For the $z$-test / $z$-CI (mean):
- Population standard deviation $\sigma$ is known
- Either: population is normal, or $n$ is large enough for CLT ($n \geq 30$ typically)
- Sample is random and independent
For the $t$-test / $t$-CI (mean):
- Population is approximately normal (especially important for small $n$)
- Sample is random and independent
- Observations are independent (no repeated measures)
For proportion tests / CIs:
- $np_0 \geq 10$ and $n(1-p_0) \geq 10$ (for normality of $\hat{P}$)
- Random sample
- Population much larger than sample (finite population correction negligible)
If assumptions are violated, results may be misleading:
| Violation | Consequence |
|---|---|
| Non-random sample | Bias; results may not represent population |
| Small $n$, non-normal data with $z$-test | Incorrect p-values |
| $np < 10$ for proportion test | Normal approximation poor |
| Dependent observations | Underestimated standard error |
A nutritionist measures daily calcium intake (mg) for a random sample of $n=25$ adult women.
$\bar{x} = 820$ mg, $s = 120$ mg. Is the mean intake below the recommended 900 mg?
Assumptions: Calcium intake approximately normally distributed; random sample; $n=25$ so use $t$-test.
$H_0: \mu = 900$; $H_1: \mu < 900$. $\alpha = 0.05$.
$$t = \frac{820-900}{120/\sqrt{25}} = \frac{-80}{24} \approx -3.33$$
$t_{0.05, 24} = 1.711$ (left-tailed: compare with $-1.711$).
Since $-3.33 < -1.711$, reject $H_0$.
Conclusion: There is sufficient evidence at the 5% level that the mean daily calcium intake is below 900 mg.
95% CI: \$820 \pm 2.064 \times 24 = 820 \pm 49.5 = (770.5,\ 869.5)$ mg.
Effect size $d = \dfrac{|\bar{x}-\mu_0|}{s}$ measures the practical magnitude of a difference.
In the example: $d = 80/120 \approx 0.67$ — a moderate-to-large effect.
KEY TAKEAWAY: Always state and check assumptions before applying inference procedures. Report both statistical significance (p-value or CI) and practical significance (effect size or contextual meaning).
EXAM TIP: VCAA inference questions frequently ask you to identify and comment on assumptions. State them specifically — not just “the data is normal” but “the sample is a random sample from the population of interest.”
REMEMBER: Statistical inference provides a framework for decision-making under uncertainty — it cannot eliminate uncertainty or guarantee correct conclusions. Understanding limitations is as important as performing calculations.
