Confidence Intervals for Population Proportions
A confidence interval uses sample data to estimate an unknown population parameter while explicitly representing sampling uncertainty.
For a population proportion—such as a presidential approval rating—the interval provides a range of plausible values rather than reporting only one sample percentage.
Population Proportion and Sample Proportion
Suppose the target population consists of all likely voters, and let\[ p \]
denote the proportion who approve of the president’s job performance.
Assume, for illustration, that the true population proportion is\[ p=0.60. \]
In a real poll, this value is unknown. Estimating it is the purpose of collecting the sample.
Suppose a random sample of\[ n=1{,}000 \]
likely voters is selected. Let\[ \hat{p} \]
denote the sample proportion who approve.
If 580 of the 1,000 sampled voters approve, then\[ \hat{p} = \frac{580}{1{,}000} = 0.58. \]
The sample estimate is therefore 58%.
Indicator-Variable Representation
Define an indicator variable for each sampled voter:\[ X_i= \begin{cases} 1, & \text{if voter }i\text{ approves},\\ 0, & \text{otherwise}. \end{cases} \]
If the population approval proportion is \(p\), then\[ P(X_i=1)=p \]
and\[ P(X_i=0)=1-p. \]
The expected value is\[ E(X_i)=p. \]
The variance is\[ \operatorname{Var}(X_i)=p(1-p), \]
so the standard deviation is\[ \boxed{ \sigma=\sqrt{p(1-p)} }. \]
When \(p=0.60\),\[ \sigma = \sqrt{0.60(0.40)} = \sqrt{0.24} \approx 0.4899. \]
Thus, the correct population standard deviation of the 0–1 indicators is approximately 0.49.
Sampling Distribution of the Sample Proportion
The sample proportion is the average of the indicator variables:\[ \hat{p} = \frac{X_1+X_2+\cdots+X_n}{n}. \]
Its expected value is\[ E(\hat{p})=p. \]
Under independent sampling, its standard error is\[ \boxed{ SE(\hat{p}) = \sqrt{\frac{p(1-p)}{n}} }. \]
For \(p=0.60\) and \(n=1{,}000\),\[ SE(\hat{p}) = \sqrt{ \frac{0.60(0.40)}{1{,}000} }. \]
Therefore,\[ SE(\hat{p}) = \sqrt{0.00024} \approx 0.01549. \]
Expressed in percentage points,\[ SE(\hat{p}) \approx 1.55\text{ percentage points}. \]
This is the typical scale on which sample approval percentages vary across repeated samples of 1,000 voters.
Normal Approximation
When the expected numbers of successes and failures are sufficiently large, the sampling distribution of \(\hat{p}\) is approximately normal:\[ \hat{p} \approx N\left( p,\frac{p(1-p)}{n} \right). \]
A common guideline is\[ np\geq10 \]
and\[ n(1-p)\geq10. \]
For \(n=1{,}000\) and \(p=0.60\),\[ np=600 \]
and\[ n(1-p)=400. \]
The normal approximation is excellent under the independent Bernoulli sampling model.
From Standard Error to a Confidence Interval
The standard normal distribution satisfies\[ P(-1.96\leq Z\leq1.96)\approx0.95. \]
Therefore,\[ P\left( -1.96 \leq \frac{\hat{p}-p} {\sqrt{p(1-p)/n}} \leq 1.96 \right) \approx0.95. \]
Rearranging gives the idea that\[ \hat{p} \]
usually falls within approximately 1.96 standard errors of \(p\). Equivalently, \(p\) usually lies within approximately 1.96 standard errors of \(\hat{p}\).
This motivates an interval of the form\[ \text{estimate} \pm \text{critical value}\times\text{standard error}. \]
For a 95% confidence interval,\[ \boxed{ \hat{p}\pm1.96\,SE(\hat{p}) }. \]
The quantity\[ 1.96\,SE(\hat{p}) \]
is the margin of error.
The Unknown Standard Error Problem
The exact standard error contains the unknown population proportion:\[ SE(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}. \]
Because \(p\) is the quantity being estimated, it cannot normally be inserted into the confidence interval calculation.
The simplest solution is to replace \(p\) with the sample proportion \(\hat{p}\):\[ \widehat{SE}(\hat{p}) = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }. \]
This produces the familiar Wald confidence interval:\[ \boxed{ \hat{p} \pm z^* \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } }. \]
For a 95% confidence level,\[ z^*=1.96. \]
Example: A Sample Approval Rating of 58%
Suppose\[ \hat{p}=0.58 \]
and\[ n=1{,}000. \]
The estimated standard error is\[ \widehat{SE}(\hat{p}) = \sqrt{ \frac{0.58(0.42)}{1{,}000} }. \]
Calculate the product:\[ 0.58(0.42)=0.2436. \]
Therefore,\[ \widehat{SE}(\hat{p}) = \sqrt{0.0002436} \approx 0.01561. \]
This is approximately 1.56 percentage points.
The 95% margin of error is\[ 1.96(0.01561) \approx 0.0306. \]
Thus, the confidence interval is\[ 0.58\pm0.0306. \]
The lower endpoint is\[ 0.58-0.0306=0.5494, \]
and the upper endpoint is\[ 0.58+0.0306=0.6106. \]
Therefore, the approximate 95% confidence interval is\[ \boxed{ (0.549,\ 0.611) }. \]
Expressed as percentages:\[ \boxed{ 54.9\%\text{ to }61.1\% }. \]
Rounding the critical value to 2 and the standard error to 1.6 percentage points gives the rough interval\[ 58\%\pm3.2\text{ percentage points}, \]
or approximately\[ 54.8\%\text{ to }61.2\%. \]
The small difference between these versions is caused by rounding.
Why the Interval Is Called a Confidence Interval
After a particular sample has been observed and its interval calculated, the population proportion \(p\) is fixed. The interval either contains \(p\) or it does not.
Under the conventional frequentist interpretation, it is therefore not strictly correct to say:\[ P(p\text{ is in this realized interval})=0.95. \]
The randomness belongs to the sampling process and the interval-producing procedure, not to the fixed population parameter.
Before sampling, the endpoints are random because they depend on \(\hat{p}\). After sampling, the endpoints become fixed numbers.
Correct Frequentist Interpretation
Imagine repeating the following procedure many times:
- Select a random sample of 1,000 voters.
- Calculate the sample approval proportion.
- Construct a 95% confidence interval using the same method.
- Record whether the interval contains the true population proportion.
- Repeat.
In the long run, approximately 95% of the intervals produced by a valid 95% confidence procedure will contain the true value \(p\).
Approximately 5% will miss it.
The 95% describes the long-run coverage rate of the interval-building procedure.
It does not mean that 95% of the population lies inside the interval. The interval concerns a parameter, not individual voters.
What “95% Confident” Means
A common report is:
We are 95% confident that the population approval rating is between 54.9% and 61.1%.
This means that the interval was produced by a method designed to cover the true population proportion in approximately 95% of repeated applications, assuming the statistical conditions are satisfied.
It does not mean:
- There is a 95% probability that this fixed interval contains \(p\).
- 95% of voters have approval values between 54.9% and 61.1%.
- There is a 95% chance the sample was collected correctly.
- The result is 95% likely to be free of bias.
- The president’s approval will remain in this range in the future.
Confidence Level
The confidence level determines the long-run coverage rate.
Common confidence levels and normal critical values include:
| Confidence level | Critical value \(z^*\) |
|---|---|
| 80% | 1.282 |
| 90% | 1.645 |
| 95% | 1.960 |
| 98% | 2.326 |
| 99% | 2.576 |
A higher confidence level requires a larger critical value and therefore produces a wider interval.
For the same sample:\[ 99\%\text{ interval} \]
is wider than\[ 95\%\text{ interval}, \]
which is wider than\[ 90\%\text{ interval}. \]
Greater confidence comes at the cost of lower precision.
Confidence Level Versus Confidence-Interval Width
The approximate interval width is\[ 2z^* \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }. \]
The interval becomes wider when:
- The confidence level increases.
- The estimated variability increases.
- The sample size decreases.
It becomes narrower when:
- The confidence level decreases.
- The sample size increases.
- The proportion approaches 0 or 1, under the Wald formula.
Effect of Sample Size
The estimated standard error is\[ \widehat{SE}(\hat{p}) = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }. \]
It decreases according to\[ \frac{1}{\sqrt{n}}. \]
To cut the margin of error in half, the sample size must be multiplied by four.
For a proportion near 0.5, the approximate 95% margin of error is\[ 1.96\sqrt{\frac{0.25}{n}} = \frac{0.98}{\sqrt{n}}. \]
In percentage points, this is approximately\[ \frac{98}{\sqrt{n}}. \]
Examples include:
| Sample size | Approximate maximum 95% margin of error |
|---|---|
| 400 | 4.9 percentage points |
| 1,000 | 3.1 percentage points |
| 2,500 | 2.0 percentage points |
| 10,000 | 1.0 percentage point |
These figures account only for random sampling error under the basic model.
Why the Total Population Size Often Has Little Effect
For independent sampling with replacement, the standard error depends on \(n\), not directly on the population size \(N\):\[ SE(\hat{p}) = \sqrt{ \frac{p(1-p)}{n} }. \]
A sample of 1,000 can therefore have approximately the same sampling precision whether the target population contains one million or one hundred million people.
When sampling without replacement, the exact standard error includes the finite population correction:\[ SE(\hat{p}) = \sqrt{ \frac{p(1-p)}{n} } \sqrt{ \frac{N-n}{N-1} }. \]
If \(n\) is tiny relative to \(N\), then\[ \sqrt{ \frac{N-n}{N-1} } \approx1. \]
For a poll of 1,000 voters drawn from a population numbering in the millions, the correction is negligible.
Confidence Intervals Do Not Measure Every Source of Error
The confidence interval describes random sampling uncertainty under the assumed design. It does not automatically account for:
- Selection bias
- Undercoverage
- Nonresponse bias
- Question wording
- Response-order effects
- Social-desirability bias
- Misclassification
- Incorrect likely-voter models
- Data-processing errors
- Changes in public opinion after the poll
- Dependence created by cluster sampling
A narrow interval can be precisely wrong if the sampling or measurement process is biased.
Complex Survey Designs
Real opinion polls often use:
- Stratification
- Clustering
- Unequal selection probabilities
- Survey weights
- Post-stratification
- Likely-voter adjustments
These features can make the standard error different from the simple random-sampling formula.
The design effect is commonly defined as\[ DEFF = \frac{ \operatorname{Var}_{\text{actual design}}(\hat{p}) }{ \operatorname{Var}_{\text{SRS}}(\hat{p}) }. \]
An effective sample size can be approximated by\[ n_{\text{eff}} = \frac{n}{DEFF}. \]
If\[ DEFF>1, \]
the effective sample size is smaller than the nominal sample size, and the confidence interval should be wider than the basic formula suggests.
Limitations of the Wald Interval
The interval\[ \hat{p} \pm z^* \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \]
is easy to calculate but can perform poorly when:
- The sample is small.
- \(\hat{p}\) is near 0 or 1.
- The expected success or failure count is small.
- The interval extends below 0 or above 1.
A population proportion must satisfy\[ 0\leq p\leq1, \]
but the Wald interval does not automatically respect these bounds.
Wilson Score Interval
The Wilson score interval often provides better coverage than the Wald interval.
Its center is\[ \frac{ \hat{p}+\frac{z^{*2}}{2n} }{ 1+\frac{z^{*2}}{n} }, \]
and its half-width is\[ \frac{ z^* }{ 1+\frac{z^{*2}}{n} } \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} + \frac{z^{*2}}{4n^2} }. \]
For \(n=1{,}000\) and \(\hat{p}=0.58\), the Wilson interval is very close to the Wald interval because the sample is large and the estimated proportion is not near a boundary.
For smaller or more extreme samples, the difference can be important.
Margin of Error
A confidence interval is often written as\[ \text{estimate}\pm\text{margin of error}. \]
For the basic 95% proportion interval,\[ ME = 1.96 \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }. \]
In the polling example,\[ ME\approx0.0306, \]
or approximately 3.1 percentage points.
Thus, the result can be reported as\[ 58.0\%\pm3.1\text{ percentage points}. \]
This margin of error normally reflects sampling variability only. It should not be interpreted as a bound on total survey error.
Example of Repeated Intervals
Suppose the true population approval rating is 60%. Repeated samples might produce intervals such as:\[ (56.1\%,62.2\%), \]\[ (58.0\%,64.1\%), \]\[ (54.9\%,61.1\%), \]\[ (50.7\%,56.8\%). \]
The first three intervals contain 60%; the fourth does not.
A valid 95% procedure is designed so that, over many repetitions, approximately 95% of the intervals contain the true proportion and approximately 5% do not.
We do not know which individual intervals are among the successful 95%.
Key Takeaway
For a sample proportion \(\hat{p}\), the estimated standard error under independent sampling is\[ \widehat{SE}(\hat{p}) = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }. \]
An approximate 95% confidence interval is\[ \hat{p} \pm 1.96 \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }. \]
For a sample approval rating of 58% based on 1,000 voters, this gives approximately\[ \boxed{54.9\%\text{ to }61.1\%}. \]
The 95% confidence level describes the long-run performance of the interval-building procedure: approximately 95% of intervals constructed this way will contain the true population proportion under the stated assumptions.
The interval represents sampling uncertainty, not every possible source of survey error. Its reliability still depends on representative sampling, accurate measurement, appropriate handling of the survey design, and a suitable interval method.