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Approximate confidence intervals for a population proportion

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Which expression gives the estimated margin of error for a two-sided confidence interval for a population proportion?

Year 12, VCE Units 3 & 4, Chapter 14, 4 difficulty levels, Calculator allowedBuild a worksheet

Confidence intervals for a population proportion

Use the observed sample proportion as the point estimate, then estimate its standard error. Multiply that estimate by the confidence-level quantile and subtract and add the margin of error

\[\hat p\pm z\sqrt{\frac{\hat p(1-\hat p)}{n}}\]
  1. Find the point estimate The sample proportion is the observed number of successes divided by the sample size \(\displaystyle \hat p=\frac{20}{100}=0.2\)
  2. Estimate the standard error Use the sample proportion in the standard-error formula and take the square root \(\displaystyle SE=\sqrt{\frac{0.2(1-0.2)}{100}}=\sqrt{0.0016}=0.04\)
  3. Find the margin of error Multiply the estimated standard error by the two-sided quantile for the confidence level \(\displaystyle M=1.96\times0.04=0.0784\)
  4. Form both endpoints Subtract the margin for the lower endpoint and add it for the upper endpoint \(\displaystyle L=0.2-0.0784=0.1216,\quad U=0.2+0.0784=0.2784\)

Worked example

A random sample has 60 successes in 150 observations. Assuming the large-population approximation is appropriate, find the approximate 90% confidence interval for the population proportion

\[\hat p=\frac{60}{150},\quad z=1.645\]
  1. \(\displaystyle \hat p=\frac{60}{150}=0.4\)The point estimate is the sample proportion
  2. \(\displaystyle SE=\sqrt{\frac{0.4(1-0.4)}{150}}=\sqrt{0.0016}=0.04\)Use the observed sample proportion to estimate the standard error
  3. \(\displaystyle M=1.645\times0.04=0.0658\)The 90% quantile times the standard error gives the margin of error
  4. \(\displaystyle L=0.4-0.0658=0.3342,\quad U=0.4+0.0658=0.4658\)Subtract and add the margin to form the lower and upper endpoints

Common mistakes

Forgetting the square root
✕\(\displaystyle SE=\frac{0.2(1-0.2)}{100}=0.0016\)✓\(\displaystyle SE=\sqrt{\frac{0.2(1-0.2)}{100}}=0.04\)
The standard error is the square root of the estimated variance
Using the population proportion
✕\(\displaystyle SE=\sqrt{\frac{p(1-p)}{100}}\)✓\(\displaystyle SE=\sqrt{\frac{\hat p(1-\hat p)}{100}}\)
An approximate confidence interval uses the observed sample proportion because the population proportion is unknown
Giving only one endpoint
✕\(\displaystyle L=\hat p-M\)✓\(\displaystyle L=\hat p-M,\quad U=\hat p+M\)
A confidence interval has a lower endpoint and an upper endpoint

Difficulty levels

introFind a margin of error and interval from a friendly exact standard error
standardCalculate both endpoints at a stated precision and interpret the interval method
challengeDetermine a required sample size or recover a confidence level from interval information
examSelect the correct interval construction or interpretation and complete a reverse calculation

Questions about this topic

Why is the sample proportion used in the standard error?+

The population proportion is the unknown quantity being estimated. The observed sample proportion supplies its estimate for the interval calculation

What does the margin of error represent?+

It is the distance from the sample proportion to either endpoint of a symmetric approximate interval

Why is the estimated standard error multiplied by the confidence-level quantile?+

The standard error measures the spread of sample proportions, and the confidence-level quantile sets how many standard errors are included on each side

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 14: Statistical inference

In this chapter

Populations, samples, parameters, statistics and sampling methodsAllocate a stratified random sampleFind a percentage of a quantityCalculate a sample proportion and list its possible valuesExact sampling distribution of a sample proportionFinite events, probability assignments and complementsConditional probability from exact representationsEvaluate factorial and combination expressionsConvert between fractions, decimals and percentagesLarge-sample conditions and the sample-proportion distributionCalculate a standard score or actual valueUse z-score boundaries to estimate normal-model percentages and countsCalculate the standard deviation from the formulaSolve a linear equationSolve a quadratic exactly with the quadratic formulaApproximate confidence intervals for a population proportionRearrange literal equations and recognise equivalent line formsInterpret simulated sampling distributions and interval variationDesign and interpret a probability simulation