Chi-squared tests for goodness of fit and independence

Learning objectives

Calculate the test statistic Σ(O − E)²/E from observed and expected frequencies, and explain that its distribution is only approximately χ², the approximation improving as the expected frequencies grow

Carry out a goodness of fit test for a proposed model such as a uniform, binomial, Poisson, geometric or normal distribution, computing each expected frequency as the total frequency multiplied by the model probability

Determine the degrees of freedom as the number of classes, minus one for the fixed total, minus one for each parameter estimated from the data itself

Combine adjacent classes until every expected frequency is at least 5, recalculate the degrees of freedom after pooling, and explain why small expected frequencies distort the test statistic

Carry out a test for independence in an r by c contingency table, computing each expected frequency as (row total × column total) / grand total, using (r − 1)(c − 1) degrees of freedom, with no Yates continuity correction (the syllabus does not require it), even in a 2 by 2 table

Compare the test statistic with the upper-tail critical value, state whether the null hypothesis is rejected, and interpret the outcome as evidence about association or fit in the context given