Inference using normal and t-distributions — FAQ

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.