4.3 χ²-tests

2026 Syllabus Objectives

Fit a theoretical distribution, as prescribed by a given hypothesis, to given data
Use a χ²-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis (classes should be combined so that each expected frequency is at least 5)
Use a χ²-test, with the appropriate number of degrees of freedom, for independence in a contingency table (Yates' correction is not required; where appropriate, either rows or columns should be combined so that the expected frequency in each cell is at least 5)

10.1 Forming Hypotheses

Understanding Goodness of Fit Tests

We test how well observed data from an experiment fits the expected values from a theoretical distribution. This is crucial for determining whether our data follows a particular probability model.

🔑 Key Concept: A discrete uniform distribution occurs when each outcome is equally likely and has the same probability.

Setting Up Hypotheses

For a hypothesis test, we define:

Null Hypothesis ( $H_0$ ): The theoretical distribution is a good-fit model for the observed data
Alternative Hypothesis ( $H_1$ ): The theoretical distribution is not a good-fit model for the observed data

Critical: Always refer to the specific distribution you are testing in your hypotheses, and ensure your conclusions address the initial problem.

Key Terms 📌

Constraint: A condition that reduces the number of free variables in a system by one
Free variables: Independent variables in a system
Degrees of freedom ( $\nu$ ): The number of independent values in the system after constraints and estimated parameters are accounted for
Chi-squared ( $\chi^2$ ): A test statistic used to measure the difference between observed and expected frequencies

Calculating Expected Frequencies

Expected frequency for outcome $i$ :

$E_i = N \times P(X = x_i)$

where $N$ is the total number of trials and $P(X = x_i)$ is the probability of outcome $x_i$ .

Degrees of Freedom Formula

\nu = \text{number of expected values} - 1 - \text{number of parameters estimated}

The $-1$ accounts for the constraint that the total observed frequency equals the total expected frequency.

The Chi-Squared Test Statistic ⚡

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

where:

$O_i$ = observed frequency for category $i$
$E_i$ = expected frequency for category $i$

Alternative computational formula:

$\chi^2 = \sum \left( \frac{O_i^2}{E_i} \right) - N$

This form is often more convenient for calculations.

Worked Example: Testing a Fair Die

Problem: An experiment is carried out to test whether a die is biased. A die is rolled 180 times with the following observations:

Number ( $n$ )	1	2	3	4	5	6
Observed frequency	29	31	34	39	23	24

Test, at the 5% significance level, whether the die is biased.

Solution:

Step 1: Set up hypotheses

$H_0$ : A discrete uniform distribution is a good fit
$H_1$ : A discrete uniform distribution is not a good fit

Step 2: Calculate expected frequencies

For a fair die, each outcome has probability $\frac{1}{6}$ .

E_i = 180 \times \frac{1}{6} = 30 \text{ for each outcome}

Step 3: Calculate the test statistic

Number	Observed ( $O_i$ )	Expected ( $E_i$ )	$\frac{(O_i - E_i)^2}{E_i}$
1	29	30	0.0333
2	31	30	0.0333
3	34	30	0.5333
4	39	30	2.7
5	23	30	1.6333
6	24	30	1.2

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} = 6.1333$

Step 4: Determine degrees of freedom

$\nu = 6 - 1 - 0 = 5$

(6 categories, no parameters estimated)

Step 5: Find critical value

From tables: $\chi^2_5(0.95) = 11.07$

Step 6: Make decision

Since $6.1333 < 11.07$ , there is insufficient evidence to reject $H_0$ .

Conclusion: There is insufficient evidence to suggest that the die is biased. A discrete uniform distribution is a good fit for the data.

Menu

Dashboard

Notes & Flashcards

Topical Past Papers

Yearly Past Papers

AI Assistant

Test / Mock Builder

Feedback

All Subtopics