AP Syllabus focus:
‘A confidence interval provides a range of values used to support a claim about a population proportion. When interpreting a confidence interval, it's crucial to reference the sample and the broader population it represents, ensuring that claims are grounded in the context of the specific study.’
A confidence interval allows statisticians to evaluate whether sample evidence supports a claim about a population proportion, offering a structured way to connect sample results to broader population statements.
Making Claims Using Confidence Intervals
Confidence intervals are essential tools for determining whether sample data provide enough support for a claim about a population proportion. A claim is a statement about the value of the true proportion in the population, and the confidence interval helps evaluate whether that claim is reasonable based on the sample results. Because sample data are subject to random variation, the interval provides a range of plausible values rather than a single point estimate. This aligns directly with the syllabus emphasis on grounding claims in both the sample and the population it represents.
Understanding the Purpose of Claims
When a study produces a confidence interval, that interval must be interpreted specifically in the context of the question being researched. A claim may assert that the population proportion equals some value, exceeds a value, or lies within a particular range. The central purpose of the interval is to determine whether the claim is supported, not supported, or inconsistent with the sample evidence.
Key Ideas for Claim Evaluation
Students must recognize that a confidence interval:
Represents a plausible range of values for the true population proportion.
Should always be tied back to the study population.
Can support a claim only when the entire claim is consistent with the interval.
Cannot prove a claim true; it can only provide evidence that is consistent or inconsistent with it.
Interpreting an Interval in Context
Interpreting a confidence interval requires referencing both the sample and the population. The syllabus emphasizes that interpretations must remain anchored in the actual context. This means describing what the interval suggests for the population characteristic being studied, not simply restating numerical endpoints.
Components of a Strong Interpretation
A contextually grounded interpretation:
Mentions the confidence level, the population, and the proportion being estimated.
Avoids claiming that the interval contains the true parameter with certainty.
Clarifies that the method—if repeated many times—produces intervals capturing the true proportion approximately C% of the time.
A bar chart showing sample estimates with 95% confidence intervals as vertical error bars. The intervals display the plausible ranges of population values associated with each estimate. Although these intervals are shown for means, the same structure applies to confidence intervals for proportions when evaluating statistical claims. Source.
Claims Supported by Confidence Intervals
A claim is considered supported when the value asserted by the claim lies within the confidence interval. If the claim proposes a single value, the task is to check whether that number falls between the interval bounds. If the claim proposes a range, all values in the claim’s range must fall inside the interval for it to be fully supported.
Types of Claims and Their Alignment With Intervals
Common claim structures include:
Point claims, such as “The true proportion is 0.42.”
Comparative claims, such as “More than half of the population has this characteristic.”
Range claims, such as “The proportion is between 0.35 and 0.45.”
Each type of claim requires matching the proposed value or range against the interval.
Claims Not Supported by Confidence Intervals
A claim is not supported when it contradicts the plausible values for the population proportion. If the confidence interval does not include the value asserted by the claim, the sample provides insufficient evidence to support that claim. This does not prove the claim false; rather, it means the sample evidence does not align with it.
Situations Leading to Lack of Support
A claim may fail to be supported when:
The claimed value lies entirely outside the interval.
Only part of a claimed range falls within the interval.
The direction of the claim (greater than, less than) is inconsistent with interval bounds.
A sentence explaining how the sample results relate to the population supports clarity when evaluating unsupported claims.
Using Confidence Intervals to Frame Statistical Evidence
Confidence intervals supply a structured basis for reasoning about population parameters. Because intervals represent uncertainty and variation in sampling, they strengthen claims only when the interval aligns with the proposed population statement. Strong statistical reasoning requires students to relate the claim to the entire interval, not merely to the sample proportion itself.
Guidelines for Using Intervals to Justify Claims
To justify a claim effectively:
Identify the parameter clearly, using population language.
Examine whether the confidence interval contains or supports the value(s) specified by the claim.
Connect the reasoning explicitly to the context of the study.
Recognize that the interval reflects uncertainty and sample variability.
A brief reflection on the direction and magnitude of the interval can help reinforce how the evidence relates to the claim being evaluated.
A set of 20 confidence intervals illustrating long-run coverage: most intervals contain the true parameter, while one does not. This visual demonstrates how confidence levels express the proportion of intervals expected to succeed across repeated samples. Although the graphic is general rather than proportion-specific, the same reasoning applies to intervals for population proportions when evaluating claims. Source.
Communicating Claims and Evidence Clearly
Because statistical claims are often used in real-world decisions, clear communication is essential. Students should present claims and supporting evidence using precise, context-rich statements. Confidence intervals help structure this communication by giving a mathematically justified range that guides interpretation.
Elements of Effective Communication
Use population-focused language.
Reference the confidence level when relevant.
State whether the claim is supported, not supported, or uncertain based on interval placement.
Maintain consistency with the scope and limits of the original study.
Confidence intervals thus serve as a bridge connecting sample-based findings to broader population claims, enabling statistically sound and contextually grounded reasoning.
FAQ
A confidence interval focuses on identifying a range of plausible values for the population proportion, while a hypothesis test evaluates evidence against a specific hypothesised value.
When making claims, an interval allows you to check directly whether the claimed value lies inside or outside the plausible range, offering a more intuitive comparison. A hypothesis test, by contrast, relies on a p-value and a significance level, which are not part of this subsubtopic.
Differences in sample size, confidence level, or sampling variability can produce intervals with different widths or centres.
For example:
• A smaller sample produces a wider interval, which may include or exclude a claimed value purely due to increased uncertainty.
• A higher confidence level widens the interval and may alter whether a claim appears supported.
Yes. Even reasonable claims may fall outside the observed interval because of sampling variability.
This reflects the fact that any one interval, even at a high confidence level, has a chance of missing the true proportion. An unsupported claim does not imply the claim is impossible; only that the sample evidence does not align with it.
Confidence interval interpretation always refers to a specific population, not generic percentages.
Clear contextualisation ensures:
• The claim is evaluated for the correct population.
• The conclusion avoids implying certainty.
• The reasoning refers to the characteristic being measured, rather than the numbers alone.
If multiple claims fall within the interval, the sample evidence does not distinguish between them.
In such cases:
• The interval supports each claim as plausible.
• Additional data or a larger sample may be required to refine the estimate.
• Emphasise that the interval reflects uncertainty, not a definitive value.
Practice Questions
Question 1 (1–3 marks)
A study reports that a 95% confidence interval for the proportion of adults in a town who support a new recycling programme is (0.41, 0.55). A local councillor claims that the majority of adults in the town support the programme.
Based on the confidence interval, comment on whether the councillor’s claim is supported by the data.
Question 1
1 mark: States whether the claim is supported or not.
Example: The claim is not supported.
1 mark: Justifies using the position of 0.50 relative to the interval.
Example: 0.50 (a majority) is within the interval, but the interval also contains values below 0.50.
1 mark: Provides clear reasoning linked to population inference.
Example: Because the interval includes values below 0.50, the data do not provide evidence that a majority support the programme.
Question 2 (4–6 marks)
Researchers conduct a survey to estimate the proportion of students at a college who regularly use a bike for commuting. They obtain a 90% confidence interval of (0.18, 0.27).
A campus organisation claims that at least 30% of the student population commute by bike.
(a) Using the confidence interval, assess whether the claim is supported by the data.
(b) Explain, in context, what the 90% confidence level means.
(c) Briefly state one limitation of using this confidence interval to justify the organisation’s claim.
Question 2
(a): 2 marks
1 mark: Correctly states whether the claim is supported (it is not supported).
1 mark: Uses the interval to justify: 0.30 lies outside the interval (0.18, 0.27).
(b): 2 marks
1 mark: States that the confidence level refers to the long-run proportion of intervals capturing the true population proportion.
1 mark: Places this in context (students at the college and bike commuting).
(c): 1–2 marks
1 mark: Identifies a valid limitation (e.g., sampling bias, non-response, poor representativeness).
1 mark: Explains how this limitation affects the claim (e.g., the interval may not reflect the true population proportion if the sample is not representative).
