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VICMath MethodsQuick questions
Unit 4
Quick questions on Sample proportions and sampling distributions: VCE Math Methods Unit 4
11short 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 of $\hat{p}$?Show answer
The expected value of the sample proportion equals the population proportion. The sample proportion is an unbiased estimator of $p$.
What is standard deviation of $\hat{p}$?Show answer
$$\text{SD}(\hat{p}) = \sqrt{\frac{p (1 - p)}{n}}$$
What is conditions for the formula?Show answer
These are the conditions of the binomial distribution: $X \sim \text{Bin}(n, p)$.
What is why the normal approximation works?Show answer
$X \sim \text{Bin}(n, p)$ for large $n$ is approximately $N(n p, n p (1 - p))$ by the central limit theorem. Dividing by $n$ gives $\hat{p} = X / n$ approximately $N(p, p(1-p)/n)$.
What is worked example?Show answer
A factory produces 60 percent of items meeting specification. A sample of $n = 150$ items is taken. Find the probability that the sample proportion meeting spec is at least 0.55.
What is confusing $\hat{p}$ with $p$?Show answer
$p$ is the (unknown) population proportion; $\hat{p}$ is the random sample-based estimate. Different objects with different statistical roles.
What is wrong formula for SD?Show answer
$\sqrt{p (1 - p) / n}$, not $\sqrt{p (1 - p) n}$ or $\sqrt{p / n}$. The factor of $n$ is in the denominator.
What is using sample SD without checking conditions?Show answer
If $n p$ or $n(1-p)$ is too small (below 10 by VCAA convention), the normal approximation is unreliable and the binomial distribution is needed instead.
What is forgetting that $\hat{p}$ is a random variable?Show answer
Treating $\hat{p}$ as a fixed number ignores the entire point of the sampling distribution. Probabilities require treating it as random.
What is using $\hat{p}$ instead of $p$ in the SD formula?Show answer
When the true population proportion $p$ is known, use $p$. When $p$ is unknown (confidence-interval setting), substitute $\hat{p}$ as an estimate.
What is calculator without set-up?Show answer
Paper 2 expects the explicit standardisation set-up. A naked normCdf call without the distribution statement loses set-up marks.