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Large-sample conditions and the sample-proportion distribution

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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

\[E(\hat p)=p,\quad\operatorname{Var}(\hat p)=\frac{p(1-p)}{n},\quad SE(\hat p)=\sqrt{\frac{p(1-p)}{n}}\]
  1. 1Check all three conditions The expected successes, expected failures and population-size multiple must all meet the large-sample rule \(\displaystyle np=60\times\frac{1}{2}=30\ge10,\quad n(1-p)=30\ge10,\quad N=800\ge10n=600\)
  2. 2Use the population proportion as the mean The sampling distribution is centred at the population proportion, not at a selected sample result \(\displaystyle p=\frac{3}{5}\Rightarrow E(\hat p)=\frac{3}{5}\)
  3. 3Calculate variance before standard error Divide p(1-p) by n for the variance, then take the square root for the standard error \(\displaystyle p=\frac{1}{2},\ n=100\Rightarrow\operatorname{Var}(\hat p)=\frac{1}{400},\quad SE(\hat p)=\frac{1}{20}\)
  4. 4Standardise a boundary Subtract the mean and divide by the standard error before reading a normal probability \(\displaystyle z=\frac{\hat p-p}{SE(\hat p)}=\frac{0.55-0.5}{0.05}=1\)
Year 12, VCE Units 3 & 4, Chapter 14, 4 difficulty levels, Calculator allowedBuild a worksheet

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

\[E(\hat p)=p,\quad\operatorname{Var}(\hat p)=\frac{p(1-p)}{n},\quad SE(\hat p)=\sqrt{\frac{p(1-p)}{n}}\]
  1. Check all three conditions The expected successes, expected failures and population-size multiple must all meet the large-sample rule \(\displaystyle np=60\times\frac{1}{2}=30\ge10,\quad n(1-p)=30\ge10,\quad N=800\ge10n=600\)
  2. Use the population proportion as the mean The sampling distribution is centred at the population proportion, not at a selected sample result \(\displaystyle p=\frac{3}{5}\Rightarrow E(\hat p)=\frac{3}{5}\)
  3. Calculate variance before standard error Divide p(1-p) by n for the variance, then take the square root for the standard error \(\displaystyle p=\frac{1}{2},\ n=100\Rightarrow\operatorname{Var}(\hat p)=\frac{1}{400},\quad SE(\hat p)=\frac{1}{20}\)
  4. 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

\[N=5000,\quad n=400,\quad p=\frac{1}{2}\]
  1. \(\displaystyle np=400\times\frac{1}{2}=200\ge10,\quad n(1-p)=200\ge10,\quad N=5000\ge10n=4000\)The expected counts and the population-size condition satisfy the large-sample rule
  2. \(\displaystyle E(\hat p)=p=\frac{1}{2}\)The population proportion gives the centre of the sampling distribution
  3. \(\displaystyle \operatorname{Var}(\hat p)=\frac{p(1-p)}{n}=\frac{(1/2)(1/2)}{400}=\frac{1}{1600}\)Use p and 1-p in the variance formula and divide by the sample size
  4. \(\displaystyle SE(\hat p)=\sqrt{\frac{1}{1600}}=\frac{1}{40}=0.025\)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 variance
Using 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 result
Checking 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 used

Difficulty 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