Page 568 - Probability and Statistical Inference

12. Large-Sample Inference 545

with approximate confidence coefficient 1 − α.
Tests of Hypotheses for the Population Mean
Let us first pay attention to the one-sample problems. With preassigned α
∈ (0, 1), we wish to test a null hypothesis H : µ = µ at an approximate level
0
0
α where µ is a specified real number.. The alternative hypotheses will be
0
given shortly. One considers the test statistic
approximately distributed as N(0,1) (12.3.8)
for large n when µ = µ .
0
Upper-Sided Alternative Hypothesis
Here we test H : µ = µ versus H : µ > µ . Along the lines of Section 8.4,
0
0
0
1
we can propose the following upper-sided approximate level a test:
Reject H in favour of H : µ > µ if and only if U > Z
0 1 0 α

Figure 12.3.2. Upper-Sided Standard Normal Rejection Region
Lower-Sided Alternative Hypothesis
Here we test H : µ = µ versus H : µ < µ . Along the lines of Section 8.4,
0
0
0
1
we can propose the following lower-sided approximate level α test:

Figure 12.3.3. Lower-Sided Standard Normal Rejection Region