Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability | Topic 4.18
📋 Learning Intentions
- Define the critical region, critical value, and p-value for a hypothesis test and state their relationship to the significance level \(\alpha\).
- Carry out a one-tailed or two-tailed hypothesis test for a population mean using the normal distribution or t-distribution.
- Carry out a hypothesis test for a population proportion using the binomial distribution.
- Carry out a hypothesis test for a Poisson mean \(\lambda\).
- Interpret p-values and critical regions in context, stating conclusions with reference to the significance level.
- Justify whether a result is statistically significant and communicate conclusions using correct statistical language.
Real-World Introduction
A pharmaceutical company claims its new drug reduces average recovery time from 14 days to fewer than 10 days. Before approving the drug, a regulator tests a sample of 30 patients. The question is not simply "did the sample mean come out below 10?" — it is "could this result have occurred by chance even if the drug has no effect?" Hypothesis testing provides a rigorous mathematical framework for answering exactly that question.
The same logic applies across science, business, and public policy. A factory claims its machines produce bolts with a mean diameter of 5 mm; a quality inspector measures 40 bolts and tests whether the mean has shifted. A city council claims a road improvement reduced the monthly number of accidents, which follow a Poisson distribution. In every case, the method is the same: state hypotheses, choose a significance level, calculate a test statistic or p-value, and reach a conclusion.
🌍 Why This Matters Every clinical trial, quality-control process, and scientific study published in a peer-reviewed journal uses hypothesis testing. The p-value is arguably the most widely cited — and most widely misunderstood — number in modern science. Understanding it rigorously is both an IB exam skill and a genuine tool for interpreting the world.
Core Concepts & Definitions
Before performing any test, two competing statements about a population parameter must be formalised. One statement assumes "nothing has changed" or "the claim is true"; the other asserts a specific direction or difference. These are the null and alternative hypotheses.
📖 Key Definition — Null and Alternative Hypotheses
- The null hypothesis \(H_0\) states that the population parameter equals a specific value: e.g. \(\mu = \mu_0\), \(p = p_0\), or \(\lambda = \lambda_0\).
- The alternative hypothesis \(H_1\) states the direction of the test:
- One-tailed (left): \(\mu \mu_0\)
- Two-tailed: \(\mu \neq \mu_0\)
- The hypotheses must be stated in terms of the population parameter , not the sample statistic.
The significance level \(\alpha\) is the probability threshold below which a result is considered too unlikely to have occurred by chance under \(H_0\). Common values are \(\alpha = 0.05\) (5%) and \(\alpha = 0.01\) (1%). A smaller \(\alpha\) demands stronger evidence before rejecting \(H_0\).
📖 Key Definition — Significance Level \(\alpha\) The significance level \(\alpha\) is the probability of rejecting \(H_0\) when it is in fact true (a Type I