Hypothesis testing: χ² and t-tests

Maths: Applications and Interpretation · Statistics and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A hypothesis test asks whether data give enough evidence against a starting claim. The null hypothesis H₀ says nothing special is happening: the variables are independent, the data fit the model, or the two means are equal. The alternative hypothesis H₁ says the opposite.

The calculator gives a p-value: the probability of data this extreme if H₀ is true. A small p-value is evidence against H₀.

Use a χ² test for frequencies (counts in categories) and a t-test to compare the means of two groups. The χ² test compares the observed frequencies with the expected frequencies you would get if H₀ were true.

Rules to remember

If p-value < significance level, reject H₀; otherwise do not reject H₀
If χ²_calc > critical value, reject H₀; otherwise do not reject H₀
Expected frequency in a contingency table = (row total × column total) ÷ grand total
Degrees of freedom, test for independence: (rows − 1) × (columns − 1)
Degrees of freedom, goodness of fit: number of categories − 1
χ²_calc = sum of (observed − expected)² ÷ expected
t-test: H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂ (two-tailed) or μ₁ > μ₂ or μ₁ < μ₂ (one-tailed); the pooled test assumes normal populations with equal variances

Common mistake

Students reverse the decision. A p-value BELOW the significance level means reject H₀. And in a χ² test for independence H₀ is always that the variables are independent, so rejecting it means there is evidence that they are related.

Worked example

A 2 by 3 contingency table has grand total 150. One cell has row total 60 and column total 45. Find its expected frequency and the degrees of freedom. If χ²_calc = 4.10 and the 5% critical value is 5.991, what is the conclusion?

Expected frequency = (60 × 45) ÷ 150 = 2700 ÷ 150 = 18
Degrees of freedom = (2 − 1) × (3 − 1) = 1 × 2 = 2
Compare: 4.10 < 5.991, so χ²_calc is below the critical value

Answer: Expected frequency 18, 2 degrees of freedom; do not reject H₀, so there is not enough evidence that the variables are related.

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Topic 4: Statistics and Probability

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