Unit 4 deepens understanding of hypothesis testing and confidence intervals, with emphasis on practical application and interpretation.
| Concept | Symbol | Role |
|---|---|---|
| Population mean | $\mu$ | True (unknown) parameter |
| Population proportion | $p$ | True (unknown) parameter |
| Sample mean | $\bar{x}$ | Point estimate of $\mu$ |
| Sample proportion | $\hat{p}$ | Point estimate of $p$ |
| Standard error of $\bar{x}$ | $\sigma/\sqrt{n}$ | Spread of sampling distribution |
| Standard error of $\hat{p}$ | $\sqrt{p(1-p)/n}$ | Spread of sampling distribution |
One-sample $z$-test ($\sigma$ known):
$$H_0: \mu = \mu_0, \quad z = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$
One-sample $t$-test ($\sigma$ unknown, estimated by $s$):
$$t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}, \quad t \sim t_{n-1}$$
The $t$-distribution has heavier tails than the normal and approaches $N(0,1)$ as $n\to\infty$.
$$H_0: p = p_0, \quad z = \frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}$$
Use $p_0$ (not $\hat{p}$) in the standard error under $H_0$.
| Parameter | CI formula |
|---|---|
| $\mu$ ($\sigma$ known) | $\bar{x} \pm z_{\alpha/2}\sigma/\sqrt{n}$ |
| $\mu$ ($\sigma$ unknown) | $\bar{x} \pm t_{\alpha/2,n-1}\cdot s/\sqrt{n}$ |
| $p$ | $\hat{p} \pm z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}$ |
A new teaching method is trialled with 50 students. Mean score $\bar{x} = 72$, $s = 15$. Previous mean was $\mu_0 = 68$. Test at $\alpha = 0.05$.
$H_0: \mu = 68$; $H_1: \mu > 68$ (one-tailed).
$$t = \frac{72-68}{15/\sqrt{50}} = \frac{4}{2.121} \approx 1.886$$
Critical value $t_{0.05, 49} \approx 1.677$. Since \$1.886 > 1.677$, reject $H_0$.
Conclusion: There is sufficient evidence at the 5% level that the new method improves scores.
KEY TAKEAWAY: Further inference extends the Unit 3 framework to include the $t$-distribution for unknown variance and reinforces correct interpretation of all outputs.
VCAA FOCUS: VCAA questions on inference test both calculation and interpretation. Always state hypotheses formally, calculate the test statistic, find (or compare with) the critical value, and draw a conclusion in context.
COMMON MISTAKE: Using the normal $z$-test when $\sigma$ is unknown and $n$ is small. When $\sigma$ is estimated by $s$, use the $t$-test.
