Page 603 - Probability and Statistical Inference

580 13. Sample Size Determination: Two-Stage Procedures

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which remains unknown since σ is assumed unknown. We recall that this is
the frequentist risk discussed earlier in Section 10.4.
It is clear that based on the data X , ..., X we may estimate R (µ, ) by
1
n
n
In Table 13.3.1, we present a summary from a simple simu-
lated exercise. Using MINITAB Release 12.1, we generated n random samples
from a N(0, σ ) population where we let a = 1, n = 5, 10 and σ = 5, 2, 1. Since
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the distribution of the sample variance and hence that of is free from
the parameter µ, we fixed the value µ = 0 in this illustration. With a fixed pair
of values of n and σ, we gathered a random sample of size n, one hundred
times independently, thereby obtaining an observed value of each time.
Consequently, for a fixed pair of values of n and s, we came up with one

hundred values of the random variable . Table 13.3.1 lists some descrip-
tive statistics derived from these one hundred observed values of .
Table 13.3.1. Simulated Description of the Estimated
Values : 100 Replications

n = 5
minimum = .232, maximum = 16.324, mean = 4.534
standard deviation = 3.247, median = 3.701
σ = 5 n = 10
minimum = .457, maximum = 9.217, mean = 2.597
standard deviation = 1.443, median = 2.579
n = 5
minimum = .0569, maximum = 3.3891, mean = .7772
standard deviation = .6145, median = .6578
σ = 2 n = 10
minimum = .1179, maximum = 1.0275, mean = .3927
standard deviation = .1732, median = .3608
n = 5
minimum = .0135, maximum = .7787, mean = .1904
standard deviation = .1281, median = .1684
σ = 1 n = 10
minimum = .02843, maximum = .24950, mean = .10113
standard deviation = .05247, median = .09293
From Table 13.3.1, it is obvious that for a fixed value of σ, the estimated
risk goes down when n increases and for a fixed value of n, the same goes
down when σ decreases. But, it may feel awkward when we realize that for
any fixed pair n and σ, an experimenter may get stuck with an estimated risk
which is relatively “large”.