Quick Summary
This guide will equip you to use a confidence interval as a powerful tool for statistical decision-making. You will learn how to interpret a given confidence interval for a population proportion to determine if a specific claim about that proportion is plausible. By the end of this lesson, you will be able to confidently justify a conclusion about a claim by checking whether the claimed value falls inside or outside the calculated interval and articulating that conclusion in the context of the problem.
Key Concepts
The core idea of this topic is to connect the concept of a confidence interval—a range of plausible values for a parameter—to the process of evaluating a claim. Instead of running a full significance test, we can often use a pre-existing confidence interval to make a similar judgment.
The Logic: Plausible vs. Not Plausible
A confidence interval provides a range of values for the true population proportion, p, that are considered plausible based on our sample data. We can use this range to test a claim about p. Let's call the claimed or hypothesized value p₀.
The Central Question: Is the claimed value, p₀, a plausible value for the true population proportion, p, given our sample evidence?
The Decision Rule: We check if p₀ falls inside or outside our calculated confidence interval.
[Image: A number line showing a confidence interval from a lower bound to an upper bound. Below it, two scenarios are depicted. Scenario A shows a point for the claimed value p₀ located inside the interval. Scenario B shows a point for the claimed value p₀ located outside the interval.]
Case 1: The Claimed Value is INSIDE the Interval
If the claimed value, p₀, falls within the bounds of the confidence interval, it is considered a plausible value.
What this means: Our sample data does not provide strong evidence to contradict the claim. The claim is consistent with our findings.
The Conclusion: We state that we do not have convincing statistical evidence that the true population proportion is different from the claimed value.
Example: A software company claims that 70% of users are satisfied with a new update. A survey of 200 users results in a 95% confidence interval for the true proportion of satisfied users of (0.65, 0.73).
The claimed value is p₀ = 0.70.
Since 0.70 is inside the interval (0.65, 0.73), the claim is plausible.
Our conclusion: We do not have convincing evidence that the true proportion of satisfied users is different from 70%.
Case 2: The Claimed Value is OUTSIDE the Interval
If the claimed value, p₀, falls outside the bounds of the confidence interval, it is considered not a plausible value.
What this means: Our sample data provides strong evidence to contradict the claim. The claim is inconsistent with our findings.
The Conclusion: We state that we do have convincing statistical evidence that the true population proportion is different from the claimed value.
Example: A national pollster claims that 50% of voters support a certain candidate. A random sample of 1,000 voters yields a 95% confidence interval for the true proportion of supportive voters of (0.52, 0.58).
The claimed value is p₀ = 0.50.
Since 0.50 is outside the interval (0.52, 0.58), the claim is not plausible.
Our conclusion: We have convincing evidence that the true proportion of voters who support the candidate is different from 50%. In fact, since the entire interval is above 0.50, we have evidence that their support is greater than 50%.
Connection to Two-Sided Significance Tests
This method of using a confidence interval to evaluate a claim is directly related to a two-sided significance test.
A C% confidence interval contains all the plausible values for a parameter.
A two-sided significance test with a significance level of α = 1 - C (as a decimal) will test a single one of these values (the null hypothesis).
The Rule of Thumb:
If a value p₀ is not in a 95% confidence interval, then you would reject the null hypothesis H₀: p = p₀ in a two-sided test at the α = 0.05 significance level.
If a value p₀ is in a 95% confidence interval, then you would fail to reject the null hypothesis H₀: p = p₀ in a two-sided test at the α = 0.05 significance level.
This connection holds for other confidence levels as well (e.g., 99% CI corresponds to α = 0.01, and a 90% CI corresponds to α = 0.10).
Key Vocabulary
Confidence Interval: An interval of plausible values for an unknown population parameter, calculated from sample data. The interval has an associated confidence level.
Plausible Value: A value for a population parameter that is believable based on the sample data. Any value inside a confidence interval is considered a plausible value.
Claimed Value (p₀): A specific, pre-existing value asserted for a population proportion. This is the value we are testing for plausibility.
Convincing Statistical Evidence: The standard phrasing used to indicate that the sample results are strong enough to reject a claim. This is used when a claimed value falls outside the confidence interval.
Point Estimate (p̂): The sample proportion, calculated as x/n. It serves as the best single guess for the true population proportion and is always the center of the confidence interval.
Calculator Tech (TI-84)
No major calculator functions are required for this topic. The primary skill is interpreting a pre-calculated confidence interval.
The calculation of the interval itself is covered in Topic 6.2. As a reminder, you can calculate a one-proportion z-interval using the following command:
STAT -> TESTS -> A:1-PropZInt...
You would need to input `x$ (number of successes), (sample size), and the (confidence level).
How to Show Work on the FRQ
On the AP Exam, justifying a claim using a confidence interval requires a clear, two-step conclusion. You are typically given the interval or the data to calculate it. Your job is to write the conclusion. Use this template to ensure you earn full credit.
Template for Justifying a Claim:
State the Decision (The "Link"): Explicitly compare the claimed value to the calculated interval.
- "Because the claimed value of [p₀] is [inside / outside] the 95% confidence interval of ([lower bound], [upper bound]), ..."
State the Conclusion in Context (The "Conclusion"): Write a formal conclusion based on your decision, using the language of statistical evidence and including the context of the problem.
- "...we [do not have / have] convincing statistical evidence that the true proportion of [describe the population and success attribute in context] is different from [p₀]."
Example using the template:
Scenario: A city claims that 60% of its residents support a new recycling program. A 95% confidence interval based on a sample of residents is (0.63, 0.69).
FRQ Response:
- "Because the claimed value of 0.60 is outside the 95% confidence interval of (0.63, 0.69), we have convincing statistical evidence that the true proportion of residents who support the new recycling program is different from 0.60."
Practice Problems
Problem 1:
A large university claims that 80% of its undergraduate students are satisfied with the campus dining options. The student newspaper conducts a survey of a simple random sample of 250 undergraduate students and finds that 190 of them are satisfied. A 99% confidence interval for the true proportion of all undergraduate students who are satisfied with campus dining is calculated to be (0.691, 0.829).
Does this interval provide convincing evidence that the true proportion of satisfied students is different from the university's claim of 80%? Justify your answer.
Solution:
Using the FRQ template:
State the Decision: The university's claimed proportion of satisfied students is p₀ = 0.80. We compare this value to the calculated 99% confidence interval, which is (0.691, 0.829). The value 0.80 is between 0.691 and 0.829.
State the Conclusion in Context:
"Because the university's claimed value of 0.80 is inside the 99% confidence interval of (0.691, 0.829), we do not have convincing statistical evidence that the true proportion of all undergraduate students who are satisfied with campus dining is different from 0.80. The university's claim is a plausible value."
Problem 2:
A pharmaceutical company develops a new flu vaccine and claims that it is effective for more than 90% of the population. In a clinical trial, a random sample of 2,000 volunteers were given the vaccine, and 1,850 of them did not get the flu. A 95% confidence interval for the proportion of the population for whom the vaccine is effective is (0.914, 0.936).
Does this confidence interval support the company's claim? Justify your answer.
Solution:
The company's claim is that the true proportion p is more than 0.90. We can evaluate this by looking at the plausible values in our interval.
State the Decision: The company's claim is that p > 0.90. The calculated 95% confidence interval is (0.914, 0.936). Every single plausible value in this interval (from 91.4% to 93.6%) is greater than 0.90.
State the Conclusion in Context:
"Because the entire 95% confidence interval of (0.914, 0.936) lies completely above the claimed threshold of 0.90, we have convincing statistical evidence that the true proportion of the population for whom the vaccine is effective is greater than 0.90. The data supports the company's claim."
Self-Correction Note: While the standard template works for testing if a proportion is different from a value, for a one-sided claim like "more than 90%", we check if the entire interval is above (or below) the claimed value.
Common Mistakes to Avoid
"Accepting" the Claim: When a claimed value is inside the interval, it's a common mistake to say "we accept the claim" or "this proves the claim is true." Never use the word "accept" or "prove." The correct phrasing is that we "do not have convincing evidence to reject the claim" or that the claim is "plausible." An interval contains many plausible values, not just one true one.
Misstating the Conclusion: A frequent error is confusing the logic. Students might say "because 0.55 is in the interval, we have evidence the true proportion is different from 0.55." This is backward. Inside = Plausible = No evidence of a difference.Outside = Not Plausible = Evidence of a difference.
Forgetting Context: A conclusion like "Because 0.50 is outside the interval, we have convincing evidence the true proportion is not 0.50" will not receive full credit. You MUST include the context from the problem. For example: "...the true proportion of voters who support the new law is not 0.50."
Ignoring the Entire Interval for One-Sided Claims: For claims like "more than 50%," it's not enough that the point estimate (p̂) is above 50%. The entire interval of plausible values must be above 50% to provide convincing evidence for the claim. If the interval was (0.48, 0.56), it contains values both above and below 0.50, so we would not have convincing evidence of a majority.