Step by Step
I
Type I error — a false alarm
A Type I error occurs when H₀ is actually true, but you mistakenly reject it anyway — a "false positive," like crying wolf when there's no wolf. The probability of a Type I error is α, the significance level.
II
Type II error — a missed detection
A Type II error occurs when H₀ is actually false, but you fail to reject it — missing a real effect that was actually there. The probability of a Type II error is β.
Pwr
Statistical power
Power = 1 − β — the probability of correctly rejecting a false H₀ (correctly detecting a real effect when one exists). Higher power means a better chance of catching a true effect.
Trade
The trade-off between the two error types
Lowering α (making it harder to reject H₀) reduces the risk of a Type I error but increases the risk of a Type II error, and vice versa — there's an inherent trade-off between the two.
Applied Walkthrough
1
A medical test is designed to detect a disease. H₀ is "the patient does not have the disease."
2
A Type I error here means the test says the patient has the disease, but they actually don't — a false positive, causing unnecessary worry or treatment.
3
A Type II error means the test says the patient doesn't have the disease, but they actually do — a false negative, missing a real diagnosis that needed treatment.
4
If the test is made more conservative (harder to trigger a positive result) to reduce false positives (Type I), this necessarily increases the risk of missing real cases (Type II) — illustrating the fundamental trade-off between the two error types.
Exam Application
Exams test whether you can correctly distinguish Type I (false positive, probability α) from Type II (false negative, probability β) errors in a given scenario, and whether you understand the inherent trade-off between them and how it relates to power.
⚠ Common Trap
The most common trap is mixing up which error is which — remember Type I is rejecting a TRUE H₀ (a false alarm), while Type II is failing to reject a FALSE H₀ (missing a real effect).
✓ Quick Self-Check
1. What is a Type I error?
Rejecting H₀ when it is actually true — a false positive, with probability α.
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2. What is a Type II error?
Failing to reject H₀ when it is actually false — a false negative, with probability β.
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3. What is statistical power, in terms of β?
Power = 1 − β, the probability of correctly rejecting a false H₀.
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4. What happens to the risk of a Type II error if you lower α to reduce Type I error risk?
It increases — there's a trade-off between the two error types.
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5. In a medical test scenario, what would a Type II error represent?
A false negative — the test says the patient doesn't have the disease, but they actually do.
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