6. Scope of Inference: Generalization

Math 115

Yurk and Martin

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

Sampling Words

  • Use the words in the Gettysburg Address as a population to explore the concept of sampling.
  • We’ll categorize the words as long or short (less than 4 letters)

Let’s try it!

Take a sample of 10 representative words from the Gettysburg Address

Four score and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty, and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battlefield of that war. We have come to dedicate a portion of that field, as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this. But, in a larger sense, we cannot dedicate, we cannot consecrate, we cannot hallow this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion that we here highly resolve that these dead shall not have died in vain that this nation, under God, shall have a new birth of freedom and that government of the people, by the people, for the people, shall not perish from the earth.

Sampling Frame

Go to www.random.org/sequences/

  • 1 Four score and seven years ago our fathers brought forth
  • 11 on this continent a new nation, conceived in liberty, and
  • 21 dedicated to the proposition that all men are created equal.
  • 31 Now we are engaged in a great civil war, testing
  • 41 whether that nation, or any nation so conceived and so
  • 51 dedicated, can long endure. We are met on a great
  • 61 battlefield of that war. We have come to dedicate a
  • 71 portion of that field, as a final resting place for
  • 81 those who here gave their lives that that nation might
  • 91 live. It is altogether fitting and proper that we should
  • 101 do this. But, in a larger sense, we cannot dedicate,
  • 111 we cannot consecrate, we cannot hallow this ground. The brave
  • 121 men, living and dead, who struggled here, have consecrated it,
  • 131 far above our poor power to add or detract. The
  • 141 world will little note, nor long remember what we say
  • 151 here, but it can never forget what they did here.
  • 161 It is for us the living, rather, to be dedicated
  • 171 here to the unfinished work which they who fought here
  • 181 have thus far so nobly advanced. It is rather for
  • 191 us to be here dedicated to the great task remaining
  • 201 before us that from these honored dead we take increased
  • 211 devotion to that cause for which they gave the last
  • 221 full measure of devotion that we here highly resolve that
  • 231 these dead shall not have died in vain that this
  • 241 nation, under God, shall have a new birth of freedom
  • 251 and that government of the people, by the people, for
  • 261 the people, shall not perish from the earth.

The Bottom Line

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

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

References