Sample proportions and confidence intervals: QCE Maths Methods Unit 4

What this dot point is asking

QCAA wants you to treat the sample proportion $\hat{p}$ as a random variable, apply the normal approximation to its sampling distribution, construct confidence intervals for a population proportion, and interpret the interval correctly. The dot point bridges Unit 3 binomial probability with Unit 4 statistical inference, and is heavily examined in PSMT and EA.

Sample proportion

If a population has true proportion $p$ of "successes" and a random sample of $n$ is drawn with $X$ successes, the sample proportion is:

Because $X$ is random, $\hat{p}$ is a random variable: it varies from sample to sample.

Sampling distribution of $\hat{p}$

Mean. $E(\hat{p}) = p$. The sample proportion is an unbiased estimator of $p$.

Standard deviation. $\text{SD}(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}$.

Two takeaways:

• SD falls as $\sqrt{n}$: quadruple $n$ to halve SD.
• SD is maximised at $p = 0.5$; minimised at $p = 0$ or $p = 1$.

Normal approximation

For large $n$:

Conditions for the approximation (QCAA convention):

• $n p \geq 10$
• $n(1-p) \geq 10$

When these conditions hold, $\hat{p}$ is approximately normal with mean $p$ and SD $\sqrt{p(1-p)/n}$. Standardise via $Z = (\hat{p} - p)/\sqrt{p(1-p)/n}$ to compute probabilities.

Confidence intervals

A confidence interval for a population proportion combines:

• The point estimate $\hat{p}$ (centre).
• The standard error: $\sqrt{\hat{p}(1-\hat{p})/n}$ (using $\hat{p}$ in place of unknown $p$).
• The critical value $z^*$ for the confidence level.

The formula:

The margin of error is $\text{MoE} = z^* \times \text{SE}$.

Standard $z^*$ values

Interpretation

A $C \%$ confidence interval has the long-run interpretation:

Approximately $C \%$ of intervals constructed by this procedure across repeated samples would contain the true population proportion.

This is NOT:

• "There is a $C \%$ probability that $p$ is in this interval." (Once the interval is constructed, $p$ either is or is not in it.)
• "$C \%$ of the population have proportions in this interval." (The interval is about the parameter, not about individuals.)

The correct language refers to the long-run success rate of the procedure.

Sample size design

To achieve a margin of error at most $E$ at $C \%$ confidence:

If $\hat{p}$ is unknown in advance, use $\hat{p} = 0.5$ (worst case) for a conservative design.

Always round up to the next integer (cannot sample a fractional person).

Trade-offs

Confidence vs precision. Higher confidence (99 percent) requires a wider interval. Lower confidence (90 percent) gives a narrower interval. To improve both, increase $n$.

Sample size economics. Doubling $n$ reduces SE by a factor of $\sqrt{2}$. Quadrupling $n$ halves SE. Diminishing returns above $n \approx 1000$ for opinion polling.

When the normal approximation fails

For very small samples or proportions near 0 or 1, the approximation can be poor. QCAA conventions ($n p \geq 10$ and $n(1-p) \geq 10$) ensure validity. Outside these conditions, an exact (binomial-based) interval would be needed, beyond Methods scope.