Key takeaway: Type I and Type II errors are the two ways a significance test can be wrong: a Type I error rejects a true null hypothesis, and a Type II error fails to reject a false one.
Grade 12 – AP StatisticsAP Statistics Unit 3Statistics and Data
Definition of Type I and Type II Errors
Type I and Type II errors describe the two possible mistakes in a hypothesis test, because the decision is based on a sample, not on the whole population. A Type I error is a false alarm: the data look convincing, but the null hypothesis was actually true. A Type II error is a missed detection: a real effect exists, but the sample failed to reveal it.
The two risks pull against each other. Making the test stricter lowers the chance of a false alarm and raises the chance of a miss. Only a larger sample can reduce both.
Formula and notation
The probabilities have their own symbols:
\[ P(\text{Type I})=\alpha,\qquad P(\text{Type II})=\beta,\qquad \text{power}=1-\beta. \]
Power is the probability of correctly rejecting a false null hypothesis. Both \(\beta\) and power depend on how large the true effect is.
Key facts
- The Type I error rate equals the significance level alpha chosen before the test.
- Increasing the sample size raises power and lowers beta without changing alpha.
- Lowering alpha makes a Type II error more likely if everything else stays the same.
- Which of the type I and type II errors is worse depends on context; judge the real-world cost of each.
Type I and Type II Errors example
A bank tests each card purchase with \(H_0\): the purchase is legitimate, against \(H_a\): it is fraud. The system blocks a purchase when it rejects \(H_0\) at \(\alpha=0.05\). Describe each error and what \(\alpha\) means here.
-
Type I: \(H_0\) is true but is rejected, so a legitimate purchase gets blocked.
-
Type II: \(H_0\) is false but is not rejected, so a fraudulent purchase goes through.
-
With \(\alpha=0.05\), about 5% of legitimate purchases will be blocked by mistake.
Blocking a legitimate purchase is a Type I error (probability 0.05), and approving fraud is a Type II error.
Common mistake
Watch out
A student writes that the bank commits a Type II error whenever it blocks a customer’s real purchase.
Why it is wrong
Blocking a real purchase means rejecting a null hypothesis that was true, which is the false alarm known as Type I. Memory aid for type I and type II errors: Type I is the error about a true null hypothesis, and Type II is the error about a false one that went unnoticed.
Related terms and lessons
- Significance Level (Alpha)
- Hypothesis Testing
- P-Value
- Null and Alternative Hypotheses
- Conditional Probability
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data