Hypothesis testing: chi-squared and t-tests

IB Mathematics Applications and Interpretation SL· Unit 4: Statistics and Probability, Section 12· 25 min read

1. Core Hypothesis Testing Framework★★☆☆☆⏱ 5 min

All 3 tests follow the same 5-step standard workflow required for full marks in IB exams: 1) State H₀ and H₁, 2) Define the significance level α, 3) Calculate the test statistic and p-value via GDC, 4) Compare p-value to α, 5) Write a contextual conclusion.

📘 Definition

Significance Level

\alpha

The pre-defined threshold for rejecting the null hypothesis, most commonly 5% (0.05) for IB AI SL questions

📐 Worked Example

A café owner claims average daily sales are $1200. A staff member suspects the true average is lower. Write the null and alternative hypotheses for a t-test.

1
Define the population parameter first: μ = true average daily sales at the café
2
Write H₀ as the statement of no effect, matching the owner's claim:
$H_0: \mu = 1200$
3
Write H₁ as the staff's directional suspicion:
$H_1: \mu < 1200$

✓ Quick check

What is the maximum standard significance level used in IB AI SL hypothesis tests?
- 0.01
- 0.05
- 0.10
- 0.20
Reveal answer
0.05 —
5% (0.05) is the default threshold specified in almost all IB AI SL assessment questions

2. Chi-Squared Goodness of Fit Test★★★☆☆⏱ 7 min

This test checks if an observed frequency distribution matches a specified theoretical model (e.g. uniform, binomial, normal). Expected frequencies are calculated by multiplying the total sample size by the theoretical probability for each category.

📐 Worked Example

A 6-sided die is rolled 60 times, with observed frequencies [7, 12, 11, 9, 10, 11]. Test at α=0.05 if the die is fair.

1
State hypotheses: H₀ = die is fair (each outcome has 1/6 probability), H₁ = die is not fair
2
Calculate expected frequencies: 60 * 1/6 = 10 for all 6 outcomes
3
Calculate degrees of freedom: 6 categories - 1 = 5
4
Run test on GDC to get χ² = 2.0, p-value = 0.849
5
Conclusion: 0.849 > 0.05, so there is insufficient evidence to reject H₀, the die can be assumed fair

Exam tip:

IB exam markers award 1 mark for correctly stating the degrees of freedom, so always write this value explicitly before running your GDC calculation

3. Chi-Squared Test for Independence★★★☆☆⏱ 7 min

This test checks if two categorical variables are associated, using a contingency table of observed frequencies. Expected frequency for each cell = (row total * column total) / grand total.

📐 Worked Example

A survey tests if snack preference is independent of age group, with a 3-row (age group) x 4-column (snack type) contingency table. Calculate degrees of freedom for the test.

1
Identify number of rows r = 3, number of columns c = 4
2
Apply the independence test df formula:
$df = (r-1)(c-1) = (3-1)(4-1) = 2 * 3 = 6$

4. One-Sample t-Test for Population Mean★★★★☆⏱ 6 min

This test compares a sample mean to a hypothesized population mean, used when the population standard deviation is unknown and the data is approximately normally distributed. Degrees of freedom = sample size n - 1.

📐 Worked Example

Sample of 15 students has a mean test score of 72, sₙ₋₁ = 8. Test at α=0.05 if the population mean is different from 75.

1
State hypotheses: H₀: μ =75, H₁: μ ≠75 (two-tailed test)
2
Calculate t-statistic:
$t = \frac{72 -75}{8 / \sqrt{15}} \approx -1.452$
3
df = 15 - 1 =14, GDC returns p-value = 0.168
4
Conclusion: 0.168 > 0.05, insufficient evidence to reject H₀

5. Common Pitfalls

Wrong move:

Stating 'the null hypothesis is proven true' when p > α

Why:

You can never confirm the null hypothesis, only find insufficient evidence to reject it

Correct move:

Write 'there is insufficient evidence at the α significance level to reject the null hypothesis'

Wrong move:

Using population standard deviation instead of sₙ₋₁ for t-test inputs

Why:

This produces an incorrect z-statistic instead of the required t-statistic, losing method marks

Correct move:

Always select the unbiased sample standard deviation sₙ₋₁ from your GDC menu

Wrong move:

Forgetting to adjust degrees of freedom after merging categories with expected frequencies <5

Why:

Unadjusted df will return an invalid p-value that does not match IB mark scheme values

Correct move:

Recalculate df using the new smaller number of categories after merging

Wrong move:

Using a one-tailed test for a t-test when no direction is specified in the question

Why:

IB AI SL defaults to two-tailed tests for unstated alternative hypotheses

Correct move:

Confirm the number of tails from the question prompt before running the test on your GDC

Wrong move:

Mixing up goodness of fit and independence test df formulas

Why:

This is the most common mark-losing error for chi-squared questions

Correct move:

Write the df formula explicitly on your working page before calculating the value

6. Quick Reference Cheatsheet

Test Type	Primary Use Case	Degrees of Freedom Formula	Core GDC Inputs
Chi-squared Goodness of Fit	Match observed distribution to theoretical model	$k - 1 - m$	Observed frequencies, theoretical probabilities
Chi-squared Independence	Test association between two categorical variables	$(r - 1) (c - 1)$	Full contingency table of observed values
One-sample t-test	Compare sample mean to hypothesized population mean	$n - 1$	Sample data, hypothesized population mean

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2024 · Paper 2
Chi-squared independence test for survey data
2023 · Paper 2
One sample t-test for population mean of test scores
2022 · Paper 2
Chi-squared goodness of fit for uniform distribution

What's Next

These three hypothesis tests make up ~15% of your total IB Math AI SL assessment marks, and appear on nearly every recent Paper 2. Next, you will practice scenario identification to select the correct test type for mixed question prompts, a skill that trips up 40% of candidates. You will also refine your GDC workflow to avoid common input errors, and practice writing full, mark-aligned conclusions that meet IB examiner standards. This foundation will also support your upcoming work on hypothesis testing for Pearson's correlation coefficient later in the unit.