Say the Interval in Percentage Form
A confidence interval for a population proportion can be written as proportions, such as \((0.46,0.59)\), or as percentages, such as 46% to 59%. The values are equivalent: multiply each proportion endpoint by 100 to express it as a percentage. What matters most is that your interpretation makes clear which population and characteristic the interval concerns.
In Interpreting the Confidence Interval for a Proportion, you learned the standard interpretation: “We are \(C\)% confident that the true proportion of [population] who [characteristic] is between [lower endpoint] and [upper endpoint].” This tutorial practices expressing those endpoints as percentages and keeping the population proportion \(p\) distinct from the sample proportion \(\hat p\).
For example, if a sample proportion is \(\hat p=0.52\) and a confidence interval for \(p\) is \((0.46,0.58)\), the sample estimate is 52%, while the interval estimates that the population proportion is between 46% and 58%. The sample proportion is already known from the sample; it is not a value that the interval needs to estimate.
A clear percentage interpretation names the confidence level, population, characteristic, and both endpoints. One useful structure is: “We are [confidence level]% confident that between [lower percentage]% and [upper percentage]% of [population] have [characteristic].” Keep the population and characteristic specific. If the question is about households that recycle, do not write only “people who recycle.”
Keep the Parameter and Statistic in Their Proper Roles
The symbols help you choose accurate wording. Use \(\hat p\) when describing the sample result: “In the sample, 52% reported using the service.” Use \(p\) when describing the population quantity being estimated: “The interval estimates the proportion of all eligible residents who use the service.” A confidence statement refers to \(p\), even though the interval is calculated from sample data.
Converting to percentage form changes only the units used to display the numbers. For instance, an interval of \((0.46,0.58)\) is the same interval as 46% to 58%. Multiplying the endpoints by 100 does not change the confidence level, the population, or what the interval estimates.
In ordinary prose, you can avoid writing \(100p\) by saying “the population proportion is between 46% and 58%.” The word “proportion” refers to a number from 0 to 1, while a percentage expresses the same quantity per 100. Either form is appropriate if the values and units are clear.
Do not change the subject of the sentence midway. If the interval is for the percentage of all members of a population who have a characteristic, the endpoints do not describe a percentage of the sample, a range of individual outcomes, or a probability that applies separately to each person. They are plausible values for one population proportion.
Percentages and Percentage Points
The distance from the sample estimate to an interval endpoint is usually described in percentage points when the values are written as percentages. If the sample estimate is 52% and the interval extends from 46% to 58%, each endpoint is 6 percentage points from the estimate. This is an absolute difference between percentage values, not a relative percent change.
As explained in Reading Confidence Intervals from Survey Reports, a margin of error of 6 percentage points around an estimate of 52% gives endpoints of 46% and 58%. It would be incorrect to call this “a 6% increase” from the estimate to the upper endpoint: 6 percentage points is the difference between 52% and 58%. In a standard confidence-interval interpretation, you usually do not need to mention the margin of error at all; give the confidence level and the two endpoints.
Worked Examples
Worked Example: Convert Proportion Endpoints to Percentages
A fictional city survey produces a 95% confidence interval of \((0.462,0.588)\) for the proportion of the city’s adult residents who used public transit at least once during the past month. The sample included 240 randomly selected adults, of whom 126 reported using transit. Interpret the interval in percentage form and distinguish the sample result from the population estimate.
First, calculate the sample proportion. There were 126 successes among 240 sampled adults, so \(\hat p=126/240=0.525\). As a percentage, the sample estimate is \(0.525(100)=52.5\%\). This describes the surveyed adults, not all adults in the city.
Now convert the interval endpoints. The lower endpoint is \(0.462(100)=46.2\%\), and the upper endpoint is \(0.588(100)=58.8\%\). The interval in percentage form is 46.2% to 58.8%. As a quick check, the sample estimate of 52.5% falls between those endpoints.
A complete interpretation is: “We are 95% confident that between 46.2% and 58.8% of all adult residents of the city used public transit at least once during the past month.” The interval estimates the citywide population proportion \(p\). The 52.5% value is the sample proportion \(\hat p\) used as the estimate; it is not what the interval is describing as uncertain.
Worked Example: Interpret Endpoints Already Given as Percentages
A fictional technology company reports a 90% confidence interval from 31.8% to 40.2% for the proportion of its eligible customers who would recommend a new online support tool. The sample proportion was 36%. Write an interpretation and explain what each percentage describes.
The sample result is \(\hat p=36\%\), or \(0.36\) as a proportion. The reported endpoints are already percentages, so no conversion is needed. If written as proportions, they are \(31.8/100=0.318\) and \(40.2/100=0.402\). Thus, the same interval is \((0.318,0.402)\).
The estimate is centered in the reported interval: \((31.8\%+40.2\%)/2=72.0\%/2=36.0\%\). The lower endpoint is \(36.0\%-31.8\%=4.2\) percentage points below the estimate, and the upper endpoint is \(40.2\%-36.0\%=4.2\) percentage points above it. This confirms that the sample estimate lies at the center; it does not change the population quantity the interval estimates.
A suitable interpretation is: “We are 90% confident that between 31.8% and 40.2% of all eligible customers would recommend the new online support tool.” The 36% figure describes the sample estimate. The two endpoints give a range of plausible values for the population proportion. Do not replace the stated 90% confidence level with 95%.
Worked Example: Phrase an Interval for a Small Percentage
In a fictional regional survey, 37 of 500 randomly selected households reported having a home battery system. A 95% confidence interval for the proportion of all households in the region with a home battery system is \((0.051,0.097)\). Express the estimate and interval as percentages, then interpret them.
The sample proportion is \(\hat p=37/500=0.074\), which is \(0.074(100)=7.4\%\). The sample result is therefore 7.4% of surveyed households. Convert each interval endpoint separately: \(0.051(100)=5.1\%\) and \(0.097(100)=9.7\%\). The interval in percentage form is 5.1% to 9.7%.
The endpoints are consistent with the estimate’s position: \(7.4\%-5.1\%=2.3\) percentage points and \(9.7\%-7.4\%=2.3\) percentage points. This shows the interval is centered at the sample estimate, allowing for the displayed rounding. It does not mean that 5.1% to 9.7% of the sample had a home battery; the sample percentage is 7.4%.
A complete interpretation is: “We are 95% confident that between 5.1% and 9.7% of all households in the region have a home battery system.” Because the characteristic is relatively uncommon, keep the decimal conversion especially clear: \(0.051\) is 5.1%, not 0.051%.
A Reliable Wording Routine
Before writing your interpretation, identify what the interval is estimating and translate each endpoint into the requested units. Then check that your sentence refers to the same population and characteristic as the question. This brief routine helps prevent errors even when the interval itself has already been calculated.
Identify \(p\) as the population proportion of a specified group with a specified characteristic. Do not substitute the sample proportion \(\hat p\).
For proportion endpoints, multiply each endpoint by 100 to obtain percentages. If the endpoints already have percent signs, keep them as percentages.
Name the stated confidence level, population, characteristic, and lower and upper endpoints.
Make sure the sentence describes a population proportion, not the sample percentage or a range for individual people.
For example, the phrase “We are 95% confident that 46% to 59% of residents use the service” is incomplete if it does not say which residents or which service. A stronger sentence identifies both: “We are 95% confident that between 46% and 59% of adult residents of the district use the evening bus service.”
Common Mistakes and AP Exam Tips
- Writing an interval for \(\hat p\) instead of \(p\). The sample proportion is calculated from the observed sample. Say that the interval estimates the population proportion.
- Giving the sample result as the interpretation. “52.5% of the sample used transit” reports \(\hat p\); it does not interpret the interval. Include the population and both endpoints.
- Dropping or misplacing the percent sign. \(0.051\) is 5.1%, whereas \(0.051\%\) is \(0.00051\) as a proportion. Multiply a proportion by 100 once, then label the result with %.
- Mixing proportion and percentage units. Do not write that \(p\) is between 0.46% and 0.58% when the endpoints are \(0.46\) and \(0.58\). Those endpoints equal 46% and 58%.
- Calling percentage points “percent.” The difference from 52% to 58% is 6 percentage points. Avoid describing that absolute difference as a 6% relative increase.
- Leaving out context or confidence level. A full-credit sentence identifies the population, the characteristic, both endpoints, and the stated confidence level.
- Describing individual outcomes. The interval is for a population proportion, not a range in which individual people’s responses fall.
Key Takeaway
A one-proportion confidence interval can be expressed in proportions or percentages, but its target remains the population proportion \(p\). Convert endpoints by multiplying proportions by 100, then interpret them for the specified population and characteristic. Use \(\hat p\) for the observed sample estimate, and use percentage points—not percent—to describe an absolute difference between percentages.
Check Your Understanding
For each item, focus on the units and on whether the statement describes the sample or the population.
- A confidence interval is \((0.274,0.346)\) for the proportion of eligible residents who use a community garden. Write the endpoints as percentages.
- A sample proportion is \(\hat p=0.41\), and a confidence interval for \(p\) is \((0.36,0.46)\). Which value describes the sample, and which values estimate the population proportion?
- Write a contextual interpretation for a 90% confidence interval of 23% to 31% for the proportion of students at a school who participate in a music group.
- An estimate is 62%, and the upper endpoint is 68%. Describe the difference using the appropriate units.
- Explain why “We are 95% confident that between 5% and 10% of the sample has the characteristic” is not an appropriate interpretation of a confidence interval for \(p\).