AP Syllabus focus:
‘Understanding that a confidence interval may or may not contain the true population proportion, reflecting the random variation inherent in sample data. - The interpretation of confidence intervals involves stating the degree of confidence (C%) that the interval captures the true population proportion. - It's emphasized that, with repeated sampling, approximately C% of such intervals would contain the population proportion, highlighting the concept of confidence in statistical inference.’
A confidence interval for a proportion provides a statistical estimate of where the true population proportion is likely to fall, reflecting sample variability and repeated-sampling behavior.
Understanding Confidence Intervals for a Proportion
Interpreting a confidence interval correctly is essential for linking sample evidence to claims about a population. A confidence interval is a range of plausible values for a population proportion based on a single sample. Because every sample is affected by random variation, the interval constructed from that sample may or may not contain the true population proportion. This inherent uncertainty is central to meaningful statistical inference.
When statisticians refer to a C% confidence interval, they express how frequently the method used to generate that interval will capture the true population proportion in the long run. This perspective emphasizes the behavior of the method, not the probability that the specific interval contains the true value.
Repeated Sampling and Confidence
The syllabus stresses that interpreting a confidence interval requires understanding its behavior across repeated samples of the same size from the same population. If we repeatedly draw samples and compute confidence intervals using the same procedure:
Each line represents a confidence interval from a different sample, with the vertical reference line marking the true parameter. Most intervals contain the true value, illustrating the long-run C% capture rate of the confidence-interval method. Source.
Approximately C% of those intervals will contain the true population proportion.
The remaining (100 − C)% will miss the true value due to ordinary sampling variability.
Any single interval produced from one sample is either correct (contains the parameter) or incorrect (does not), even though we do not know which is the case.
This focus on long-run performance reinforces that confidence is about the method, not the specific interval.
Components of Confidence Interval Interpretation
Interpreting a confidence interval for a proportion relies on several foundational ideas emphasized by AP Statistics.
Confidence Level
The confidence level is the percentage of all intervals, constructed using the same procedure, that would capture the true population proportion. A higher confidence level reflects greater certainty in the method but does not alter the underlying population proportion itself.
The shaded region shows the confidence level C% as the middle area under the normal curve, with the tails representing α/2. This illustrates how confidence levels correspond to probability areas used in constructing confidence intervals. Source.
Sampling Variability
Sampling variability refers to the natural fluctuations in sample statistics from one sample to the next.
Sampling Variability: The natural variation in sample statistics, such as sample proportions, that occurs because different random samples yield different results.
This variability explains why each confidence interval differs slightly and why not all intervals capture the true population proportion.
A key part of interpreting intervals is recognizing that such variability is expected and does not imply error in the sampling method.
True Population Proportion
The true population proportion is a fixed, unknown parameter describing the population. While sample data inform us about this proportion, they do not change its value. A confidence interval estimates a range of plausible values for this fixed parameter, based on evidence from a particular sample.
Correct Interpretation of a C% Confidence Interval
A correct interpretation always follows the same structure:
It refers to the method’s long-run success rate rather than the probability a specific interval is correct.
It is phrased in terms of confidence in the procedure, not in the particular interval itself.
It connects the sample result to the broader population under study.
A C% confidence interval means that we used a method that, in repeated sampling, produces intervals that capture the true population proportion about C% of the time. This does not guarantee success for the specific interval obtained.
Avoiding Misinterpretations
Students often confuse interval interpretation with statements about probability or about future samples. Important misinterpretations to avoid include:
Saying the probability that the true proportion lies within the interval is C%. The true proportion is not random.
Concluding that C% of sample observations fall within the interval, which is unrelated to the concept.
Suggesting that the interval captures C% of the population values. Confidence intervals describe parameters, not individuals.
Recognizing these common errors supports deeper conceptual understanding.
Why Confidence Intervals May or May Not Contain the True Proportion
Because each sample provides only one estimate, some intervals inevitably miss the true proportion:
Each horizontal segment represents a 95% confidence interval from a different sample. Most intervals cross the true parameter line, while a few do not, illustrating why a single interval may or may not contain the population proportion. Source.
This possibility is not a flaw of the method; it is a consequence of drawing samples from a population where variation is unavoidable. A single confidence interval reflects the randomness of the particular sample used.
Interpreting a Confidence Interval in Context
To interpret a confidence interval properly within a real-world scenario, include:
The confidence level, clearly stated.
The parameter of interest, described in population terms.
The understanding that the interpretation refers to the method, not the specific interval alone.
For example, one would interpret a C% confidence interval by explaining that the interval was constructed using a method that captures the true population proportion in about C% of repeated samples and that the obtained interval provides a reasonable range of plausible values for that proportion based on the sample collected.
FAQ
Non-overlapping intervals indicate that the samples produced noticeably different estimates, but this does not necessarily imply a real difference in population proportions.
Sampling variability alone can generate intervals that do not overlap, especially when sample sizes are small.
Overlapping or non-overlapping intervals should not be used as a formal test of difference; they merely suggest whether estimates appear distinct.
Choosing the level beforehand ensures the interval reflects a pre-specified level of uncertainty, avoiding bias based on the sample’s appearance.
Selecting the level after observing the data undermines the meaning of confidence because the long-run success rate no longer corresponds to a fixed procedure.
Predetermined confidence levels help maintain objectivity and comparability across studies.
A wider interval increases the likelihood of containing the true proportion, but this comes at the cost of reduced precision.
Reliability depends on the analytical goal:
• If the priority is certainty, a wider interval helps.
• If the priority is precise estimates, a narrower interval may be preferable.
Thus, wider intervals are not inherently better; they serve a different purpose.
Larger samples tend to produce intervals that vary less from sample to sample, making the interpretation of confidence more stable.
With small samples, intervals fluctuate more dramatically, increasing the chance that a single interval may be unrepresentative.
More stable intervals strengthen the practical usefulness of repeated-sampling interpretations.
The interval contains values consistent with the observed data under the chosen confidence method, but it does not guarantee that all plausible values fall within it.
Some values outside the interval may still be reasonable, especially if the sample is small or contains sampling irregularities.
Confidence intervals summarise evidence rather than listing every possible plausible value.
Practice Questions
Question 1 (1–3 marks)
A school surveys a random sample of students to estimate the proportion who support extending the lunch break. The resulting 95% confidence interval for the true proportion is (0.42, 0.58).
a) Explain, in context, what this 95% confidence interval means.
b) The school claims that a majority of students support extending the lunch break. Based on the interval, comment on whether this claim is supported.
Question 1
a) (2 marks)
• 1 mark: States that the confidence level refers to the method, not the specific interval.
• 1 mark: Correct contextual interpretation, e.g., “We are 95% confident that the true proportion of students who support extending the lunch break lies between 0.42 and 0.58.”
b) (1 mark)
• 1 mark: Recognises that 0.50 is inside the interval, so the interval does not provide strong evidence that a majority (more than half) support the change.
Question 2 (4–6 marks)
A charity conducts a random survey to estimate the proportion of local residents who regularly volunteer. Using the sample data, the researchers construct a 90% confidence interval for the true proportion and report it as (0.18, 0.27).
a) Describe what the phrase “90% confidence” means in terms of repeated sampling.
b) A researcher interprets the interval by saying, “There is a 90% chance that the true proportion lies between 0.18 and 0.27.” Explain why this interpretation is incorrect.
c) Another researcher claims that exactly 90% of residents volunteer regularly. Use the interval to assess this claim.
d) The charity argues that the interval is too wide and wants a narrower one. Give one method (without calculation) that would produce a narrower interval.
Question 2
a) (2 marks)
• 1 mark: States that confidence refers to the long-run success rate of the interval procedure.
• 1 mark: Correctly describes that about 90% of intervals from repeated random samples would contain the true population proportion.
b) (2 marks)
• 1 mark: Identifies that the true proportion is fixed, not random.
• 1 mark: Explains that the probability statement is about the method, not the specific interval.
c) (1 mark)
• 1 mark: States that the claim is not supported because 0.90 is far outside the reported interval (0.18, 0.27).
d) (1 mark)
• 1 mark: Suggests a valid method for narrowing the interval, such as increasing the sample size or lowering the confidence level.
