Using an Interval to Evaluate a Claimed Proportion

Use the Interval to Evaluate a Claim

In Interpreting the Confidence Interval for a Proportion, we used a one-proportion \(z\)-interval to estimate an unknown population proportion \(p\). The interval also helps assess a specific proposed value. For example, if a company claims that 30% of its customers use a particular service, we can ask whether \(0.30\) is consistent with an interval calculated from a random sample of its customers.

The basic comparison is direct: check whether the claimed value falls between the interval’s lower and upper endpoints. A value inside the interval is plausible given the sample and the interval method. A value outside the interval is not among the plausible values supported by that interval. This is an evidence-based judgment, not proof that a claim is true or false.

Key rule: If a claimed population proportion \(p_0\) is between the endpoints of a confidence interval for \(p\), the claim is plausible at that confidence level. If \(p_0\) is outside the interval, the interval provides evidence against the claim, assuming the data and interval conditions are appropriate.

The comparison is meaningful only when the interval is appropriate for the population, characteristic, and sample. As in Constructing a One-Proportion \(z\)-Interval by Hand, check that the data come from a random sample or suitable random process, check the 10% condition when sampling without replacement, and verify the Large Counts condition using the observed success and failure counts. The interval’s endpoints are estimates; use enough digits during the comparison to avoid making a decision based on premature rounding.

What “Plausible” Does—and Does Not—Mean

When a claim falls inside a confidence interval, the sample does not give convincing evidence against that particular value at the level represented by the interval. It does not establish that the value is correct. Other values inside the interval are plausible too, and the true population proportion remains unknown.

When a claim falls outside the interval, it is not supported by the range of values produced by this interval method. The farther the claim is beyond an endpoint, the farther it is from the interval’s estimates, but the interval-membership decision itself is simply inside or outside. Avoid describing an outside value as impossible: sampling variation and limitations of the data still matter.

Important distinction: A confidence interval gives a range of plausible values for a population parameter under a method with a stated long-run confidence level. It does not assign a probability that one particular value inside the interval is true. As explained in Interpreting a 95% Confidence Level Correctly, the confidence level describes the long-run capture rate of the interval method.

For a standard two-sided one-proportion \(z\)-interval, excluding a claimed value is closely connected to a two-sided test at the corresponding significance level. The formal connection between confidence intervals and two-sided tests is the subject of the next tutorial. Here, focus on making and explaining the interval comparison.

Worked Example: Is a 30% Claim Plausible?

A fictional meal-delivery company claims that 30% of its customers order a vegetarian meal at least once a month. In a random sample of 250 customers, 78 report doing so. The customer population contains 12,000 people. Use a 95% one-proportion \(z\)-interval to assess the claim.

State.
Let \(p\) be the true proportion of the company’s customers who order a vegetarian meal at least once a month. The claim is \(p_0=0.30\). We will compare this claimed value with a 95% confidence interval for \(p\).

Plan and check conditions.
The sample is described as random. The 10% condition is met because \(250\leq0.10(12{,}000)=1{,}200\). There are \(x=78\) successes and \(n-x=250-78=172\) failures; both counts are at least 10, so the Large Counts condition is met. Use a one-proportion \(z\)-interval with \(z^*=1.96\) for 95% confidence.

Do.
The sample proportion is \(\hat{p}=78/250=0.312\). The estimated standard error is \(\sqrt{(0.312)(0.688)/250}=\sqrt{0.0008586}\approx0.02930\), using \(0.0008586\) rounded from the product and division. The margin of error is \(1.96(0.02930)\approx0.05743\). Thus, the interval is \(0.312\pm0.05743\), or approximately \((0.2546,0.3694)\). The claimed value \(0.30\) lies between these endpoints.

Conclude.
The 30% claim is plausible given this sample and the 95% confidence interval. The interval does not provide convincing evidence against \(p=0.30\), but it does not prove that the true proportion is exactly 30%.

When the Claimed Value Falls Outside

The same comparison applies when the claimed value is below the lower endpoint or above the upper endpoint. State which side of the interval contains the claim, then explain what that means in context. Do not merely write “the claim is wrong.” A careful conclusion says that the interval provides evidence against the claimed population proportion, under the sampling and interval conditions.

Worked Example: A Claim Above the Interval

A fictional neighborhood internet provider claims that 55% of its subscribers use automatic data backups. A random sample of 200 subscribers includes 90 who use automatic backups. There are 8,000 subscribers in the population. Construct a 95% confidence interval and evaluate the claim.

Let \(p\) be the true proportion of the provider’s subscribers who use automatic data backups. The sample is random. The 10% condition is met because \(200\leq0.10(8{,}000)=800\). The observed counts are 90 users and \(200-90=110\) nonusers, so both are at least 10 and the Large Counts condition is met.

The sample proportion is \(\hat{p}=90/200=0.45\). Using \(z^*=1.96\), the estimated standard error is

SE_{\hat{p}}=\sqrt{\frac{(0.45)(0.55)}{200}} =\sqrt{0.0012375} \approx0.03518

The margin of error is \(1.96(0.03518)\approx0.06895\), so the interval is \(0.45\pm0.06895\), or approximately \((0.3811,0.5189)\). The claimed value \(0.55\) is greater than the upper endpoint, \(0.5189\).

The 55% claim is not plausible given this sample and the 95% confidence interval. The interval provides evidence against the claim that 55% of the provider’s subscribers use automatic backups. This conclusion relies on the stated random sampling and the conditions checked above; it does not rule out every possible source of error in the survey.

Confidence Level Can Affect the Comparison

For the same sample, a higher confidence level produces a wider interval, as discussed in How Confidence Level Affects Interval Width. A wider interval can include a claimed value that a narrower interval excludes. Therefore, when evaluating a claim, name the confidence level and use the endpoints from that specific interval.

This change does not mean that the sample data changed. The estimate stays centered at \(\hat{p}\); the critical value changes, which changes the margin of error and endpoints. A claim near an endpoint is especially likely to be classified differently when the confidence level changes.

Worked Example: One Claim, Two Confidence Levels

Use the sample from the internet-provider example: 90 of 200 subscribers report using automatic backups, so \(\hat{p}=0.45\), and \(SE_{\hat{p}}\approx0.03518\). Assess a claim that \(p=0.515\) using both a 90% and a 95% confidence interval.

The random-sample description, 10% condition, and Large Counts condition are the same as before: \(200\leq800\), with 90 successes and 110 failures. For 90% confidence, \(z^*\approx1.645\). The margin of error is \(1.645(0.03518)\approx0.05787\), giving

0.45\pm0.05787=(0.3921,0.5079)

The claim \(0.515\) is above the 90% interval’s upper endpoint, \(0.5079\), so it is outside that interval. For 95% confidence, \(z^*=1.96\), and the margin of error is \(1.96(0.03518)\approx0.06895\). The interval is approximately \((0.3811,0.5189)\), which includes \(0.515\).

Thus, the claim is outside the 90% interval but inside the 95% interval. The 95% interval is wider because its method is designed to capture the true proportion in a larger share of repeated samples. Report the confidence level with the evaluation; do not describe the two comparisons as if they used identical intervals.

Common Mistakes and AP Exam Tips

Checking only whether the claim equals the sample proportion. The sample proportion is the interval’s center, but the claim should be compared with both endpoints. A claim different from \(\hat{p}\) can still fall inside the interval.
Calling an inside claim proven or accepted. An inside value is plausible given the data and interval method. It is not established as the true value, and other values in the interval are plausible too.
Calling an outside claim impossible. Say the interval provides evidence against the claim. Keep the conclusion tied to the sample, the confidence level, and the checked conditions.
Ignoring the direction of the comparison. Identify whether the claimed value is below the lower endpoint or above the upper endpoint. This makes the reasoning clear and helps prevent endpoint mix-ups.
Rounding too early. Keep extra digits while calculating the standard error and endpoints. Round the reported interval consistently, and make sure a claim near an endpoint is compared using unrounded or sufficiently precise endpoints.
Leaving out context or conditions. A full-credit response identifies \(p\), describes the population and characteristic, supports the interval conditions, compares the claim with the interval, and states the conclusion in context.

AP Exam Tip: Write both interval endpoints, then explicitly state whether the claimed value is inside or outside them. Finish with a contextual sentence: “The claim that [proportion] is [value] is plausible given the [confidence level]% interval,” or “The interval provides evidence against the claim that [proportion] is [value].” Avoid saying the claim has been proved or disproved with certainty.

Key Takeaway

To evaluate a claimed population proportion, construct or use an appropriate confidence interval and compare the claimed value with both endpoints. A value inside the interval is plausible given the sample and method; a value outside provides evidence against the claim. The conclusion depends on the interval’s confidence level and conditions, and it does not prove a parameter value.

Key takeaway: Inside the interval means plausible, not proven; outside the interval means evidence against the claim, not impossibility. Always make the comparison at the stated confidence level and in context.

Check Your Understanding

Use the interval-membership rule and the condition checks in this tutorial to answer each question.

A 95% confidence interval for a population proportion is \((0.24,0.38)\). Is a claim of \(p=0.30\) plausible? Explain what the interval does and does not establish.
A 90% confidence interval is \((0.41,0.53)\). A stated claim is \(p=0.55\). Is the claim inside or outside the interval, and what is an appropriate conclusion?
In a random sample of 150 people from a population of 5,000, there are 18 successes and 132 failures. Does the Large Counts condition hold for a one-proportion \(z\)-interval?
Why can the same claimed proportion be inside a 95% interval but outside a 90% interval calculated from the same sample?
Give one reason why an outside claim should not be described as impossible.