Type I and Type II Errors

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Type I and Type II Errors: meaning, method and solved example (AP Statistics) – Maths PDF

In short: Type I and Type II errors name the only two mistakes a hypothesis test can make: sounding an alarm when nothing is wrong, or staying silent when something real is going on.

AP StatisticsStatisticsAP Statistics Unit 3

What is Type I and Type II Errors?

A verdict from a test can miss the truth in exactly two ways, called Type I and Type II errors. A Type I error throws out \(H_0\) although it is true: a false positive. A Type II error holds on to \(H_0\) although it is false: a false negative.

The odds of a Type I error equal the significance level \(\alpha\). The odds of a Type II error are named \(\beta\), and \(1 – \beta\) is the power, the test’s ability to catch a real effect. Pushing \(\alpha\) down by itself pushes \(\beta\) up; the usual cure for both is a larger sample.

How to spot it

  • A question asks what a false alarm or a missed detection would mean in context, naming Type I and Type II errors.
  • You see alpha, beta or power mentioned in a description of a test.
  • The wording asks about the consequences of a wrong decision, such as approving a drug that does not work.
  • A problem asks how a change in sample size or significance level affects the risk of error.

The method

  1. To sort out Type I and Type II errors, state the null hypothesis in words, such as “the transaction is legitimate”.
  2. Type I error: describe rejecting that statement when it is actually true.
  3. Type II error: describe failing to reject it when the alternative is actually true.
  4. Connect to probabilities: \(P(\text{Type I}) = \alpha\), \(P(\text{Type II}) = \beta\), power \(= 1 – \beta\).
H0 is trueH0 is falseReject H0Fail toreject H0Type I errorfalse alarmprobability alphaCorrecteffect detectedpower = 1 – betaCorrectno false alarmprobability 1 – alphaType II errormissed effectprobability beta
Two columns for the truth, two rows for the decision: the green cells are correct decisions, the raspberry and amber cells are the two errors.

Type I and Type II Errors: solved example

Problem

A bank tests each card transaction with \(H_0\): the transaction is legitimate. The system blocks a transaction when the evidence against \(H_0\) is strong, using \(\alpha = 0.02\) and power 0.90. Describe both errors, and estimate how many occur on a day with 5,000 legitimate and 80 fraudulent transactions.

Solution

Type I error: blocking a legitimate transaction. Expected number: \(0.02 \times 5000 = 100\) blocked purchases.

Type II error: letting a fraudulent transaction through. With power 0.90, \(\beta = 0.10\), so about \(0.10 \times 80 = 8\) frauds slip by.

Tightening \(\alpha\) would annoy fewer customers but let more fraud through, unless more data per transaction improves the test.

Try it

Question 1

In a drug trial, \(H_0\) says the new drug is no better than the old one. Describe a Type I error.

Show the answer

Concluding the new drug is better when it is not, perhaps leading to an ineffective treatment being approved.

Question 2

What single change can lower the chances of both errors at the same time?

Show the answer

Increase the sample size: it narrows the sampling distribution, so the test can keep \(\alpha\) small and still detect real effects.

Type I and Type II Errors vs. Significance Level

The significance level only prices one of these mistakes, the false alarm. The other, a missed effect, carries its own probability \(\beta\) that \(\alpha\) does not control. Choosing \(\alpha\) therefore trades one risk against the other.

Significance Level (Alpha)

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