Revise Hypothesis testing: chi-squared and t-tests for IB Mathematics: Applications and Interpretation (4.8)

Every hypothesis test, whatever its name, runs on the same five-step logic. You state a null hypothesis $H_0$, the cautious 'nothing is going on' claim, and an alternative hypothesis $H_1$, the effect you hope to detect. You fix a significance level, the bar the evidence must clear, before you look at the data. You compute a test statistic from the data, which the GDC converts into a p-value: the probability of data at least this extreme if $H_0$ were true. Finally you apply the decision rule and state the outcome in context.

The critical-value method is an exactly equivalent route to the same decision, used when a question quotes a critical value instead of expecting you to read a p-value. Each test statistic has a distribution under $H_0$, and the critical value marks the point beyond which results are counted as surprising. If $\chi^2_{calc}$ lands beyond the critical value you reject $H_0$; this is precisely the same as the p-value falling below the significance level. Keep both methods to hand, because IB questions use either.

Use the $\chi^2$ test for independence when you have two categorical variables cross-tabulated in a contingency table and want to know whether they are related. For example: is a shopper's age group associated with their preferred payment method? The test compares the observed counts with the expected counts, that is, the counts you would predict if the two variables were completely independent. The bigger the gap between observed and expected, the larger $\chi^2_{calc}$, and the stronger the evidence of an association.

Hypotheses for a test of independence: \n $H_0$: the two variables are independent. \n $H_1$: the two variables are not independent (they are associated).

The expected frequency of any cell comes from its margins. Under independence the proportion in a column should be the same in every row, which gives the compact rule below. The degrees of freedom count how many cells are free to vary once the totals are fixed, which for an $r \times c$ table is $(r-1)(c-1)$. You then either compare $\chi^2_{calc}$ with the critical value or read the p-value from the GDC.

Expected frequency of a cell: $E = \dfrac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$. \n Degrees of freedom: $\text{df} = (r-1)(c-1)$ for an $r \times c$ table. \n Decision: reject $H_0$ if $\chi^2_{calc} >$ critical value, equivalently if p-value $<$ significance level.

The goodness-of-fit test handles a single categorical variable rather than a relationship between two. You have observed frequencies across several categories and a hypothesised distribution, and you ask whether the data is consistent with it. For instance, a manufacturer claims its sweets come in equal numbers of four colours; a sample lets you test that claim. The mechanics mirror the independence test: expected frequencies come from the claimed distribution, the GDC returns $\chi^2_{calc}$ and a p-value, and the same decision rule applies. Here the degrees of freedom are (number of categories) $- 1$, reduced by a further 1 for each parameter you had to estimate from the data.

When the data is numerical and you want to compare the averages of two independent groups, use a two-sample t-test. 'Independent' means membership of one group has no bearing on the other, for example the exam marks of a class taught with a new method versus a separate class taught traditionally. The null hypothesis is that the two population means are equal; the alternative is that they differ, in either direction for a two-tailed test or in a stated direction for a one-tailed test. As always on Paper 2 the GDC carries the load: enter the two data lists, set the alternative, and read off $t$ and the p-value.

Hypotheses for a two-sample t-test: \n $H_0$: $\mu_1 = \mu_2$ (the two population means are equal). \n $H_1$: $\mu_1 \neq \mu_2$ (two-tailed), or $\mu_1 > \mu_2$ / $\mu_1 < \mu_2$ (one-tailed).

On Paper 2 the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach; an accuracy mark (A) rewards a correct value and, where it follows an M, depends on that method being present. Follow-through (FT) means an earlier slip need not cost the later marks that depend on it, provided the later step is done correctly on your own figure. A reasoning mark (R) rewards a correct comparison or decision. The engine also accepts any equivalent form and any correctly rounded value. All of that protection exists only if the method is written down. The example below shows every mark earned by a specific line.

Hypothesis testing is the decision-making layer that sits on top of the rest of the statistics course. The contingency tables and p-values here build directly on the probability and distributions you have already met, and the same logic, a cautious null, a test statistic, a p-value, a decision in context, reappears whenever you judge whether a correlation, a regression slope or a difference of means is real or just noise. Master the shared five-step logic, keep the two decision methods and the two tail choices straight, and every hypothesis-test question on Paper 2 becomes a variation on a routine you already know how to write.