Topic 6.9 Notes – Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions

What a Confidence Interval for $p_1 - p_2$ Tells You

When you build a confidence interval for $p_1 - p_2$, you’re estimating the true difference between two population proportions.

• Parameter: $p_1 - p_2$
  (The true difference in proportions, in a specific order.)
• Statistic: $\hat{p}_1 - \hat{p}_2$
  (The observed difference from your samples.)

From Topic 6.8, the interval has the form:

$$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$$

That formula accounts for sampling variability, which is why we use an interval instead of just reporting the sample difference.

What the confidence level means

If you repeated the same random sampling process over and over, about C% of the intervals you construct would capture the true value of $p_1 - p_2$.

So for a 95% confidence interval:

• About 95% of intervals from repeated random samples would contain the true difference.
• The parameter is fixed. The interval is what changes from sample to sample.

This is one of the most commonly tested ideas. Avoid saying “There is a 95% probability the true difference is in this interval.” The true difference isn’t random.

Order matters

If you define the parameter as $p_1 - p_2$, your interpretation must follow that order.

If you reverse the groups, the entire interval changes sign. A positive interval can become negative just from switching labels.

How to Correctly Interpret the Interval

A full-credit interpretation has three parts:

1. The confidence level
2. A reference to the true population proportions
3. Clear context

Here’s what that sounds like:

“We are 95% confident that the true difference in the proportion of registered voters in City 1 who support Candidate A and the proportion of registered voters in City 2 who support Candidate A ( $p_1 - p_2$ ) is between 0.04 and 0.12.”

Notice:

• It says true difference.
• It names the groups in context.
• It keeps the subtraction order consistent.

Reading the interval

• Entirely positive → evidence that $p_1 > p_2$
• Entirely negative → evidence that $p_1 < p_2$
• Contains 0 → a difference of 0 is plausible

If 0 is inside the interval, you cannot rule out “no difference.”

Using the Interval to Justify a Claim

Confidence intervals can be used to evaluate claims about two proportions.

Step 1: Identify the claim

Common forms:

• “There is no difference.” → $p_1 - p_2 = 0$
• “There is a difference.” → $p_1 - p_2 \neq 0$
• “Group 1 has a higher proportion.” → $p_1 - p_2 > 0$
• “Group 1 has a lower proportion.” → $p_1 - p_2 < 0$

Step 2: Check whether 0 is in the interval

• 0 is in the interval
  • A true difference of 0 is plausible.
  • There is not convincing evidence of a difference.
  • In test language, you would fail to reject $H_0: p_1 - p_2 = 0$.
• 0 is not in the interval
  • A difference of 0 is not plausible.
  • There is convincing evidence of a difference.
  • You would reject $H_0$.

Step 3: Match the direction

Suppose a 95% CI for $p_1 - p_2$ is (-0.03, 0.08).

• 0 is inside.
• There is not convincing evidence that the two population proportions differ.

If the interval were (0.02, 0.11):

• Entirely positive.
• There is convincing evidence that the proportion in population 1 is greater than in population 2.

Always state your final conclusion in context, not just “reject” or “fail to reject.”

Connecting Confidence Intervals and Significance Tests

There’s a direct link between intervals and two-sided hypothesis tests.

A two-sided test at significance level $\alpha$ matches a $(1 - \alpha)$ confidence interval.

• 95% CI ↔ 5% significance test
• If the CI excludes 0 → $p\text{-value} < 0.05$
• If the CI includes 0 → $p\text{-value} > 0.05$

That’s why checking for 0 works.

This equivalence is for two-sided tests. If the claim is directional, you still use the interval’s sign to decide whether the data support that direction.