The parameter of interest is p₁ − p₂, the difference between two population proportions.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Conditions doubled
Both samples must be random and independent of each other, and all four success and failure counts must reach 10.
The interval
The difference of sample proportions plus or minus a margin. Variances add, so the standard error combines both samples.
Reading the interval
If the interval contains zero, no difference is established. That single check answers most questions.
Pooling in a test
A test assumes the two proportions are equal, so the samples are pooled into one estimate for the standard error.
Not in an interval
A confidence interval makes no assumption of equality, so it does not pool. Pooling in an interval is a standard error.
Is the difference bigger than chance?
Two sample proportions will almost always differ. The question is whether the difference is larger than sampling variation would routinely produce, which is exactly what the test computes.
Pooling for the test, not the interval
A two-proportion test pools the samples to estimate a common proportion under the null. The confidence interval does not pool, because it assumes no common value. Using the wrong standard error is a frequent slip.
Conditions for both groups
Randomness, independence and large counts must hold in each sample separately. Checking one group and assuming the other is fine is an incomplete verification.
An interval containing zero
If a confidence interval for the difference includes zero, no difference is a plausible value. That connects intervals to tests, and the exam frequently asks for the link to be made explicitly.
Step 2: Try It Yourself
Tap and try it out.
Group 1
Group 2
Group 2 has the most. It has 15 more than Group 1.
Step 3: Watch an Example
One step at a time.
Watch Ines Read an Interval
A 95% interval for p₁ − p₂ runs from −0.02 to 0.14.
- Step 1
The interval spans both negative and positive values.
Step 4: Your Turn
Practice makes it stick.
The Interval
Problem 1 of 2
An interval for p₁ − p₂ runs from −0.02 to 0.14. Does it contain zero? 1 yes, 0 no.
The Difference
Problem 2 of 2
Sample proportions 0.6 and 0.45. What is the difference?
Two Groups
1 of 8
Sample proportions 0.7 and 0.5. Difference?
2 of 8
An interval runs from 0.03 to 0.19. Does it contain zero? 1 yes, 0 no.
3 of 8
That interval establishes a difference? 1 yes, 0 no.
4 of 8
How many success and failure counts must be checked for two groups?
5 of 8
Do you pool for a significance test? 1 yes, 0 no.
6 of 8
Do you pool for a confidence interval? 1 yes, 0 no.
7 of 8
Sort each procedure by whether it pools the samples.
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- Empty
- Empty
8 of 8
Sample proportions 0.4 and 0.55. Difference?
Step 5: Quick Check
Show what you know.
Question 1 of 2
An interval for p₁ − p₂ runs from 0.05 to 0.21. Is a difference established? 1 yes, 0 no.
Question 2 of 2
Why does a two-proportion test pool the samples?
What You Learned
- The parameter is p₁ − p₂, and the conditions apply to both samples.
- An interval containing zero establishes no difference.
- A test pools the samples; a confidence interval does not.