AP Statistics Topic 3.4: Justifying a Claim Based on a Confidence Interval for a Population Proportion

01 · What can the interval support?

The concept in 60 seconds

A school takes a simple random sample of $500$ students from its $10{,}000$ students, without replacement. Of the sampled students, $320$ prefer an earlier lunch. The sample proportion is $\hat{p} = 0.64$ .

After checking the conditions, a one-sample $z$ -interval gives an approximate $95\%$ confidence interval of $(0.5979, 0.6821)$ for $p$ , the proportion of all students at this school who prefer an earlier lunch.

Visual guide 1:

1

interval can support different claims

Estimate $64\%$ in the sample

$320$ of $500$ students prefer earlier lunch.

95\%

interval $59.79\%\text{–}68.21\%$

A range of plausible values for the whole-school proportion.

Compare the claimAll above $50\%$ ; crosses $60\%$

Supports a majority. Leaves the stronger $60\%$ threshold uncertain.

A claim needs a comparison with the whole interval, not just the center. The school population proportion is unknown.

The school asks two questions:

Does a majority prefer earlier lunch? Yes, the interval provides convincing evidence: every value in the interval is above $0.50$ .
Do more than $60\%$ prefer it? The interval does not provide convincing evidence for that stronger claim: it includes values just below $0.60$ as well as above it.

The key habit: Compare the claim with the whole interval. The estimate $0.64$ alone does not tell you how much uncertainty remains.

Scenarios are fictional teaching examples. The repeated-sampling chart is a documented simulation; the other charts show calculations from stated sample values.

Quick check: does “

64\%

in the sample” prove “more than

60\%

in the population”?

No. $\hat{p}$ is $1$ sample result. The $95\%$ interval contains population proportions below $0.60$ , so this interval does not support that population claim convincingly.

02 · Turn endpoints into a meaningful sentence

Interpret the interval

A confidence interval estimates a population parameter. Here the parameter is a proportion, not an individual student’s response and not the already observed sample proportion.

Sentence pattern: “We are [confidence level] confident that the proportion of [population] who [response category] is between $L$ and $U$ .”

School interpretation: “We are $95\%$ confident that the proportion of all students at this school who prefer an earlier lunch is between $0.5979$ and $0.6821$ .”

You can say about $59.79\%\text{ to }68.21\%$ instead. Both versions describe the same range on different scales. Include the confidence level, both endpoints, the population and the response category.

Known sample result, unknown population value.
Quantity	Meaning	School example
$p$	Fixed unknown population proportion	Proportion of all $10{,}000$ students who prefer earlier lunch
$\hat{p}$	Observed sample proportion	$\frac{320}{500}= 0.64$
$L$ and $U$	Lower and upper interval endpoints	About $0.5979$ and $0.6821$ at $95\%$ confidence
Confidence level	Long-run confidence level of the procedure	$95\%$
$p_{0}$ or $k$	A proposed value or comparison threshold	$0.60$ or $0.50$ , depending on the claim

Keep the sample and population separate

The sample proportion is known: exactly $\frac{320}{500}= 0.64$ for these responses. We do not need an interval to learn that value. The interval uses sample information to estimate the unknown proportion in the whole school.

“ $95\%$ of students prefer earlier lunch” is a different statement and is not supported by this interval. “ $95\%$ of student responses are between $59.79\%$ and $68.21\%$ ” also makes no sense: each student’s response is a category, such as yes or no.

Write for the actual target population: A sample from this school supports inference about this school when the sampling conditions are met. It does not automatically support a statement about all students in the country.

Quick check: repair “We are

95\%

confident that

\hat{p}

lies between

0.5979

and

0.6821

.”

Replace $\hat{p}$ with the population proportion $p$ , defined in context. $\hat{p} = 0.64$ is already known from the sample and is the center of this interval.

03 · The population value stays fixed

Explain confidence and coverage

Imagine repeatedly taking random samples of the same size from the same population and constructing an interval by the same method each time. The sample proportion changes, and the interval endpoints change. The population proportion $p$ stays fixed.

Under suitable conditions, a $95\%$ confidence procedure produces intervals that capture the true $p$ in about $95\%$ of repeated samples in the long run. Some intervals miss it.

Visual guide 2: changing intervals, fixed population proportion

Simulation: $5{,}000$ independent binomial samples, $n = 200$ and

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Justifying a Claim Based on a Confidence Interval for a Population Proportion

By the end of this lesson, you should be able to:

The concept in 60 seconds

Interpret the interval

Keep the sample and population separate

Explain confidence and coverage

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The concept in 60 seconds

Interpret the interval

Keep the sample and population separate

Explain confidence and coverage

Main Menu

Quick Links​

Quick Links