Type II Error

Study guide: Exam MAS-I

A Type II Error occurs when a hypothesis test fails to reject a false null hypothesis $H_0$ . It is also called a false negative.

$\beta = P(\text{Fail to Reject } H_0 \mid H_0 \text{ is false})$

$\text{Power} = 1 - \beta = P(\text{Reject } H_0 \mid H_0 \text{ is false})$

$\beta$ depends on the true value of the parameter, the significance level $\alpha$ , and the sample size $n$
Increasing sample size $n$ decreases $\beta$ (increases power) for a fixed $\alpha$
Decreasing $\alpha$ generally increases $\beta$ — there is a trade-off between Type I and Type II errors
In actuarial contexts, a Type II Error might mean failing to detect an inadequate premium rate

Power of a Test A test of $H_0: \mu = 100$ vs. $H_1: \mu = 110$ uses a sample of $n = 25$ with $\sigma = 20$ and $\alpha = 0.05$ (one-sided, upper). Find the power (probability of rejecting $H_0$ when the true mean is 110).

Answer The critical value is $c = 100 + 1.645 \cdot (20/\sqrt{25}) = 100 + 6.58 = 106.58$ . Power $= P(\bar{X} > 106.58 \mid \mu = 110) = P\!\left(Z > \frac{106.58 - 110}{20/5}\right) = P(Z > -0.855) = \Phi(0.855) \approx 0.804$ . So $\beta = 1 - 0.804 = 0.196$ ; there is about a 20% chance of a Type II Error.