Inference using normal and t-distributions — common mistakes

Always state your hypotheses clearly, defining any parameters used. When using the t-distribution, you must state the assumption that the underlying population is normal. Pay close attention to whether the question gives you the sample variance or the unbiased estimate of the population variance. If you are given $\sum(x-\bar{x})^2$, you must calculate $s^2 = \frac{1}{n-1}\sum(x-\bar{x})^2$ yourself. Finally, always write your conclusion in the context of the original problem.

A two-tailed test checks for a difference in either direction (e.g., $H_1: \mu \neq 50$). The significance level $\alpha$ is split between the two tails. A one-tailed test checks for a difference in a specific direction (e.g., $H_1: \mu > 50$ or $H_1: \mu < 50$). The entire significance level $\alpha$ is in one tail.

It relates to the calculation of the sample variance, $s^2$. When we calculate the sum of squared deviations from the sample mean, $\sum(x_i - \bar{x})^2$, only $n-1$ of these deviations are free to vary. Once the first $n-1$ values and the sample mean are known, the last value is fixed. This 'loss' of one degree of freedom is reflected in the t-distribution.

If the sample size is small and you cannot assume the population is normal, neither the z-test nor the t-test is appropriate. In this situation, you would need to use non-parametric tests (e.g., a sign test or Wilcoxon signed-rank test), which are covered in other parts of the syllabus and do not assume a particular distribution for the population.

A common rule of thumb in A-Level Further Mathematics is that a sample size $n > 30$ is considered large enough for the Central Limit Theorem to apply, allowing you to use the normal distribution even if the population variance is unknown. For exams, if $n$ is around this value or higher, this is the expected approach.