Page 593 - Probability and Statistical Inference

570 13. Sample Size Determination: Two-Stage Procedures

Now, recall that the support for the random variable S is the whole posi-
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tive half of the real line, that is S has a positive density at the point s if and
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only if s > 0.
How wide can the customary confidence interval J be?
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First, one should realize that P{W > k} > 0 for any fixed k > 0, that is
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there is no guarantee at all that the width W is necessarily going to be “small”.
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In Table 13.1.1, we present a summary from a simple simulated exercise.
Using MINITAB Release 12.1, we generated n random samples from a N(0,
σ ) population where we let n = 5, 10, α = .05 and σ = 1, 2, 5. Since the
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distribution of the sample standard deviation S , and hence that of W , is free
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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 the confidence interval J for the pa-
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rameter µ each time. Consequently, for a fixed pair of values of n and σ, we
came up with one hundred values of the random variable W , the width of the
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actually constructed 95% confidence interval for µ. Table 13.1.1 lists some
descriptive statistics found from these one hundred observed values of W .
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Table 13.1.1. Simulated Description of 95% Confidence Interval’s
Width W Given by (13.1.2): 100 Replications
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n=5
minimum = 2.841, maximum = 24.721, mean = 11.696,
standard deviation = 4.091, median = 11.541
σ = 5 n = 10
minimum = 2.570, maximum = 10.606, mean = 6.661,
standard deviation = 1.669, median = 6.536
n = 5
minimum = 1.932, maximum = 9.320, mean = 4.918,
standard deviation = 1.633, median = 4.889
σ = 2 n = 10
minimum = 1.486, maximum = 5.791, mean = 2.814,
standard deviation = 0.701, median = 2.803
n = 5
minimum = 0.541, maximum = 5.117, mean = 2.245,
standard deviation = 0.863, median = 2.209
σ = 1 n = 10
minimum = 0.607, maximum = 2.478, mean = 1.394,
standard deviation = 0.344, median = 1.412