The large-sample distribution of a sample proportion
For a sufficiently large population and sample, the sample proportion is modelled with mean p and standard error based on p(1-p)/n. Check all three large-sample conditions before using the normal approximation
\[E(\hat p)=p,\quad\operatorname{Var}(\hat p)=\frac{p(1-p)}{n},\quad SE(\hat p)=\sqrt{\frac{p(1-p)}{n}}\]
- Check all three conditions The expected successes, expected failures and population-size multiple must all meet the large-sample rule \(\displaystyle np=60\times\frac{1}{2}=30\ge10,\quad n(1-p)=30\ge10,\quad N=800\ge10n=600\)
- Use the population proportion as the mean The sampling distribution is centred at the population proportion, not at a selected sample result \(\displaystyle p=\frac{3}{5}\Rightarrow E(\hat p)=\frac{3}{5}\)
- Calculate variance before standard error Divide p(1-p) by n for the variance, then take the square root for the standard error \(\displaystyle p=\frac{1}{2},\ n=100\Rightarrow\operatorname{Var}(\hat p)=\frac{1}{400},\quad SE(\hat p)=\frac{1}{20}\)
- Standardise a boundary Subtract the mean and divide by the standard error before reading a normal probability \(\displaystyle z=\frac{\hat p-p}{SE(\hat p)}=\frac{0.55-0.5}{0.05}=1\)
Worked example
A population has 5000 binary units. A sample of 400 units is taken. State the mean and standard error of the sample proportion
\[N=5000,\quad n=400,\quad p=\frac{1}{2}\]
- \(\displaystyle np=400\times\frac{1}{2}=200\ge10,\quad n(1-p)=200\ge10,\quad N=5000\ge10n=4000\)The expected counts and the population-size condition satisfy the large-sample rule
- \(\displaystyle E(\hat p)=p=\frac{1}{2}\)The population proportion gives the centre of the sampling distribution
- \(\displaystyle \operatorname{Var}(\hat p)=\frac{p(1-p)}{n}=\frac{(1/2)(1/2)}{400}=\frac{1}{1600}\)Use p and 1-p in the variance formula and divide by the sample size
- \(\displaystyle SE(\hat p)=\sqrt{\frac{1}{1600}}=\frac{1}{40}=0.025\)Taking the square root of the variance gives the standard error
Common mistakes
Using the variance as the standard error
✕\(\displaystyle SE(\hat p)=\frac{p(1-p)}{n}\)✓\(\displaystyle SE(\hat p)=\sqrt{\frac{p(1-p)}{n}}\)
The standard error is the square root of the varianceUsing one observed sample as the centre
✕\(\displaystyle E(\hat p)=\hat p\)✓\(\displaystyle E(\hat p)=p\)
The sampling distribution is centred at the population proportion, whereas a sample proportion is one observed resultChecking only one condition
✕\(\displaystyle np\ge10\Rightarrow\text{the large-sample rule is satisfied}\)✓\(\displaystyle np\ge10,\quad n(1-p)\ge10,\quad N\ge10n\)
All three large-sample conditions must hold before the approximation is usedDifficulty levels
introState the mean and calculate variance and standard error for a friendly exact-square case
standardTest all three large-sample conditions and calculate a stated-precision standard error
challengeUse standardised boundaries to find a normal-approximation probability
examRecover a population proportion or minimum sample size from mean or standard-error information
Questions about this topic
Why must all three large-sample conditions be checked?+
The approximation requires enough expected successes, enough expected failures and a population sufficiently large compared with the sample
Why is the mean of the sample proportion p?+
Across repeated samples from the same population, the sample proportion is centred on the population success proportion p
What is the difference between variance and standard error?+
The variance is the square of the standard error. The standard error is the square root of the variance
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