AP Statistics (AP Stats): Confidence Intervals for Difference of Two Proportions

AP Statistics – Confidence Intervals for Difference of Two Proportions

AP Statistics: Confidence Intervals for Difference of Two Proportions

Exam-Ready Study Guide

What This Is

A confidence interval for the difference of two proportions (p? – p?) estimates the true difference between two population proportions using sample data. This is essential on the AP exam because it allows us to compare two groups (e.g., treatment vs. control in a medical trial, success rates of two teaching methods, or defect rates in two factories). For example, a researcher might use this interval to determine if a new vaccine reduces infection rates more effectively than a placebo.

Key Terms & Formulas

• Two-proportion z-interval for p? – p?: [ (\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}} ]
• (\hat{p}_1, \hat{p}_2) = sample proportions
• (n_1, n_2) = sample sizes

• (z^*) = critical z-value (e.g., 1.96 for 95% confidence)

• Pooled sample proportion (for hypothesis tests, not intervals): [ \hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2} ]

• (X_1, X_2) = number of successes in each sample

• Conditions for inference (BINS):

• Binary? Outcomes are success/failure.
• Independent? Samples are independent (or 10% condition if sampling without replacement).
• Normal? (n_1\hat{p}_1 \geq 10), (n_1(1-\hat{p}_1) \geq 10), (n_2\hat{p}_2 \geq 10), (n_2(1-\hat{p}_2) \geq 10).

• SRS? Both samples are random (or treatments are randomly assigned).

• Calculator command (TI-84): 2-PropZInt(x?, n?, x?, n?, C-Level)

• (x_1, x_2) = number of successes

• (C-Level) = confidence level (e.g., 0.95 for 95%)

• Interpretation of a confidence interval: "We are [C-Level]% confident that the true difference in proportions (p? – p?) is between [lower bound] and [upper bound]."

• Hypotheses for a two-proportion z-test (not an interval, but related):

• (H_0: p_1 = p_2) (or (p_1 - p_2 = 0))
• (H_a: p_1 \neq p_2) (or (p_1 > p_2) or (p_1 < p_2))

Step-by-Step / Process Flow

How to construct a confidence interval for p? – p? on an FRQ:

1. State the parameter and context: Define (p_1) and (p_2) (e.g., "Let (p_1) = proportion of patients cured with Drug A, (p_2) = proportion cured with Drug B").

2. Check conditions (BINS):

3. Binary? Yes (success/failure).
4. Independent? Samples are independent (or 10% condition holds).
5. Normal? Verify (n_1\hat{p}_1 \geq 10), (n_1(1-\hat{p}_1) \geq 10), etc.

6. SRS? Samples are random (or treatments are randomly assigned).

7. Name the procedure: "We will construct a two-proportion z-interval for (p_1 - p_2)."

8. Compute the interval:

9. Use the formula or 2-PropZInt on the TI-84.

10. Example: 2-PropZInt(45, 100, 30, 100, 0.95)? (0.012, 0.288).

11. Interpret the interval in context: "We are 95% confident that the true difference in cure rates (Drug A – Drug B) is between 1.2% and 28.8%."

12. Answer the question (if applicable): If the interval is entirely positive, conclude (p_1 > p_2); if entirely negative, (p_1 < p_2); if it includes 0, there’s no significant difference.

Common Mistakes

• Mistake: Forgetting to check the Normal condition (e.g., (n\hat{p} \geq 10)). Correction: Always verify all four BINS conditions. The AP exam will deduct points for missing this.

• Mistake: Using the pooled proportion ((\hat{p}_c)) in the confidence interval formula. Correction: (\hat{p}_c) is only for hypothesis tests, not intervals. Use the separate sample proportions ((\hat{p}_1, \hat{p}_2)) in the interval formula.

• Mistake: Misinterpreting the interval as "the probability that (p_1 - p_2) is in the interval." Correction: The interval either contains (p_1 - p_2) or it doesn’t. The confidence level (e.g., 95%) refers to the method’s long-run success rate.

• Mistake: Ignoring the order of subtraction (e.g., (p_1 - p_2) vs. (p_2 - p_1)). Correction: Define (p_1) and (p_2) clearly at the start. The interval’s sign depends on the order.

• Mistake: Not using the correct critical value (e.g., using (t^) instead of (z^)). Correction: For proportions, always use (z^) (from invNorm on the TI-84). (t^) is for means.

AP Exam Insights

• Tricky distinction: The AP exam often tests whether to use a two-proportion z-interval or a two-sample t-interval. Proportions-z; means-t.
• Common FRQ setup: A scenario comparing two groups (e.g., "Does a new fertilizer increase crop yield more than the old one?"). You’ll need to:
• Define (p_1) and (p_2).
• Check conditions.
• Compute the interval.
• Interpret it in context.
• Calculator pitfall: The 2-PropZInt command requires number of successes (x), not proportions. If given (\hat{p}), multiply by (n) to get (x).
• Confidence level vs. interval: The exam may ask, "What does 95% confidence mean?" Answer: "If we repeated this process many times, 95% of the intervals would contain the true (p_1 - p_2)."

Quick Check Questions

1. Multiple Choice: A 90% confidence interval for (p_1 - p_2) is calculated as (-0.05, 0.12). Which of the following is the best interpretation? (A) There is a 90% probability that (p_1 - p_2) is between -0.05 and 0.12. (B) We are 90% confident that (p_1) is between 5% less and 12% more than (p_2). (C) The method used to construct the interval will capture the true difference 90% of the time. (D) (p_1) is significantly greater than (p_2) at the 10% level.

Answer: (B) Explanation: The interval is about the difference (p_1 - p_2), not probability. (C) is correct but less specific than (B).

1. FRQ Part: A study compares the proportion of students who pass an exam after using two different study methods. Method A has 80 out of 100 students passing, and Method B has 65 out of 100 passing.
2. Construct a 95% confidence interval for the difference in pass rates (Method A – Method B).
3. Interpret the interval in context.

Answer: - Conditions: Binary, independent samples, (n_1\hat{p}_1 = 80 \geq 10), (n_1(1-\hat{p}_1) = 20 \geq 10), (n_2\hat{p}_2 = 65 \geq 10), (n_2(1-\hat{p}_2) = 35 \geq 10), SRS assumed. - 2-PropZInt(80, 100, 65, 100, 0.95)? (0.015, 0.285). - Interpretation: We are 95% confident that the true difference in pass rates (Method A – Method B) is between 1.5% and 28.5%.

Last-Minute Cram Sheet

1. Formula: ((\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}})
2. Conditions: BINS (Binary, Independent, Normal, SRS).
3. Normal check: (n\hat{p} \geq 10) and (n(1-\hat{p}) \geq 10) for both samples.
4. Calculator: 2-PropZInt(x?, n?, x?, n?, C-Level) (use number of successes, not proportions).
5. Interpretation: "We are [C-Level]% confident that the true difference in proportions (p? – p?) is between [LB] and [UB]."
6. Order matters: Define (p_1) and (p_2) clearly (e.g., "treatment – control").
7. Don’t pool for intervals! Use (\hat{p}_1) and (\hat{p}_2) separately.
8. 10% condition: Check if sampling without replacement (e.g., "10n-N" for both samples).
9. Critical value: (z^*) for proportions (e.g., 1.96 for 95% confidence).
10. Hypothesis test vs. interval: Intervals estimate (p_1 - p_2); tests check if (p_1 = p_2).