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TwoSample Estimation: Paired Observation, Difference

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Between Two Means. Therefore. IEEM 320 IEEM151. Notes 19, Page 5 ... Two Samples: Estimating the Difference for Unknown but Equal s ... – PowerPoint PPT presentation

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Title: TwoSample Estimation: Paired Observation, Difference


1
Two-Sample Estimation Paired Observation,
Difference
2
Outline
  • confidence intervals of difference
  • known variance
  • unknown but equal variance
  • Unknown, unequal variance
  • paired observations


3
Procedure for Interval Estimation
  • determine a, the amount of risk that we want to
    bear, usually being 0.05 or 0.01

4
Estimations of the Difference Between Two Means
  • population means m1 m2 known variances s12
    s22
  • approximately normally distributed
    mean
  • and variance
  • Therefore

5
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.
6
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
7
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
8
Two Samples Estimating the Difference for
Unknown but Equal s
9
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.
10
Two Samples Estimating the Difference for
Unknown but Equal s
The 96 confidence interval is
11
Two Samples Estimating the Difference for
Unknown, Unequal Variances
  • a (1-a)100 confidence interval of m1-m2 s12?s22
    unknown

12
Two Samples Estimating the Difference for
Unknown, Unequal Variances
13
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

14
Two Samples Estimating the Difference for Paired
Observations
15
Two Samples Estimating the difference for Paired
Observations
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