Topic 3.11 Notes – Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

What a Confidence Interval for $p_1-p_2$ Means

Here, the parameter is $p_1-p_2$, the true difference between two population proportions. You already built this interval in the previous topic. Now the job is to read it correctly and use it as evidence.

A confidence interval gives a range of plausible values for the true difference. The most important value to check is 0 because $p_1-p_2=0$ means the two population proportions are equal.

That leads to three quick cases on a number line.

Confidence intervals for $p_1-p_2$

Intervals entirely above 0 support $p_1>p_2$, intervals entirely below 0 support $p_1<p_2$, and intervals that include 0 do not give strong evidence of a difference.

Order matters the whole time:

• If the interval for $p_1-p_2$ is positive, population 1 likely has the larger proportion.
• If it is negative, population 2 likely has the larger proportion.
• If you reverse the order, you reverse the signs and flip the endpoints.

Example:

• For $p_1-p_2$, suppose the interval is $(0.03, 0.19)$
• Then for $p_2-p_1$, it becomes $(-0.19, -0.03)$

A full interpretation has to name:

• population 1
• population 2
• the response/success
• the subtraction order, $p_1-p_2$

Also, differences in proportions are often best said in percentage points.

• $0.10 = 10$ percentage points
• That does not automatically mean a 10% increase

Interpreting the Interval and the Confidence Level

A correct interpretation sounds like this:

“We are $C\%$ confident that the interval from $a$ to $b$ contains the true difference $p_1-p_2$, where $p_1$ is the proportion of ___ in population 1 and $p_2$ is the proportion of ___ in population 2.”

That interval is about population proportions, not sample proportions. The sample proportions were used to build it, but they are not what the interval is estimating.

The confidence level is about the method’s long-run success rate:

$$\text{In repeated random sampling, about } C\% \text{ of intervals from this method would capture } p_1-p_2.$$

That means:

• the parameter $p_1-p_2$ is fixed
• the samples change
• the interval changes from sample to sample
• any one interval either does contain the parameter or does not

Common wrong interpretations:

• “There is a $C\%$ chance the true difference is in this interval.”
• “$C\%$ of the data are in the interval.”
• “The sample proportions are in the interval.”
• “Each population proportion is between the endpoints.”

Using the Interval to Judge Claims

This is where the interval becomes evidence.

1. Identify the claim as a claim about $p_1-p_2$.
2. Check whether the claimed value is inside or outside the interval.
3. Match the conclusion to what the interval says.

Claims about whether the proportions differ

• Entire interval above 0
  • convincing evidence that $p_1>p_2$
  • also convincing evidence the population proportions differ
• Entire interval below 0
  • convincing evidence that $p_1<p_2$
  • also convincing evidence the population proportions differ
• Interval contains 0
  • insufficient evidence to conclude the population proportions differ
  • this does not mean they are equal

If 0 is exactly an endpoint, it still counts as contains 0.

Claims about a specific difference

Suppose a claim says $p_1-p_2=0.10$.

• If $0.10$ is inside the interval, that claim is plausible
• If $0.10$ is outside the interval, there is evidence against it

Claims about size

Suppose the claim is “$p_1$ exceeds $p_2$ by more than 0.05.”

• If the whole interval is above $0.05$, the data support that stronger claim
• If the interval includes values $\le 0.05$, you cannot support it

Writing the Conclusion in Context

Your conclusion has to match the sign and order of the interval.

Good AP Stats wording:

• “provides convincing evidence”
• “supports the claim”
• “insufficient evidence to conclude”

Example in context:

“We are 95% confident that the true difference in the proportion of adult residents who support the policy, $p_A-p_B$, is between 0.027 and 0.193. Because the entire interval is above 0, the data provide convincing evidence that support is higher in City A by about 2.7 to 19.3 percentage points.”

The study design matters too:

• Random samples let you generalize to those populations
• Randomized experiments can support cause-and-effect
• Observational comparisons can compare groups, but not claim causation

All of this only works if the interval procedure was valid when built. That means randomization, independence, the 10% condition, and large counts were met.

Common Mistakes and Test-Day Checks

Students lose points here for wording more than math.

• Mixing up $p_1-p_2$ and $p_2-p_1$
• Saying “different” when the interval contains 0
• Saying “equal” when the interval contains 0
• Forgetting to name both populations and the response
• Saying “percent” instead of “percentage points”
• Making a causal claim without random assignment