Comparing Two Proportions

Is the difference bigger than chance?

12 problems · about 22 minutes · UNC-4.C, DAT-3.D

What this lesson teaches

The student constructs intervals and tests for a difference between two proportions.

Straightforward practice. Get the method working first.

5 problems

An interval for p₁ − p₂ runs from 0.05 to 0.21. Is a difference established? 1 yes, 0 no.
Answer 1
Why Yes.
Why does a two-proportion test pool the samples?
Answer The null assumes the proportions are equal, so one estimate is best.
Why The assumed equality justifies one combined estimate.
Sample proportions 0.7 and 0.5. Difference?
Answer 0.2
Why 0.7 − 0.5.
An interval runs from 0.03 to 0.19. Does it contain zero? 1 yes, 0 no.
Answer 0
Why Both ends are positive.
That interval establishes a difference? 1 yes, 0 no.
Answer 1
Why Zero is excluded.

The same ideas with more to keep track of.

3 problems

How many success and failure counts must be checked for two groups?
Answer 4
Why Two per group.
Do you pool for a significance test? 1 yes, 0 no.
Answer 1
Why The null assumes equality.
Do you pool for a confidence interval? 1 yes, 0 no.
Answer 0
Why No equality is assumed.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each procedure by whether it pools the samples.
Answer Pools: Two-proportion significance test, A procedure that assumes the proportions are equal · Does not pool: Two-proportion confidence interval, A procedure making no equality assumption
Why Pooling follows from assuming the two are equal.
Sample proportions 0.4 and 0.55. Difference?
Answer -0.15
Why 0.4 − 0.55.
The Interval: An interval for p₁ − p₂ runs from −0.02 to 0.14. Does it contain zero? 1 yes, 0 no.
Answer 1
Why Yes.
The Difference: Sample proportions 0.6 and 0.45. What is the difference?
Answer 0.15
Why 0.15.