Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability | Topic 4.11
📋 Learning Intentions
- State the null and alternative hypotheses for a two-sample t-test in correct mathematical notation.
- Identify the conditions required to apply a two-sample t-test, including independence and approximate normality.
- Calculate the t-test statistic and p-value using the TI-84 Plus CE and interpret the output correctly.
- Justify a conclusion by comparing the p-value to the significance level \(\alpha\) and stating the decision in context.
- Distinguish between a one-tailed and two-tailed t-test and select the appropriate form for a given scenario.
- Interpret the result of a hypothesis test in terms of the original claim, avoiding over-generalisation.
Real-World Introduction
A pharmaceutical company develops a new sleep aid and wants to know whether patients taking it sleep significantly longer than those taking a placebo. They record sleep duration for 15 patients on the drug and 18 patients on the placebo. The sample means differ — but is that difference real, or just random variation between the two groups?
This is exactly the problem the two-sample t-test is designed to answer. It quantifies how likely the observed difference in means would be if there were actually no difference in the underlying populations. The test is used in medicine, psychology, education, and engineering whenever two independent groups are compared on a numerical measurement.
🌍 Why This Matters Clinical trials, A/B testing in technology, and educational research all rely on the two-sample t-test to decide whether an observed difference between two groups reflects a genuine effect or could plausibly be explained by chance alone.
Core Concepts & Definitions
Before running any hypothesis test, it is essential to understand what question is being asked. The two-sample t-test compares the means of two independent populations. The word "independent" is critical: the two groups must have no relationship to each other — the same individual cannot appear in both groups.
📖 Key Definition — Two-Sample t-Test A two-sample t-test (also called an independent samples t-test) is a hypothesis test used to determine whether there is a statistically significant difference between the means of two independent populations, \(\mu_1\) and \(\mu_2\), when the population standard deviations are unknown.
The test uses sample data from each group to estimate the population means. Because the population standard deviations \(\sigma_1\) and \(\sigma_2\) are unknown, the test statistic follows a t-distribution rather than a normal distribution. The shape of the t-distribution depends on the degrees of freedom, which the GDC calculates automatically.
Conditions for Applying the Two-Sample t-Test
The two-sample t-test is valid only when specific