Sample Size2: 1

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Sample Size Determination In the Context of
Hypothesis Testing
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• Recall, in context of Estimation, Sample Size is
  based upon
• the width of the Confidence interval
• The confidence level (1 a)? confidence
  coefficient, z1-a/2
• The population standard error, s/?n

w
)
(
x z1-a/2(s/?n)
x z1-a/2(s/?n)
x
3

• Sample size in Context of Hypothesis Testing
• Need to consider POWER as well as confidence level

• Example Suppose we have a hypothesis on one
  mean
• Ho mo 100 vs. Ha mo ?100
• s 10
• a .05
• If the true mean is in fact ma 105,
• what size sample is required so that the power of
  the test is (1-b) .80 ?

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For our hypothesis test, we will reject Ho for x
greater than C1 or less than C2
a/2 .025
a/2 .025
mo100
C2 mo-Z1-a/2(s/?n)
C1 moZ1-a/2(s/?n)
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• Lets look at these decision points (C1 and C2)
  relative to a specific alternative.
• Suppose, in fact, that ma 105.
• We will reject Ho
• if x is greater than C1
• or x is less than C2

Distribution based on Ha
ma105
C1
C2
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We want b .20 for power.80
We want b .20 ? zb -.842
ma
C1
C2
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• Note for sample size determination
• a, b are set by the investigator
• Both a specific null (mo) and a specific
  alternative (ma) must be specified
• we assume that the same variance s2 holds for
  both the null and alternative distributions

a/2
a/2
b
ma
m0
C1 moz1-a/2(s/?n)
C1 ma-z1-? (s/?n)
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We now have C1 in terms of both the Ho and
Ha distributions Setting these equal Then
solve for n.
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• Sample Size is then
• Note
• Always use positive values for z1-a/2 and (we
  defined CI using ? positive z)

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• In our example
• s 10
• a .05 ? z1-a/2 1.96
• b .20 ? .842
• mo 100
• ma 105
• Or a sample size of n32 is needed.

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• If we change the desired power to 1-b .90
• b .10 ? 1.28
• Or a sample size of n42 is needed.

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• In the context of hypothesis testing, sample size
  is a function of
• s2, the population variance
• a .05 ? z1-a/2 , Type I error rate
• b .20 ? , Type II error rate
• Distance between mo , hypothesized mean and ma ,
  a specific alternative

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Using Minitab to estimate Sample Size Stat ?
Power and Sample Size ? 1-Sample Z
Difference between mo and ma
Desired power (separate by spaces if entering
several)
2-sided test
s
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Power and Sample Size 1-Sample Z Test Testing
mean null (versus not null) Calculating power
for mean null difference Alpha 0.05 Sigma
10 Sample Target
Actual Difference Size Power Power
5 32 0.8000 0.8074 5
43 0.9000 0.9064
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Sample size and power for comparing means of 2
independent groups.

• In the example comparing LOS for elective vs.
  emergency patients, we observed a difference
  between sample means of 3.3 days but found that
  this was NOT statistically significantly
  different from zero.
• However 3.3 days is a large, expensive difference
  in length of stay. Our data had relatively large
  observed variance, and small n.
• What was the power of our study to detect a
  difference of 3.3 days?
• What sample size would be needed per group to
  find a difference of 3 days or more significantly
  different from zero?

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In Minitab Stat ? Power and Sample Size ?
2-sample t

• To evaluate power
• Enter sample sizes
• Enter observed difference in means
• Enter standard deviation

Set a and 1- or 2- sided test, using options menu.
s
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In Minitab Stat ? Power and Sample Size ?
2-sample t
Power and Sample Size 2-Sample t Test Testing
mean 1 mean 2 (versus not ) Calculating power
for mean 1 mean 2 3.3 Alpha 0.05 Sigma
10 Sample Size Power 14 0.1342 11 0.1142
Note Minitab assumes equal ns for the 2
groups, and only gives space for one value of
s Clearly, our power to detect a difference as
large as 3.3 days was only about 12 -- not very
good!
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In Minitab Stat ? Power and Sample Size ?
2-sample t

• To estimate sample size
• Enter desired power
• Enter desired significant difference in means
• Enter standard deviation

s
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Power and Sample Size 2-Sample t Test Testing
mean 1 mean 2 (versus not ) Calculating power
for mean 1 mean 2 3 Alpha 0.05 Sigma
10 Sample Target Actual Size Power
Power 176 0.8000 0.8014 235 0.9000
0.9007
Note sample sizes are per group. If this seems
excessive is your estimate of the standard
deviation reasonable? I used the larger of the 2
observed SD here. You might want to compute a
pooled SD, and try that.