Quick Summary
This guide focuses on using a confidence interval for the difference between two population proportions (p₁ - p₂) to make a statistically sound conclusion. You will learn how to interpret this type of interval and, most importantly, how to use the value of zero as a critical benchmark to determine if there is convincing evidence of a real difference between the two groups being studied.
Key Concepts
The primary goal of a two-sample confidence interval for proportions is to estimate the true difference, p₁ - p₂, between two distinct populations. The interval provides a range of plausible values for this difference. The way we use this interval to make a claim is centered on one key question: Is zero a plausible value for the difference?
The Meaning of the Difference (p₁ - p₂):
If p₁ - p₂ = 0, it means p₁ = p₂. The two population proportions are equal; there is no difference between them.
If p₁ - p₂ > 0, it means p₁ > p₂. The proportion for the first population is greater than the second.
If p₁ - p₂ < 0, it means p₁ < p₂. The proportion for the first population is less than the second.
The "Zero Test": Justifying a Claim
A confidence interval gives us a range of plausible values for p₁ - p₂. We check if zero is inside this range.
Case 1: The interval is entirely positive.
Example: (0.02, 0.08)
Interpretation: All plausible values for the difference p₁ - p₂ are positive. This means it is not plausible that the difference is zero or negative.
Conclusion: We have convincing statistical evidence that p₁ - p₂ > 0, which means p₁ > p₂. There is a statistically significant difference.
Case 2: The interval is entirely negative.
Example: (-0.11, -0.03)
Interpretation: All plausible values for the difference p₁ - p₂ are negative. This means it is not plausible that the difference is zero or positive.
Conclusion: We have convincing statistical evidence that p₁ - p₂ < 0, which means p₁ < p₂. There is a statistically significant difference.
Case 3: The interval contains zero.
Example: (-0.04, 0.06)
Interpretation: The range of plausible values for the difference p₁ - p₂ includes positive values, negative values, and zero. Since zero is a plausible value, we cannot rule out the possibility that p₁ - p₂ = 0 (meaning p₁ = p₂).
Conclusion: We do not have convincing statistical evidence of a difference between the two population proportions. It is plausible that there is no difference.
[Image: A number line with three examples. The first interval (0.02, 0.08) is shown entirely to the right of 0. The second interval (-0.11, -0.03) is shown entirely to the left of 0. The third interval (-0.04, 0.06) is shown spanning across 0.]
Connecting to Hypothesis Tests
This method of checking for zero in a confidence interval is directly linked to a two-sided hypothesis test.
Null Hypothesis (H₀): p₁ - p₂ = 0 (There is no difference)
Alternative Hypothesis (Hₐ): p₁ - p₂ \neq 0 (There is a difference)
If a 95% confidence interval for p₁ - p₂ does not contain 0, it is equivalent to rejecting H₀ at a significance level of α = 0.05.
If a 95% confidence interval for p₁ - p₂ does contain 0, it is equivalent to failing to reject H₀ at a significance level of α = 0.05.
Key Vocabulary
Parameter of Interest: The true value being estimated, which for this topic is the difference in population proportions (p₁ - p₂).
Confidence Interval for a Difference of Proportions: A range of plausible values for the true difference between two population proportions, calculated from sample data.
Point Estimate: The single best guess for the parameter, calculated from the sample data. For this topic, it is the difference in sample proportions (p̂₁ - p̂₂).
Statistically Significant Difference: An observed difference between two groups that is too large to be reasonably attributed to random sampling variation alone. This is indicated when the confidence interval for the difference does not contain zero.
Plausible Value: Any value for the parameter that is contained within the calculated confidence interval. If a value is not in the interval, it is considered implausible at the given confidence level.
Calculator Tech (TI-84)
While this topic focuses on interpreting a pre-existing interval, you often need to calculate the interval first. The function to do this is .
To calculate a confidence interval for a difference of proportions:
Press
STAT.Navigate to the
TESTSmenu.Scroll down and select
B: 2-PropZInt....
You will be prompted for the following inputs:
x1: The number of successes in the first sample. Must be a whole number.
n1: The sample size of the first sample.
x2: The number of successes in the second sample. Must be a whole number.
n2: The sample size of the second sample.
C-Level: The desired confidence level, written as a decimal (e.g., 0.95 for 95% confidence).
Calculate: Select this and press
ENTER.
The calculator will output the confidence interval in the format , along with the sample proportions p̂₁ and p̂₂.
How to Show Work on the FRQ
On the AP exam, simply stating "yes, there's a difference" is not enough. You must provide a clear, structured justification based on the interval. This justification is the "Conclude" step of the four-step inference process.
FRQ Template for Justifying a Claim:
State the Interval: "The C% confidence interval for the true difference in proportions (p₁ - p₂) is (lower bound, upper bound)."
Check for Zero: "Because the value 0 is / is not contained within this interval..."
Link to Evidence: "...we do not have / have convincing statistical evidence of a difference..."
Conclusion in Context: "...in the true proportion of [describe population 1 in context] and the true proportion of [describe population 2 in context] at the [C%] confidence level."
If the interval is entirely positive or negative (0 is NOT in the interval): You can go one step further and state the direction of the difference.
Example (for an all-positive interval): "...We have convincing evidence that the true proportion of [population 1] is greater than the true proportion of [population 2]."
Example (for an all-negative interval): "...We have convincing evidence that the true proportion of [population 1] is less than the true proportion of [population 2]."
Practice Problems
Problem 1:
A pharmaceutical company is testing a new allergy medication. In a clinical trial, 250 subjects were randomly assigned to two groups. Group A received the new medication, and Group B received a placebo. Of the 125 subjects in Group A, 35 reported a significant reduction in symptoms. Of the 125 subjects in Group B, 22 reported a significant reduction in symptoms. A 95% confidence interval for the difference in the proportion of subjects who experience symptom reduction (Group A - Group B) was calculated to be .
Based on this interval, is there convincing evidence of a difference in the effectiveness of the new medication and the placebo? Justify your answer.
Solution:
Using the FRQ template:
State the Interval: The 95% confidence interval for the true difference in proportions of subjects who experience symptom reduction (Medication - Placebo) is (-0.011, 0.227).
Check for Zero: Because the value 0 is contained within this interval...
Link to Evidence: ...we do not have convincing statistical evidence of a difference...
Conclusion in Context: ...in the true proportion of subjects who experience symptom reduction between the new medication and the placebo. It is plausible that the medication has no effect compared to the placebo.
Problem 2:
A university's admissions office wants to know if there is a difference in the proportion of admitted students who enroll depending on whether they attend an on-campus "Admitted Students Day." They take a random sample of 200 students who attended the event and find that 130 of them enrolled. They take another random sample of 150 students who did not attend the event and find that 75 of them enrolled. Let p_A be the true proportion of all admitted students who attend the event and enroll, and p_N be the true proportion of all admitted students who do not attend and enroll. A 99% confidence interval for the difference p_A - p_N is .
Based on this interval, is there convincing evidence that attending the Admitted Students Day is associated with a higher enrollment rate? Justify your answer.
Solution:
Using the FRQ template:
State the Interval: The 99% confidence interval for the true difference in enrollment proportions (Attended - Did Not Attend) is (0.025, 0.275).
Check for Zero: Because the value 0 is not contained within this interval...
Link to Evidence: ...we have convincing statistical evidence of a difference...
Conclusion in Context: ...in the true proportion of students who enroll between those who attend the Admitted Students Day and those who do not. Furthermore, since the entire interval is positive, we have convincing evidence that the true proportion of students who enroll is higher for those who attend the event than for those who do not.
Common Mistakes to Avoid
Claiming "No Difference" Exists: When a confidence interval contains 0, you must state that you do not have convincing evidence of a difference. It is incorrect to state that you have proven there is no difference. Zero is a plausible value, but it's not the only one.
Forgetting Context: Your final conclusion must always be phrased in the context of the problem. Do not just say "we have evidence of a difference." You must specify a difference in what and between which groups.
Interpreting in Terms of Samples: The conclusion is always about the population proportions (p₁ and p₂), not the sample proportions (p̂₁ and p̂₂). We already know the sample proportions are different; the goal of inference is to see if that difference is significant enough to imply a difference in the larger populations.
Misstating the Confidence Level: Avoid saying "There is a 95% chance that the true difference is in the interval (a, b)." The correct interpretation is that we are 95% confident that the method we used to construct the interval will capture the true difference.
Confusing Direction: For an interval that does not contain zero, be careful to state the direction of the difference correctly. An all-positive interval (like in Problem 2) means p₁ - p₂ > 0, so p₁ > p₂. An all-negative interval would mean p₁ - p₂ < 0, so p₁ < p₂.