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1
Inferences Comparing Two
Population Means
Chapter
6
Confidence intervals
Statistical tests
Sample size selection
3
4
Estimation for m1-m2


Point estimator
Confidence interval
1.
2.
Normal populations with known s1,s2, or two large
samples (n1,n2>30): Z interval
Normal populations with unknown s1,s2: t interval


3.
5
s1=s2: pooled t interval
s1=s2: approximate t interval
At least one nonnormal population and at least one
small sample: out of our scope
Two Populations
Non-parametric
tests
6
7
8
Sample Size for Estimating m1-m2
2( z / 2 ) s
n=
2
E
2
2
Where E is the largest tolerable error and
s1=s2=s. n is the sample size per sample.
9
Tests for m1-m2 = d0
1.
2.
Normal populations with known s1,s2, or two
large samples (n1,n2>30): Z test
Normal populations with unknown s1,s2: t test


3.
10
s1=s2: pooled t test
s1=s2: approximate t test
At least one nonnormal population and at least
one small sample: nonparametric methods
Two Populations
Non-parametric
tests
11
12
Sample Size for Testing m1-m2
When n1=n2=n and s1=s2=s
the type II error rate must be <  if |m1-m2|>=
One-tailed tests:
n=
2( z  z  ) 2 s 2
2
2( z / 2  z  ) s
2
Two-tailed tests:
13
n=

2
2
Minitab: stat>>basic statistics>>2 sample t …
14

Two-Sample T-Test and CI: C2, C1

Two-sample T for C2

C1 N Mean StDev SE Mean
1 4 5.88 1.04 0.52
2 4 4.22 1.50 0.75







Difference = mu (1) - mu (2)
Estimate for difference: 1.66442
95% CI for difference: (-0.56935, 3.89819)
T-Test of difference = 0 (vs not =): T-Value = 1.82 P-Value = 0.118 DF = 6
Both use Pooled StDev = 1.2910


Two-Sample T-Test and CI: C2, C1

Two-sample T for C2

C1 N Mean StDev SE Mean
1 4 5.88 1.04 0.52
2 4 4.22 1.50 0.75





15

Difference = mu (1) - mu (2)
Estimate for difference: 1.66442
95% CI for difference: (-0.68224, 4.01108)
T-Test of difference = 0 (vs not =): T-Value = 1.82 P-Value = 0.128 DF = 5
Paired Samples
16
17
Non-parametric Methods

Independent samples: Wilcoxon Rank Sum
Test (also called Manny-Whitney test)
–

Paired samples: Wilcoxon Signed-Rank Test
–

18
Assumption: distributions of the same shape
Assumption: symmetric distribution of the
differences
Examples: See Lab 3