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Type II Error

A Type II error happens when you fail to reject the null hypothesis even though it is actually false. In other words, there is a real effect or difference, but your test misses it — a "false negative."

A Type II error, denoted by the probability β\beta, occurs when a statistical hypothesis test fails to reject the null hypothesis H0H_0 when the alternative hypothesis HaH_a is true. The probability of avoiding a Type II error is called the power of the test, defined as 1−β1 - \beta. A test with low power has a high chance of committing a Type II error, meaning it is unlikely to detect a real effect even when one exists.

Key Formula

β=P(fail to reject H0∣H0 is false)\beta = P(\text{fail to reject } H_0 \mid H_0 \text{ is false})
Where:
  • ββ = the probability of making a Type II error
  • H0H_0 = the null hypothesis

Worked Example

Problem: A school claims the average student score on a standardized test is 500. A researcher suspects the true mean is actually 520. She tests a sample of 25 students and gets a sample mean of 510 with a population standard deviation of 50. Using a significance level of α=0.05\alpha = 0.05 (one-tailed test), does she reject H0H_0? Could this be a Type II error?
Step 1: State the hypotheses.
H0:μ=500Ha:μ>500H_0: \mu = 500 \quad H_a: \mu > 500
Step 2: Calculate the standard error of the mean.
SE=σn=5025=10SE = \dfrac{\sigma}{\sqrt{n}} = \dfrac{50}{\sqrt{25}} = 10
Step 3: Compute the test statistic (z-score).
z=xˉ−μ0SE=510−50010=1.0z = \dfrac{\bar{x} - \mu_0}{SE} = \dfrac{510 - 500}{10} = 1.0
Step 4: Compare to the critical value. For a one-tailed test at α=0.05\alpha = 0.05, the critical z-value is 1.645. Since z=1.0<1.645z = 1.0 < 1.645, we fail to reject H0H_0.
1.0<1.645⇒fail to reject H01.0 < 1.645 \Rightarrow \text{fail to reject } H_0
Step 5: Assess the error type. If the true mean really is 520 (as the researcher suspects), then H0H_0 is false and she failed to reject it. This would be a Type II error. The test did not have enough evidence to detect the real difference.
Answer: The researcher fails to reject H0H_0. If the true mean is indeed 520, this is a Type II error — a false negative.

Why It Matters

Type II errors matter whenever missing a real effect has consequences. In medical testing, a Type II error means telling a patient they are healthy when they actually have a disease. In quality control, it means accepting a defective batch of products. Understanding β\beta helps researchers design studies with enough power — through larger sample sizes or higher significance levels — to reduce the chance of missing effects that truly exist.

Common Mistakes

Mistake: Confusing Type I and Type II errors.
Correction: A Type I error is rejecting a true H0H_0 (false positive). A Type II error is failing to reject a false H0H_0 (false negative). One way to remember: Type I is a false alarm; Type II is a missed detection.
Mistake: Thinking that failing to reject H0H_0 proves H0H_0 is true.
Correction: Failing to reject H0H_0 only means you lacked sufficient evidence against it. The null hypothesis could still be false — your test may simply not have had enough power to detect the difference.

Related Terms

  • Null Hypothesis — The hypothesis you fail to reject in a Type II error
  • p-value — Measures evidence against the null hypothesis