Turning a Survey Summary into an Interval
Survey reports often compress an estimate into a short statement such as “58% support the proposal, with a margin of error of 4 percentage points.” To interpret that statement, identify the estimate at the center, determine the margin of error, and use them to reconstruct the interval. The interval estimates a population proportion, while the sample proportion is the statistic calculated from the survey sample.
As covered in Point Estimates and Margins of Error for a Proportion, the sample proportion \(\hat p\) is a point estimate of the population proportion \(p\). The margin of error is the distance from the point estimate to either endpoint of the confidence interval. For a report giving an estimate and a margin of error, the basic reconstruction is estimate plus or minus margin of error.
The lower endpoint is \(\hat p-\text{margin of error}\), and the upper endpoint is \(\hat p+\text{margin of error}\). If a report gives a percentage estimate and a margin in percentage points, you can subtract and add directly in percentage units. For example, 58% plus or minus 4 percentage points gives endpoints of 54% and 62%.
A percentage point is an absolute difference between percentages. Moving from 58% to 62% is an increase of 4 percentage points. It is not a 4% relative increase. When using formulas written in proportions, convert percentages to decimals: 58% is \(0.58\), and 4 percentage points is \(0.04\).
What to Extract—and What to Check
Read the report for four pieces of information: the population being described, the characteristic being counted, the estimate, and the margin of error. Also identify the confidence level, such as 90% or 95%. The confidence level matters because it describes the long-run capture rate of the interval method, as explained in Interpreting a 95% Confidence Level Correctly. Do not assume the confidence level is 95% if the report states a different level or does not give one.
If the report provides the number of surveyed individuals with the characteristic and the sample size, calculate \(\hat p=x/n\) to check the stated estimate. If it gives only a rounded percentage, treat that percentage as the reported estimate; the exact underlying sample proportion may have more decimal places. Similarly, reported endpoints or margins of error may be rounded, so a reconstructed interval may be approximate.
A published margin of error commonly summarizes sampling uncertainty for a stated confidence level, under the survey’s method. It does not automatically account for every possible source of error. Nonresponse, question wording, undercoverage, inaccurate answers, or other design problems can affect a survey in ways that a stated sampling margin of error may not capture. A narrow interval is not a guarantee that the survey is unbiased or representative.
Worked Examples
Worked Example: Estimate and Margin of Error in the Report
A fictional regional survey reports that 58% of sampled residents support adding a protected bicycle lane, with a margin of error of 4 percentage points at the 95% confidence level. The survey used a random sample of 600 residents from a region with 24,000 residents. Reconstruct and interpret the interval.
The report’s point estimate is \(\hat p=0.58\), and its margin of error is \(0.04\). In percentage units, subtracting and adding gives \(58\%-4\%=54\%\) and \(58\%+4\%=62\%\). In proportion units, the same calculation is \(0.58-0.04=0.54\) and \(0.58+0.04=0.62\). Thus, the reconstructed interval is \((0.54,0.62)\), or 54% to 62%.
The survey describes a random sample. The 10% condition holds because \(600\leq0.10(24{,}000)=2{,}400\). If the reported 58% is exact, it corresponds to \(x=0.58(600)=348\) residents supporting the lane, leaving \(600-348=252\) not supporting it. Both observed counts are at least 10, so the Large Counts condition for a one-proportion interval is met. These checks are consistent with using a Normal-based interval; they do not by themselves confirm every detail of the survey design.
A contextual interpretation is: We are 95% confident that between 54% and 62% of all residents in the region support adding a protected bicycle lane. The interval is about the population proportion, not the proportion in this particular sample, which the report estimates as 58%.
Worked Example: Check a Reported Percentage from Counts
A fictional community survey reports that 296 of 800 randomly selected adults favor extending library hours. Its summary says the estimate is 37%, with a margin of error of 3.5 percentage points at the 90% confidence level. The adult population is 40,000. Check the estimate, reconstruct the interval, and interpret it.
From the counts, \(\hat p=x/n=296/800=0.37\), or 37%. This agrees with the report. The number who did not favor the change is \(800-296=504\), and the sample proportion can also be checked by \(1-504/800=1-0.63=0.37\).
The margin of error is 3.5 percentage points, or \(0.035\) in proportion units. Subtracting and adding in percentage units gives \(37\%-3.5\%=33.5\%\) and \(37\%+3.5\%=40.5\%\). Equivalently, \(0.37-0.035=0.335\) and \(0.37+0.035=0.405\). The reconstructed interval is \((0.335,0.405)\), or 33.5% to 40.5%.
The sample is stated to be random. The 10% condition holds because \(800\leq0.10(40{,}000)=4{,}000\). The observed success and failure counts are 296 and 504, both at least 10, so the Large Counts condition for a one-proportion interval is met. The report’s stated 90% confidence level is important: do not silently describe this as a 95% interval.
We are 90% confident that between 33.5% and 40.5% of adults in the community favor extending library hours. The confidence level refers to the long-run performance of the interval method when it is used repeatedly under appropriate conditions. It does not mean there is a 90% probability that this particular, already calculated interval contains the fixed population proportion.
Worked Example: Recover the Estimate and Margin from Endpoints
A fictional report gives a 95% confidence interval from 42% to 50% for the proportion of households in a town that compost food scraps. It also states that a random sample was taken from the town’s 60,000 households: 414 of 900 sampled households compost. Recover the estimate and margin of error, then interpret the interval.
The sample proportion is \(\hat p=414/900=0.46\), or 46%. As a second check, \(900-414=486\) households did not compost, and \(1-486/900=1-0.54=0.46\). The estimate is therefore 46%.
For an interval given by its endpoints, the estimate is the midpoint and the margin of error is half the interval’s width. The midpoint is \((42\%+50\%)/2=92\%/2=46\%\). The width is \(50\%-42\%=8\) percentage points, and half of that is \(8/2=4\) percentage points. The same results in proportion units are \((0.42+0.50)/2=0.46\) and \((0.50-0.42)/2=0.04\). So the report’s interval can be written as 46% plus or minus 4 percentage points.
The conditions described are consistent with a one-proportion interval. The sample is random, and the 10% condition holds because \(900\leq0.10(60{,}000)=6{,}000\). The observed counts are 414 successes and \(900-414=486\) failures, both at least 10. The interval’s midpoint also agrees with the sample estimate calculated from the counts.
We are 95% confident that between 42% and 50% of households in the town compost food scraps. Here, the recovered margin of error describes the distance from the 46% sample estimate to either endpoint. It is not the total amount by which the survey result might differ from the truth for every possible reason.
Reading Reports Carefully
Reports do not all present results in the same format. One may give an estimate and a margin of error; another may give the lower and upper endpoints; a third may provide counts and sample size. The same underlying information can often be checked or reconstructed in more than one way. Keep the units consistent as you move between percentages and proportions.
| Report wording or information | What to identify or calculate |
|---|---|
| “58%, plus or minus 4 percentage points” | Estimate \(=58\%\); endpoints \(=54\%\) and \(62\%\). |
| 296 of 800 respondents | \(\hat p=296/800=0.37\), or 37%. |
| Interval from 42% to 50% | Midpoint \(=46\%\); margin of error \(=4\) percentage points. |
| “90% confidence” | Describe the method’s long-run capture rate as 90%, not 95%. |
Be alert to what the report says the percentage is about. “Support among surveyed adults” describes the sample statistic; “support among adults in the town” is the population quantity the interval aims to estimate. A strong interpretation names the population and the characteristic, and includes both endpoints and the stated confidence level.
Common Mistakes and AP Exam Tips
- Confusing percentage points with percent. If an estimate is 58% and the margin is 4 percentage points, the endpoints are 54% and 62%. Do not multiply 58% by 4% or describe the change as a 4% relative change.
- Using the sample estimate as the whole interval. The reported estimate is the center. Add and subtract the margin of error to get the endpoints.
- Forgetting to halve the width when recovering a margin. From endpoints of 42% and 50%, the width is 8 points but the margin of error is 4 points.
- Interpreting the interval as a range for individuals. The interval estimates a population proportion. It does not say that a particular person has a probability between the endpoints of supporting the characteristic.
- Changing the stated confidence level. If the report says 90%, interpret a 90% interval. Do not assume every published interval is 95%.
- Claiming the margin covers all survey error. A margin of error does not necessarily account for nonresponse, coverage problems, leading wording, or inaccurate answers.
- Leaving out context. A full-credit interpretation identifies the population and characteristic and uses the reported confidence level and interval endpoints.
Key Takeaway
A survey estimate and its margin of error reconstruct an interval by subtracting and adding the margin to the estimate. If the report gives endpoints instead, their midpoint is the estimate and half their width is the margin of error. Interpret the resulting range for the population proportion, use the confidence level the report states, and remember that a sampling margin of error is not a guarantee against every form of survey error.
Check Your Understanding
Use the report details in each question to identify or interpret the interval information.
- A report estimates that 64% of sampled customers would recommend a service, with a margin of error of 5 percentage points. What are the interval endpoints?
- A random sample of 500 students includes 185 who walk or bike to school. Calculate \(\hat p\) as a proportion and as a percentage.
- A report gives an interval from 28% to 36%. Find its midpoint estimate and margin of error.
- A report describes its interval as having a 90% confidence level. What does that confidence level mean over repeated samples, and what does it not mean for this one interval?
- Name one kind of survey error that a reported sampling margin of error may not include.