Scope of Inference: Generalization

Topic 5
Math 115

From Sample to Population

In hypothesis testing and confidence intervals (topics 03 & 04), we:

  • Collected data from a sample
  • Made inferences about a population parameter

Key question: When can we trust these inferences to apply to the broader population?

Answer: It depends on how we selected our sample.

Generalization

Generalization is the ability to apply findings from our sample to the broader population.

We CAN generalize:

If sample represents the population

We CANNOT generalize:

If sample doesn’t represent the population

The critical factor: How we selected our sample

Hemlock Woolly Adelgid Study

Two groups study insect presence on hemlock trees at Hope College Nature Preserve:

Group A (Convenience Sample):

  • Walk trails, inspect hemlocks visible from trail
  • Easy, but may differ systematically from population

Group B (Random Sample):

  • Use random number generator to select 50 from list of all hemlocks
  • Every tree has equal chance of selection

Question: Which group can generalize their findings to all hemlocks?

Why Random Sampling Enables Generalization

Convenience samples typically don’t represent the population:

  • Easy-to-reach individuals may differ systematically from others (introduces sampling bias)
  • Sample characteristics don’t match population

Random samples produce representative samples:

  • Every individual has equal chance of selection, eliminating systematic differences
  • Sample characteristics mirror population (on average)

Result: Random sampling enables generalization; convenience sampling does not

Simple Random Sample (SRS)

Simple Random Sample (SRS): Every individual in the population has an equal and independent chance of being selected.

Implementing an SRS (Hemlock study example):

  1. Create list of all tree IDs (sampling frame)
  2. Decide sample size (e.g., 50 trees)
  3. Use random number generator to select sample
  4. Visit the selected trees using map

Tools: Random.org, calculator, Excel, drawing from hat

The Bottom Line

Sampling Method Representative? Can Generalize?
Convenience sample Usually NO NO
Random sample Usually YES YES

Key principle: Generalization requires a representative sample, which random sampling provides.

  • We will introduce other random sampling methods in addition to SRS later

References