Module 5 · Inference · Topic 12

Chi-Square Test

Two Common Uses. Goodness-of-fit: test if observed frequencies match expected frequencies from a theoretical distribution. Independence: test if two categorical variables are associated.

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Key formulas

Chi-Square Test Statistic

χ² = Σ[(O − E)² / E]

From the Inference formula sheet

Confidence Interval for a Mean: xbar +/- critical value * standard error — Estimate plus or minus a margin of error.
One-Sample Z Test: z = (xbar - mu_0) / (sigma / sqrt(n)) — Tests a sample mean against a null mean when population sigma is known.
Power: Power = 1 - beta — Probability of correctly rejecting a false null hypothesis.

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Worked example

Goodness-of-Fit Example

A die is rolled 60 times. Expected: 10 per face. Observed: {8, 12, 11, 7, 13, 9}. χ² = (8−10)²/10 + (12−10)²/10 + ... = 0.4 + 0.4 + 0.1 + 0.9 + 0.9 + 0.1 = 2.8. With df = 5, this is not significant - the die appears fair.

When to use it

Expected Count Rule

All expected cell counts should be at least 5 for the chi-square approximation to be valid. If not, consider combining categories or using an exact test.

Related glossary terms

P-value: The probability of results this extreme or more extreme, assuming the null hypothesis is true. Example: A p-value of 0.03 is evidence against H0 at alpha = 0.05.
Confidence Level: The long-run success rate of a confidence interval procedure. Example: A 95% method captures the true parameter in about 95% of repeated samples.
Type I Error: Rejecting the null hypothesis when it is actually true. Example: The significance level alpha is the Type I error rate.
Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Example: A false negative in a hypothesis test is a Type II error.
Power: The probability of rejecting the null hypothesis when it is false. Example: Power equals 1 - beta, where beta is the Type II error probability.

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