Section 10.2 Example

Suppose we have a random variable X with a normal distribution with known variance σ² = 100, but unknown mean μ.
Consider the hypotheses H₀ : μ = 40 and H₁ : μ ≠ 40.
A random sample of size 25 will be taken, and we'll use the test statistic X.
Find a rejection region for level of significance α = 0.01.

The black curve is the density function under the null hypothesis H₀ : μ = 40
The dark blue curve is the density function under various values of μ greater than 40.
The Type I error is fixed at α = 0.01.
The initial value of the alternate mean is μ = 48
The value of β, the probability of a Type II error is shown.

You can experiment with different values of μ and n to see the effect on the rejection region and the probability of a Type II error.

Russell Blyth, Created with GeoGebra