Sample Size Needed to Achieve High Confidence Means

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Sample Size Needed to Achieve High Confidence
(Means)
Considering estimating ?X, how many observations
n are needed to obtain a 95 confidence interval
for a particular error tolerance?
The error tolerance E is ½ the width of the
confidence interval
Here, ? is a conservative (high) estimate of the
true std dev ?X, often gotten by doing a
preliminary small sample
1.960 can be adjusted to get different confidences
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Derivation of Formula

• E z std. error z sigma/sqrt(n)
• Thus, sqrt(n) z sigma/E
• So, n z sigma/E 2

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Sample Size Needed to Achieve High Confidence
(Proportions)
Considering estimating pX, how many observations
n are needed to obtain a 95 confidence interval
for a particular error tolerance?
The error tolerance E is ½ the width of the
confidence interval
Here, p is a conservative (closer to 0.5)
estimate of the true population proportion pX,
often gotten by doing a preliminary small sample
1.960 can be adjusted to get different confidences
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Derivation

• E z std. error
• But now, std error sqrtp(1-p)/n, so
• E z sqrtp(1-p)/sqrt(n), and hence
• Sqrt(n) z sqrtp(1-p)/E,
• gt n z sqrtp(1-p)/E2

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Hypothesis Testing

• Formulate null hypothesis (action is associated
  with alternative)
• Compute Test Statistic
• Determine Acceptance Region

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Heart Valve Example

• Original yield 52
• Change Process (Sort in batches of 5)
• Sample 100 assemblies sample yield 79

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Heart Valve Example Formulate Null Hypothesis

• Null Hypothesis the true process yield is 52 or
  less ------ Why?????
• Note Action would be associated with the
  alternative----adopt new process only if it
  really increases the yield.

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Heart Valve Example Compute Test Statistic

• We will use the Normal Approximation
• (how many standard deviations to the right of .52
  is .79?)

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Heart Valve Example Determine Acceptance Region
Suppose we want to set the probability of
rejecting the null hypothesis given that it is
true (type 1 error) 0.0001 Compute Z
normsinv(.9999) 3.71947 Reject the null
hypothesis if Test statistic gt 3.71947
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Heart Valve Example conclusion

• Test statistic 5.404
• Reject null hypothesis if test statistic gt
  3.71947
• We should REJECT the null hypothesis

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Two-Sample Tests for Means

• Used when we wish to compare the means of two
  populations when both means are unknown
• Tools gt Data Analysis gt two sample test
• See Two Sample Spreadsheet

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Two Sample Tests

• Z-test two sample for means when std.
  deviations are known for both distributions
• T-test two sample assuming equal variances
• T-test two sample assuming unequal variances
• Usually just use unequal variance test

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 Important Note The use of the chi-square
test on variance requires that the underlying
population be normally distributed.
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Chi-Test Examples

• See 95 Murders.xls spreadsheet