EXAMPLE 6.30

Power of the lead concentration test. Example 6.17 (page 375) presented a test of

H₀: μ = 15.00

H_a: μ ≠ 15.00

at the 1% level of significance. What is the power of this test against the specific alternative μ = 15.50?

The test rejects H₀ when |z| ≥ 2.576. The test statistic is

Some arithmetic shows that the test rejects when either of the following is true:

z ≥ 2.576  (in other words, ≥ 15.37)

z ≤ −2.576 (in other words, ≤ 14.63)

These are disjoint events, so the power is the sum of their probabilities, computed assuming that the alternative μ = 15.50 is true. We find that

= P(Z ≥ −0.90) = 0.8159

= P(Z ≤ −6.03) ≐ 0

Figure 6.17 illustrates this calculation. A power of about 0.82, we are quite confident that the test will reject H₀ when this alternative is true.