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IB Maths AI 4.13 Definitions

This page contains our IB Maths AI definitions for 4.13. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Distribution tests'.

All Topic 4 Sub-topics4.1: Fundamentals Of Statistics4.2: Visualising Statistics4.3: Distribution Properties4.4: Introduction To Linear Statistics4.5: Probability4.6: Probability Distributions4.7: Hypotheses & Tests (HL)4.8: Data collection & categorisation (HL)4.9: Advanced linear statistics (HL)4.10: Linear transformations & combinations (HL)4.11: Confidence intervals (HL)4.12: Poisson distribution (HL)4.13: Distribution tests (HL)4.14: Transition matrices
HL

binomial test

A hypothesis test for a population proportion using X∼Bin(n,p)X\sim\text{Bin}(n,p) under H0H_0, where there are a fixed number of independent trials, two outcomes per trial, and constant success probability.
HL

critical region

The set of values of the test statistic for which H0H_0 is rejected; its probability under H0H_0 is chosen to be at most the significance level.
HL

critical value

The boundary point(s) that separate the critical region from the non-critical region for a chosen significance level.
HL

one-tailed test

A hypothesis test where the critical region lies entirely in one tail of the sampling distribution, matching a directional alternative such as μ<μ0\mu\lt\mu_0 or p>p0p\gt p_0.
HL

Poisson test

A hypothesis test for a population mean rate using X∼Po(λ0)X\sim\text{Po}(\lambda_0) under H0H_0, appropriate for counts of events occurring independently at a constant average rate (with scaling if the observation period changes).
HL

significance level

The maximum allowed probability of rejecting H0H_0 when H0H_0 is true, so P(Type I error)=αP\left(\text{Type I error}\right)=\alpha.
HL

two-tailed test

A hypothesis test where the critical region is split between both tails of the sampling distribution, matching a non-directional alternative such as μ≠μ0\mu\neq\mu_0.
HL

Type I error

Rejecting H0H_0 even though H0H_0 is true (a false positive), with probability α=P(Reject H0∣H0 true)\alpha=P\left(\text{Reject }H_0\mid H_0\text{ true}\right).
HL

Type II error

Failing to reject H0H_0 even though H1H_1 is true (a false negative), with probability β=P(Fail to reject H0∣H0 false)\beta=P\left(\text{Fail to reject }H_0\mid H_0\text{ false}\right) that depends on the true parameter value and the chosen critical region.

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