AP Syllabus focus:
‘Using the confidence interval for the difference between population proportions to justify a claim involves assessing whether the interval provides evidence to support the claim within the context of the study. - It is essential to recognize that the confidence interval can suggest with a certain level of confidence, say C%, that the true difference between the population proportions lies within the calculated range, thus providing a basis for decision-making regarding the claim.’
Confidence intervals for the difference between two population proportions help determine whether sample evidence supports a contextual claim about how two groups differ, using interval bounds to guide inference.
Understanding Claims in the Context of a Confidence Interval
Justifying a claim using a confidence interval requires interpreting how the interval estimate relates to the claim being evaluated. A confidence interval for a difference in proportions is built around the statistic , which estimates how two groups compare. When using the interval to support or refute a claim, the key task is to connect its range of plausible values to the study’s research question.
A claim refers to a statement about the true population difference, such as asserting that one proportion is higher, lower, or not different from another. Because the true difference is unknown, the confidence interval provides the best available evidence for judging whether the claim is justified.
Key Idea: What the Confidence Interval Represents
A C% confidence interval gives a range of values believed, with C% confidence, to capture the true difference between the population proportions. This interpretation is rooted in the idea of repeated sampling: if many intervals were computed from many samples, approximately C% of them would contain the true difference.
The figure shows repeated samples and their confidence intervals, with most intervals capturing the true parameter and a few missing it, illustrating the meaning of C% confidence in repeated sampling. Source.
The specific interval from a given study is used to assess which population differences are reasonable and which are not.
When using the interval to justify a claim, the focus is on whether the entire interval, part of the interval, or none of the interval supports the claim's direction or magnitude.
Using the Interval to Evaluate Different Types of Claims
Claims Suggesting a Direction of Difference
Many research questions propose a directional claim, such as one group having a higher or lower proportion than another. To evaluate such a claim:
Examine whether all plausible values in the interval lie on one side of 0, which represents no difference.
If the interval lies entirely above 0, it supports a claim that the first population proportion exceeds the second.
If it lies entirely below 0, it supports a claim that the first proportion is less than the second.
If the interval includes 0, the sample does not provide strong evidence of a directional difference.
This diagram illustrates that when a confidence interval crosses the line of no difference, the evidence is insufficient to conclude that one group truly differs from the other, matching the logic used when interpreting intervals for differences in proportions. Source.
Claims Suggesting No Difference
Some studies aim to justify the claim that two populations do not differ in proportion. For such claims:
The interval must contain 0, showing that no difference is a plausible value.
If the interval does not include 0, then the evidence does not support the claim of no difference.
A single interval does not prove equality; instead, it indicates whether equality is consistent with the observed data.
Interpreting Claims in Context
Because the syllabus emphasizes contextual communication, it is important to link the mathematical interpretation of the interval back to the real-world setting. Students must clearly identify:
what two proportions represent,
how the interval relates to these proportions, and
what the captured values imply about the claim being tested.
Contextual framing ensures that the justification addresses the population characteristics rather than just numerical reasoning.
How Confidence Level Relates to Claims
Confidence level affects how strongly conclusions can be framed:
Higher confidence levels produce wider intervals, capturing more plausible values but reducing precision.
This diagram compares confidence levels and shows that higher levels correspond to wider shaded regions, reinforcing that greater confidence requires a broader interval of plausible parameter values. Source.
Lower confidence levels yield narrower intervals, tightening the estimate but providing less assurance.
Justifying a claim must acknowledge that the chosen C% level influences how compelling the interval-based evidence is.
Relationship Between Interval Width and Strength of Evidence
A narrower interval gives more precise information about the likely size of the difference, which may strengthen support for a claim if the interval consistently supports it. Conversely, wide intervals may weaken justification because they contain a broader range of plausible differences, reflecting greater uncertainty.
Definition of the Interval Components
A confidence interval for a difference of proportions relies on several terms used when justifying claims.
Population Proportion Difference: The unknown parameter representing the true difference between two population proportions, often written as .
Once interpreted, this parameter anchors any claim made about how populations differ.
A confidence interval also depends on a specific estimated difference.
Sample Proportion Difference: The statistic , representing the observed difference between sample proportions used as a point estimate of the population difference.
This estimate forms the center of the computed interval and guides the claim evaluation process.
Linking Evidence to the Claim
When justifying a claim:
State how the interval aligns with the hypothesized direction or magnitude.
Reference where 0 falls relative to the interval, since the presence or absence of 0 often determines inference.
Connect the range of plausible values to real-world implications about the two populations.
Claims must be evaluated based solely on what the interval logically supports, using its endpoints to determine which population differences remain plausible under the chosen confidence level.
FAQ
The context determines whether the size and direction of all plausible values in the interval meaningfully support the claim. Even if an interval excludes 0, the implied difference may be too small to matter practically.
Context also guides whether the estimated difference aligns with the realities of the population, such as feasibility, cost, or clinical or educational importance.
A point estimate reflects only the observed difference, whereas the interval accounts for sampling variation. If the interval is wide or includes 0, the data allow multiple plausible true differences, some consistent with no effect.
This highlights why decisions must rely on the entire interval rather than a single estimated value.
A narrow interval indicates high precision, making it easier to support directional claims because fewer values remain plausible.
A wide interval suggests high uncertainty and may prevent justification of a claim even if all values lie above or below 0. Policymakers or researchers may require a minimum effect size for a claim to be actionable.
They must consider whether the difference is practically meaningful as well as statistically in one direction.
To strengthen justification, researchers often reference:
• the magnitude of the smallest plausible difference,
• whether the effect is large enough to matter in the real world,
• and whether similar studies have reported consistent findings.
Yes. Small changes in sample size, sample proportions, or chosen confidence level can shift interval bounds enough to alter whether 0 is included.
This variability reinforces that intervals reflect uncertainty and that a single study’s justification should be framed cautiously and contextually.
Practice Questions
Question 1 (1–3 marks)
A study compares the proportion of customers who prefer Brand A between two independent regions. A 95% confidence interval for p1 − p2 is reported as (−0.04, 0.07).
Based on this interval, determine whether the study provides evidence that the two regions differ in their preference for Brand A. Justify your answer using the interval.
Question 1
• 1 mark: States that the interval includes 0.
• 1 mark: Correctly concludes that there is no evidence of a difference between the two regions.
• 1 mark: Provides justification referencing plausibility of no difference (0 being within the interval).
Question 2 (4–6 marks)
A researcher investigates whether a new teaching method changes the proportion of students who pass a statistics module compared with the traditional method. Independent random samples are taken, and a 90% confidence interval for p1 − p2 (new method minus traditional method) is calculated as (0.03, 0.18).
(a) Explain what this interval suggests about the true difference in population proportions.
(b) Using the interval, assess whether the researcher’s claim that the new method results in a higher pass rate is justified.
(c) Comment on how the choice of a 90% confidence level affects the strength of the researcher’s conclusion.
Question 2
(a)
• 1 mark: States that the entire interval is above 0.
• 1 mark: Identifies that the true proportion for the new method is likely higher than that for the traditional method.
(b)
• 1 mark: Concludes that the researcher’s claim is supported.
• 1 mark: Justifies conclusion based on the interval containing only positive values (indicating a higher pass rate with the new method).
(c)
• 1 mark: Notes that a 90% confidence level yields a narrower interval than higher levels and therefore provides less certainty.
• 1 mark: Explains how this reduced confidence should moderate the strength of the conclusion, even though the interval still supports the claim.
