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.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

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