Chi squared tests

Explanation

• For goodness of fit, $H_0$ states in context that the proposed distribution fits; for a contingency table, both hypotheses must name the two classifications, for example $H_0$: travel method is independent of year group and $H_1$: travel method is associated with year group. Use $\chi^2=\sum\dfrac{(O-E)^2}{E}$.
• Goodness-of-fit degrees of freedom are $k-1-m$ after combining, where $m$ parameters were estimated from the data. For an $r\times c$ contingency table, $df=(r-1)(c-1)$ and each expected frequency is row total times column total divided by grand total.
• Cells should be combined sensibly when $E<5$.
• In a critical-value test, state the degrees of freedom and the critical value explicitly before comparing it with $\chi^2$; the critical value can carry its own mark.
• Finish with a contextual conclusion.