Degrees of freedom, expected frequencies, and the conclusion wording that is worth more than the arithmetic.
Shift the counts until the statistic crosses the critical value, then stop and ask what changed.
Nothing about the method changed. The same test, the same critical value, and the decision reversed, because the decision was always a comparison and never a property of the data.
Students who have only ever seen one worked example think the conclusion is a fact about the table. Watching it flip at a specific, visible threshold replaces that with a comparison they can carry out themselves.
Row total times column total over grand total. It is worth deriving once, because it is just “if the two were independent, multiply the proportions”, and a derived formula survives a stressful exam far better than a remembered one.
Any expected frequency below 5 is a problem, and saying so is a mark on questions that engineer one. Combining categories is the usual fix.
| 1. Degrees of freedom | (2 − 1) × (3 − 1) = 2. |
| 2. Expected frequency | 60 × 50 over 150 = 20. |
| 3. The conclusion | B. 1.3 is well below 11.07, so there is not enough evidence to reject. That is the right wording as well as the right decision. |
1 markBoth hypotheses stated in context, with independence named. “H₀: independent” alone is usually not enough.
1 markDegrees of freedom from (r − 1)(c − 1), not from the number of cells.
1 markA conclusion that compares two numbers and then answers in context, without claiming proof.
| They give | What it means |
|---|---|
| 6 (Q1) | Counted the cells. The commonest degrees of freedom error there is. |
| 5 (Q1) | Used cells minus 1, which is the goodness-of-fit rule from a different test. |
| 3000 (Q2) | Multiplied the two totals and never divided. |
| 110 (Q2) | Added the totals. |
| 25 (Q2) | Divided the wrong pair, or used a row total twice. |
| "The variables are independent" (Q3) | The right decision stated far too strongly. Failing to reject is not proving; this wording loses the mark even with correct arithmetic. |
"Do I use p or the critical value?" Either, and both earn full marks, but not a mixture. If the question gives a critical value, use it; if it asks for p, compare with the significance level. Write which you are doing.
"What does a big chi-squared mean?" That the observed counts are far from what independence predicts. Big means evidence against H₀, which is the opposite way round from a p-value and is worth labouring.
"Can I prove they are independent?" No. You can fail to find evidence against it, which is a much weaker statement, and the wording of your conclusion has to reflect that.
| What is happening | |
|---|---|
| 1 | Shift the counts until the verdict flips. Ask what changed. |
| 2 | Expected frequencies derived, not stated, with the below-5 rule. |
| 3 | Degrees of freedom and the three questions. |
| 4 | A full test written out on a real table, hypotheses to conclusion. |
| 5 | Mark each other's conclusion sentence only. It is the part that loses marks and the part never practised. |
Do not say “we accept H₀”. It is the wrong idea in the wrong words and it costs marks all the way to the final examination.
Do not let the calculator do the whole test on day one. One hand-computed expected frequency makes the formula mean something; after that, use the calculator.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.