TwoSample Estimation: Paired Observation, Difference

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Two-Sample Estimation Paired Observation,
Difference
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Outline

• confidence intervals of difference
• known variance
• unknown but equal variance
• Unknown, unequal variance
• paired observations

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Procedure for Interval Estimation

• determine a, the amount of risk that we want to
  bear, usually being 0.05 or 0.01

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Estimations of the Difference Between Two Means

• population means m1 m2 known variances s12
  s22

• approximately normally distributed
  mean
• and variance

• Therefore

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Two Samples Estimating The Difference for Known s

• a (1-a)100 confidence interval of m1-m2 s12 and
  s22 known

where za/2 is the z-value leaving an area of
a/2 to the right.
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Two Samples Estimating the Difference for Known s
Example Find a 96 confidence interval for the
difference of .
Solution
za/2 z0.02 2.05 The 96
confidence interval is
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Two Samples Estimating the Difference for
Unknown but Equal s

• a (1-a)100 confidence interval of m1-m2 s12
  s22 s2 unknown

c2 of ni-1 degrees of freedom
c2 of n1n2-2 degrees of freedom
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Two Samples Estimating the Difference for
Unknown but Equal s
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Two Samples Estimating the Difference for
Unknown but Equal s

• a (1-a)100 confidence interval of m1-m2 s12s22
  unknown

where sp is the pooled estimate of the population
standard deviation and ta/2 is the t-value with
v n1n2-2 degrees of freedom, leaving an area
of a/2 to the right.
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Two Samples Estimating the Difference for
Unknown but Equal s
The 96 confidence interval is
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Two Samples Estimating the Difference for
Unknown, Unequal Variances

• a (1-a)100 confidence interval of m1-m2 s12?s22
  unknown

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Two Samples Estimating the Difference for
Unknown, Unequal Variances
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Paired Observations

• observations of two populations come in pairs
• e.g., 15 individuals xi and yi weight before
  and after going on a diet
• previous methods on two samples give the
  confidence interval of the difference
• to reduce the variance of the statistics, we can
• define di xi-yi
• find variance on di as if a single sample

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Two Samples Estimating the Difference for Paired
Observations
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Two Samples Estimating the difference for Paired
Observations