Title: TwoSample Estimation: Paired Observation, Difference
1Two-Sample Estimation Paired Observation,
Difference
2Outline
- confidence intervals of difference
- known variance
- unknown but equal variance
- Unknown, unequal variance
- paired observations
3Procedure for Interval Estimation
- determine a, the amount of risk that we want to
bear, usually being 0.05 or 0.01
4Estimations of the Difference Between Two Means
- population means m1 m2 known variances s12
s22
- approximately normally distributed
mean - and variance
5Two 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.
6Two 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
7Two 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
8Two Samples Estimating the Difference for
Unknown but Equal s
9Two 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.
10Two Samples Estimating the Difference for
Unknown but Equal s
The 96 confidence interval is
11Two Samples Estimating the Difference for
Unknown, Unequal Variances
- a (1-a)100 confidence interval of m1-m2 s12?s22
unknown
12Two Samples Estimating the Difference for
Unknown, Unequal Variances
13Paired 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
14Two Samples Estimating the Difference for Paired
Observations
15Two Samples Estimating the difference for Paired
Observations