What is the difference between a type I and type II error

In hypothesis testing you start with a null hypothesis H₀ and an alternative H₁. Deciding whether to reject H₀ involves two possible mistakes. A Type I error (false positive) happens if you reject H₀ when it is actually true, and a Type II error (false negative) happens if you accept H₀ when it is actually false.

Formal definitions

The probability of a Type I error is denoted by $\alpha$ and is set before the test; it is the significance level. The probability of a Type II error is denoted by $\beta$ and depends on the true effect size, sample size, and variability. Power equals $1-\beta$ and measures the test’s ability to detect a real effect.

Key points to remember:

• α is the chance of rejecting a true H₀.
• β is the chance of not rejecting a false H₀.
• Reducing α usually increases β unless you increase sample size.

Consider a concrete example. You test H₀: μ = 100 versus H₁: μ > 100 with known σ = 15, n = 25, and choose α = 0.05. The critical z‑value is 1.645, giving a rejection threshold of 100 + 1.645·(15/$\sqrt{25}$ ≈ 104.94. If the true mean is 108, the test’s β is Φ((104.94‑108)/(15/5)) ≈ Φ(‑1.02) ≈ 0.154, so the Type II error probability is about 15.4 % and power is 84.6 %.

Compute the error probabilities in the example:

1. 1Calculate the standard error: SE = σ/$\sqrt{n}$ = 15/5 = 3.
2. 2Find the critical value: 100 + 1.645·SE ≈ 104.94.
3. 3Compute β: Φ((104.94‑108)/SE) ≈ 0.154.
4. 4Set α = 0.05 by construction.

The example shows how α is fixed by the researcher, while β changes with the assumed true mean. If you wanted a smaller β (higher power), you could increase n, which would lower the standard error and move the critical region closer to the true mean. This trade‑off is central to experimental design.

Summary of error probabilities:

Understanding the distinction helps you interpret results correctly. Reporting both α and the achieved power gives readers a complete picture of the test’s reliability. Ignoring β can lead to overconfidence in non‑significant findings.