Type I and Type II Errors Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Type I and Type II Errors.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Type I error (\alpha): rejecting H_0 when it is actually true (false positive). Type II error (\beta): failing to reject H_0 when it is actually false (false negative).

Think of a medical test. Type I error: the test says you have a disease when you don't (false alarm). Type II error: the test says you're healthy when you actually have the disease (missed detection). A smoke alarm that goes off when there's no fire is a Type I error; one that stays silent during a real fire is a Type II error. You can't eliminate bothβ€”reducing one tends to increase the other.

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: There is a trade-off: lowering \alpha (fewer false positives) raises \beta (more false negatives), and vice versa. Increasing sample size is the only way to reduce both simultaneously.

Common stuck point: Students often mix up which is which. Memory aid: Type I = false positive = seeing something that isn't there. Type II = false negative = missing something that is there.

Worked Examples

Example 1

medium
Define Type I and Type II errors. A court uses 'innocent until proven guilty.' Identify which type of error corresponds to (a) convicting an innocent person, (b) acquitting a guilty person.

Solution

  1. 1
    Type I error (false positive, \alpha): reject H_0 when H_0 is true; probability = \alpha
  2. 2
    Type II error (false negative, \beta): fail to reject H_0 when H_0 is false; probability = \beta
  3. 3
    (a) Convicting innocent: H_0 = innocent; rejecting H_0 (convicting) when person is actually innocent = Type I error
  4. 4
    (b) Acquitting guilty: H_0 = innocent; failing to reject H_0 (acquitting) when person is guilty = Type II error

Answer

(a) Convicting innocent = Type I error. (b) Acquitting guilty = Type II error.
Type I and II errors are trade-offs β€” reducing one typically increases the other (for fixed sample size). The legal system historically prioritizes minimizing Type I errors (wrongful conviction) by requiring 'proof beyond reasonable doubt' (very small Ξ±).

Example 2

hard
A medical test has \alpha = 0.05 and \beta = 0.20 (Power = 0.80). If the true disease rate is 5% in the population: (a) in 100 truly diseased patients, how many will be missed? (b) In 1000 truly healthy patients, how many will get false positives?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
If we set \alpha = 0.01 (stricter) instead of \alpha = 0.05, what happens to Type I error rate? What likely happens to Type II error rate?

Example 2

hard
A factory quality test accepts shipments if sample defect rate is below 5%. H_0: defect rate ≀ 5% (accept). Type I: reject good shipment. Type II: accept bad shipment. Which error is more costly for the factory, and how should this affect choice of \alpha?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

hypothesis testingp value