chapter 17 population means: population means: two-sample problems
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Chapter 17Chapter 17 Population means:Population means:
Two-Sample ProblemsTwo-Sample Problems
Chapter Outline
Two sample problems Comparing two population means Two-sample t procedures Examples of the two-sample t
procedures Robustness of the t procedures
1. Two sample problems
The units are not matched, and the samples can have different sizes.
When to use the two-sample test?
A study is performed to compare the mean resting pulse rate of adult subjects who regularly exercise with the rate of those who do not regularly exercise.
n s
Exercisers 29 66 8.6
Nonexercisers 31 75 9.0
This is an example of when to use the two-sample t procedures.
x
Recall Matched pairs t procedures
In a matched pairs design, subjects are matched in pairs and each treatment is given to one subject in each pair.
Note: subjects paired; same sample size. Method: apply the one-sample t
procedures to the observed difference.
Conditions for using two-sample t procedures
Notation:
Want to test (1 – 2).
Population
Variable
Mean Standard
deviation
Sample
Size
SampleMean
Sample
s.d.
1 x1 µ1 σ1 n1 s1
2 x2 µ2 σ2 n2 s2
1x
2x
2. Two-Sample t Procedures Definition: the standard deviation of
the observed difference : 21 xx
2
22
1
21
nn
Problem: We don’t know the population standard deviations 1 and 2.
Solution: Estimate them with s1 and s2. The estimator is called the standard error.
2
22
1
21
n
s
n
sSE
The test statistic: Two-sample t statistic:
SE
xxt 21
Using p-values:
. value-p if HReject
1).-n 1,-min(n ison distributi t theof
freedom of degree theand level, cesignifican theis where
).t T P(2value-p : :H vs: (1)
); t T P(value-p : :H vs: (2)
); t T P(value-p : :H vs: (1)
0
21
obs21a210
obs21a210
obs21a210
H
H
H
Using critical valuesUsing critical values
1).-n 1,-min(n ison distributi t theof
freedom of degree theand level, cesignifican theis where
.T t ifHreject : :H vs: (1)
; tifHreject : :H vs: (2)
;T tifHreject : :H vs: (1)
21
2-1
obs 021a210
obs 021a210
-1obs 021a210
H
TH
H
3. Examples
Ex 17.3(P445) A researcher buried polyester strips in the soil for different lengths of time, then dug up the strips and measured the force required to break them. Here are the breaking strengths:
2 weeks 118 126 126 120 129
16 weeks
124 98 110 140 110
population
Treatment
n s
1 2 weeks 5 123.80
4.60
2 16 weeks
5 116.40
16.09
x
21210 : vs.:
:hypothesisTest
aHH
Example 17.4 (page 446)
4. Robustness of t Procedures
The two-sample t procedures are more robust than the one-sample t methods, particularly when the distributions are not symmetric.
When the two populations have similar distribution shapes, the probability values from the t table are quite accurate, even when the sample sizes are as small as n1 = n2 = 5.
Robustness:
When the two populations have different distribution shapes, larger samples are needed.
In planning a two-sample study, it is best to choose equal sample sizes. In this case, the probability values are most accurate.
More examples Ex 17.5, P444
Ex 17.23, P460
Ex 17.25, P461
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