9231 · 4.3
χ²-tests flashcards
Revision flashcards for Cambridge 9231 χ²-tests (syllabus 4.3). Flip, recall, then mark a real past-paper question.
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What is the purpose of a χ²-test?
To compare observed categorical data with expected data to see if the difference is statistically significant. It tests the 'goodness of fit' of a model or the independence of two variables.
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What are the two main types of χ²-test in the Further Maths syllabus?
1. Goodness of fit test. 2. Test for independence in a contingency table.
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What is the formula for the χ² test statistic?
$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$, where O is observed frequency, E is expected frequency, and the sum is over all k categories.
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What is the key condition for a χ²-test to be considered valid?
All expected frequencies ($E_i$) must be greater than 5. If not, adjacent categories should be combined until this condition is met.
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How do you calculate degrees of freedom (ν) for a goodness of fit test?
ν = (number of categories after any pooling) - 1 - (number of parameters estimated from the data).
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How do you calculate degrees of freedom (ν) for a contingency table?
ν = (number of rows - 1) × (number of columns - 1).
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What does a large calculated χ² value suggest?
A large χ² value indicates a large discrepancy between observed and expected frequencies, suggesting that the null hypothesis is unlikely to be true.
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What are the typical null ($H_0$) and alternative ($H_1$) hypotheses for a test of independence?
$H_0$: The two variables are independent (no association). $H_1$: The two variables are not independent (there is an association).
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How do you calculate the expected frequency for a cell in a contingency table?
$E_{ij} = \frac{(\text{Row } i \text{ total}) \times (\text{Column } j \text{ total})}{\text{Grand Total}}$
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What is a 'restriction' when calculating degrees of freedom for a goodness of fit test?
A restriction is a piece of information derived from the sample data used to calculate the expected frequencies. The total frequency is always one restriction. Any estimated parameters (like λ for Poisson or μ for Normal) are additional restrictions.
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What is a common mistake when concluding a χ²-test?
Stating 'accept H₀'. We never 'accept' the null hypothesis; we state that there is 'insufficient evidence to reject H₀'. Also, forgetting to write the conclusion in the context of the original problem.