Type I and Type II errors

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Rejecting a true H₀ is a Type I error (probability α, set by the analyst); failing to reject a false H₀ is a Type II error (probability β); power = 1 − β; lowering α raises β.

Two ways to be wrong

A test ends in one of two decisions — reject $H_{0}$ or do not — and the truth is one of two states. The $2 \times 2$ grid has two correct cells and two errors:

	$H_{0}$ true	$H_{0}$ false
Do not reject $H_{0}$	correct	Type II error ( $β$ )
Reject $H_{0}$	Type I error ( $α$ )	correct (power $= 1 - β$ )

A Type I error rejects a null that is actually true — a false alarm. Its probability is $α$ , the significance level, and the analyst chooses it: $α = 0.05$ means accepting a 5 % chance of a false alarm when $H_{0}$ holds. A Type II error fails to reject a null that is actually false — a miss. Its probability is $β$ , and it is not chosen: $β$ depends on how false $H_{0}$ is, on $σ$ , and on $n$ . The complement $1 - β$ is the power of the test — the probability of detecting a real effect.

The two errors trade off. Lowering $α$ from 0.05 to 0.01 makes rejection harder, which cuts false alarms but also cuts detections: $β$ rises and power falls. The only way to reduce both at once is a larger sample.

Worked example — rejecting a batch of invoices

Harbourview Clearing screens incoming batches of invoices. $H_{0}$ : the batch is acceptable (its error rate is at the usual level); $H_{a}$ : the batch has too many errors. Rejecting a batch sends it back for rework.

Type I — rejecting an acceptable batch. Cost: unnecessary rework, delay, an annoyed client.
Type II — passing a bad batch. Cost: the errors reach the ledger, are found by the auditor, and Harbourview carries the exposure.

Which is worse depends on the business. If rework is cheap and audit exposure is expensive, the Type II error is the costly one and the firm should tolerate a higher $α$ — screen more aggressively — to keep $β$ down. If rework halts a production line, the balance tips the other way. The paper does not want a universal answer; it wants the two errors described in context, and a reason for the ranking.

A second grid — loan approval

A credit model tests $H_{0}$ : the applicant will repay. A Type I error declines a good borrower (lost interest income, a lost customer). A Type II error approves a borrower who defaults (a written-off loan). For most lenders the second dwarfs the first, so the model is tuned to a low $β$ at the cost of turning away some good applicants.

The audit version

An auditor tests $H_{0}$ : the control operates effectively. Type I: concluding it fails when it works — extra substantive testing, wasted hours. Type II: concluding it works when it fails — a misstatement goes undetected, and with it audit risk and liability. For the audit firm the Type II error is the serious one; that is why audit procedures are designed around detection risk rather than around false alarms.

Write each error as a sentence with three parts: the decision taken, the true state, and the consequence. "Type I is $α$ " is a definition, not a description in context, and earns no mark on its own.

The decision grid. α is chosen by the analyst; β depends on the true state, σ and n. Lowering α raises β unless n grows.