Page 577 - Probability and Statistical Inference

554 12. Large-Sample Inference

since its asymptotic distribution is N(0, 1) which is free from λ , λ . With
1
2
preassigned α ∈ (0, 1), we claim that

which leads to the confidence interval

for (λ − λ ) with an approximate confidence coefficient 1 − α.
1
2
Tests of Hypotheses for the Mean
Let us first pay attention to the one-sample problems. With preassigned α
∈ (0, 1), we test a null hypothesis H : λ = λ with an approximate level α
0
0
where λ (> 0) is a fixed number. The specific alternative hypotheses will be
0
given shortly. We consider the test statistic
approximately distriuted as N (0, 1) (12.3.37)

for large n when λ = λ .
0
Upper-Sided Alternative Hypothesis
Here we test H : λ = λ versus H : λ > λ . See, for example, the Figure
0
1
0
0
12.3.2. We can propose the following upper-sided approximate level α test:

Lower-Sided Alternative Hypothesis
Here we test H : λ = λ versus H : λ < λ . See, for example, the Figure
0
1
0
0
12.3.3. We can propose the following lower-sided approximate level α test:
Two-Sided Alternative Hypothesis
Here we test H : λ = λ versus H : λ ≠ λ . See, for example, the Figure
1
0
0
0
12.3.4. We can propose the following two-sided approximate level α test: