: Use Table D or technology to take an SRS of 10 grid squares, using the rows as strata, and then repeat using the columns as strata.

Other Sampling Methods: Cluster

  • Cluster Sampling: An alternative when populations are large and spread out.
  • Process:
    • Divide the population into smaller groups (clusters).
    • Randomly select some clusters and include all individuals in those selected clusters in the sample.

Example: Sampling at a School Assembly

  • Task: Describe how to select 80 students using different sampling methods:
    • (a) Simple Random Sample
    • (b) Stratified Random Sample
    • (c) Cluster Sample

Inference for Sampling

  • Purpose of Sampling: To gain information about a larger population.
  • Inference: The process of drawing conclusions about a population based on sample data.
  • Reasons to Rely on Random Sampling:
    1. Eliminates bias in selecting samples from available individuals.
    2. Laws of probability allow trustworthy inference about the population.
    • Results from random samples come with a margin of error that sets bounds on the likely error.
    • Larger random samples yield better information about the population.

Sample Surveys: What Can Go Wrong?

  • Many sample surveys are affected by errors in addition to sampling variability.
  • Good Sampling Techniques: Include strategies for reducing all sources of error.
  • Common Errors:
    • Undercoverage: Some groups in the population are left out of the sample selection process.
    • Nonresponse: Occurs when a selected individual cannot be contacted or refuses to participate.
    • Response Bias: A systematic pattern of incorrect responses, significantly influenced by the wording of questions in surveys.

Summary of Section 4.1

  • A sample survey selects a sample from the population of individuals we want information on.
  • Random sampling employs chance to select samples.
  • The basic random sampling method is a simple random sample (SRS).
  • Stratified random samples require dividing the population into strata and then selecting a separate SRS from each stratum.
  • Cluster sampling involves dividing the population into clusters, from which some are randomly selected.
  • Failure to utilize random sampling often results in bias, systematic errors in sample representation.
  • Voluntary response and convenience samples are particularly susceptible to significant bias.
  • Sampling errors arise from selection processes. Random sampling error and undercoverage are common types of these errors.
  • The most serious errors are nonsampling errors, with common types including nonresponse, response bias, and question wording.

Looking Ahead…

  • Next Sections:
    • The distinction between Observational Studies and Experiments
    • The language of experiments
    • Randomized Comparative Experiments
    • Principles of Experimental Design
    • Inference for Experiments
    • Blocking techniques
    • Matched Pairs Design
    • Overview of upcoming concepts in Section 4.2.