The sample proportion and its distribution: WACE Year 12 Mathematics Methods Unit 4

What this dot point is asking

SCSA Unit 4 introduces statistical inference by studying how the sample proportion behaves across repeated samples. This dot point asks you to describe its sampling distribution, the mean, the standard error and the approximate normality, which underpins the confidence interval that follows. It is examined in both sections.

Random sampling and the sample proportion

When a random sample of size $n$ is drawn, the number of successes $X$ follows a binomial distribution $X\sim B(n,p)$, where $p$ is the unknown population proportion. The sample proportion is

Different samples give different values of $\hat{p}$, so $\hat{p}$ is itself a random variable with its own distribution, called the sampling distribution.

Mean and standard error

Because $\hat{p}$ is $\dfrac{1}{n}$ times a binomial variable, its mean and standard deviation follow from the binomial mean $np$ and variance $np(1-p)$.

Since $E(\hat{p})=p$, the sample proportion is an unbiased estimator: on average it equals the true proportion. The standard error has $\sqrt{n}$ in the denominator, so larger samples give less variable, more precise estimates.

The standard error and sample size

The factor $\dfrac{1}{\sqrt{n}}$ in the standard error has an important consequence: precision improves only with the square root of the sample size, not with the size itself. To halve the standard error you must quadruple the sample, and to cut it to a third you need nine times as many observations. This diminishing return is why large national surveys reach a point where extra respondents add little precision, and it explains the sample sizes that appear in confidence-interval and margin-of-error questions. The standard error is largest when $p=0.5$, because $p(1-p)$ peaks there, so a survey near a $50/50$ split is the hardest case to estimate precisely.

Why approximate normality

For large $n$ the binomial distribution of $X$ is well approximated by a normal distribution, so $\hat{p}=\dfrac{X}{n}$ is approximately normal too. A common rule of thumb is that both $np$ and $n(1-p)$ should be at least about $10$. This normal approximation is what allows the confidence interval to use a $z$-multiplier.