Finding power by hand. Even though software is used in practice to calculate power, doing the work by hand builds your understanding. Return to the test in Example 15.7. There are n = 10 observations from a population with standard deviation

σ = 1 and unknown mean μ. We will test H0: μ = 0 Ha: μ > 0 with fixed significance level α = 0.05. Find the power against the alternative μ =

0.8 by following these steps.

(a) The z test statistic is z = x − μ0

σ/

√

n

= x − 0 1/



10

= 3.162x

(Remember that you won’t know the numerical value of x until you have data.) What values of z lead to rejecting H0 at the 5% significance level?

(b) Starting from your result in (a), what values of x lead to rejecting H0? The area above these values is shaded under the top curve in Figure 15.2.

(c) The power is the probability that you observe any of these values of x when

μ = 0.8. This is the shaded area under the bottom curve in Figure 15.2. What is this probability?