Page 599 - Probability and Statistical Inference

576 13. Sample Size Determination: Two-Stage Procedures

The last step in (13.2.9) is valid because (a) the random variable I(N = n) is
determined only by , (b) and are independent for all n = m, and
hence the two random variables and I(N = n) are independent for all n ≥
m. Refer to Exercise 13.2.2. Now, we can rewrite (13.2.9) as

in view part (i). Hence, is distributed as N(0, 1).
Observe that given , we have a fixed value of N. Hence, given , we
claim that the conditional distribution of But,
this conditional distribution does not involve the given value of the condition-
ing variable . Thus, and are independently distributed.
(iv) We combine facts from part (iii) and Definition 4.5.1 to conclude that
is distributed as Student’s t with m − 1 degrees of free-
dom.
(v) By the Theorem 3.3.1, part (i), we can express as

which reduces to µ since P (N < ∞) = 1. In the same fashion, one can
µ,σ 2
verify that E µ,σ 2 = µ +σ E µ,σ 2 [1/N] and the expression of can
2
2
be found easily. The details are left out as Exercise 13.2.3. !
Now we are in a position to state a fundamental result which is due to
Stein (1945, 1949).
Theorem 13.2.2 For the stopping variable N defined by (13.2.4) and the
fixed-with confidence interval for the population
mean µ, we have:

for all fixed µ ∈ ℜ, σ ∈ ℜ , d ∈ ℜ and α ∈ (0, 1).
+
+