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QLDMath MethodsQuick questions
Unit 4: Further calculus and statistical inference
Quick questions on Sample proportions and confidence intervals: QCE Maths Methods Unit 4
8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is mean?Show answer
$E(\hat{p}) = p$. The sample proportion is an unbiased estimator of $p$.
What is standard deviation?Show answer
$\text{SD}(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}$.
What is confidence vs precision?Show answer
Higher confidence (99 percent) requires a wider interval. Lower confidence (90 percent) gives a narrower interval. To improve both, increase $n$.
What is sample size economics?Show answer
Doubling $n$ reduces SE by a factor of $\sqrt{2}$. Quadrupling $n$ halves SE. Diminishing returns above $n \approx 1000$ for opinion polling.
What is probability misinterpretation?Show answer
"The probability $p$ is in this interval" is wrong. Use long-run-procedure language.
What is sample size not rounded up?Show answer
$n = 384.1$ becomes $n = 385$, not 384.
What is $p$ vs $\hat{p}$ in formulas?Show answer
In the sampling distribution SD: use $p$ (when known). In the CI standard error: use $\hat{p}$ (when $p$ unknown).
What is forgetting worst-case $p = 0.5$?Show answer
When no prior estimate of $p$ exists for sample-size design, use $p = 0.5$ to maximise $p(1-p) = 0.25$ for the most conservative $n$.