Sample Size I: 1

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Sample Size Determination In the Context of
Estimation
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• For a confidence interval on a population mean,
  m, the width of the interval depends upon
• The confidence level (1 a)? confidence
  coefficient, z1-a/2 or t1-a/2n-1
• The population standard deviation, s or its
  estimator, s
• The sample size, n
• or

Width of interval is 2 times this !
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In fact, the width of a confidence interval, w,
is w 2 z1-a/2(s/?n)
w
(
)
x z1-a/2(s/?n)
x
x z1-a/2(s/?n)
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• Sample Size in the context of Estimation
• When planning a research study, a common first
  question is
• What sample size is necessary for a good
  estimate?
• Three pieces of information are needed
• What precision is required that is, what is the
  desired width of the confidence interval? (This
  is what is meant by good)
• What is our desired confidence level, 1 a ?
• What is the underlying variability in the
  population, e.g., what is the standard deviation,
  s ?

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Since we know the width of the interval
is we can use algebra to solve for the
sample size, n
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Lets illustrate with an example Previous
studies have shown the standard deviation of test
scores to be 25 points. How large a sample is
needed to find a 95 confidence interval for the
mean test score, with a width of 3 points?
w 3
(
)
x
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• We have
• a desired width, w 3 points
• the standard deviation, s 25 points,
• a confidence level, 1 a .95 ? z1-a/2 z.975
  1.96
• We can solve for n
• n 4(z1-a/2)2(s2) /w2 4 (1.96)2(25)2 /32
• 1067.11
• Always round up for sample size ? n 1068
• We need 1068 subjects to estimate the mean test
  score to within ?1.5 points (a width of 3 points)
  with 95 confidence.

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• Notes on sample size estimation
• It is highly dependent upon the information you
  determine is important
• If you dont have a good estimate of the
  population standard deviation
• use previous studies, published articles
• you may want to calculate n for a range of
  possible values of s
• You may need to conduct a pilot study to get a
  good estimate of the standard deviation before
  starting a larger study

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• You may need to adjust your confidence level, or
  your desired precision to get a more realistic
  sample size or decide you cant do the study!
• Continuing the example
• I decide I cant possibly recruit over 1000
  subjects for my study
• I may decide that I can be content with a
  confidence width of 5 points rather than 3
• n 4(z1-a/2)2(s2) /w2 4 (1.96)2(25)2/52
• which gives me n385 subjects, a more do-able
  study.

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• I may also decide than I can accept a lower
  confidence level,
• say 90 confidence,
• so that z1-a/2 z.95 1.645,
• my sample size estimate is now
• n 4(z1-a/2)2(s2) /w2 4 (1.645)2(25)2/52
• or an estimate of n271 subjects.

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• The choice of a confidence level is arbitrary.
• By custom, the standard is usually 95
  confidence
• However, if your study is
• more exploratory in nature, or the consequences
  of an error are not great,
• ? you may choose a lower confidence level, such
  as 90 or even less.
• If the consequences of an error are great, e.g.,
  very costly in terms of risk or money,
• ? you may wish to choose a higher confidence
  level, 99, or even higher.

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• We will revisit sample size estimation later
• for studies where we wish to estimate a
  population proportion
• in the context of hypothesis testing,
• where our goal is not only estimation, but
  testing a particular hypothesis.In this context,
  issues of power of a study will be defined.