Hypotheses, expected frequencies, degrees of freedom, and the point at which a difference stops being chance.
150 people, two groups, three preferences. Move the counts and watch χ² climb. The conclusion flips at one specific value, and nowhere else.
Grey is what you would expect if the two were unrelated. Green is what happened.
H₀: the two variables are independent. H₁: they are not independent. For goodness of fit, H₀ is that the data fits the stated distribution.
H₀ is always the boring one: nothing going on, no association. You never prove it. You either have enough evidence to reject it or you do not, and "we accept H₀" is wording that loses marks. Say "there is insufficient evidence to reject H₀".
Every expected frequency should be at least 5. If one is not, the test is not reliable and the usual fix is to combine categories. Checking this is worth a mark and almost nobody does it.
| Decide by | Reject H₀ when |
|---|---|
| critical value | χ² is bigger than it |
| p-value | p is smaller than the significance level |
The two tests always agree, and the inequalities point opposite ways, which is the thing people get backwards under pressure. Big statistic, small p, reject.
1. A table has 2 rows and 3 columns. How many degrees of freedom?
2. Row total 60, column total 50, grand total 150. What is the expected frequency?
3. χ² = 1.3 with a critical value of 11.07. What do you conclude?
Stating both hypotheses in context, not just as symbols. "H₀: preference is independent of group" rather than "H₀: independent".
The degrees of freedom. One number, often one mark, and easy to get wrong by forgetting to subtract.
The conclusion in context and with the comparison stated. "16.7 is greater than 5.99, so reject H₀: there is evidence that preference depends on group."
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