Transcript sample
Two Independent Means
Unit 8
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Sampling Considerations
One sample or two?
If two samples, paired
or independent?
Is the response variable
quantitative or
categorical?
Am I interested in the
mean difference?
This chapter → two independent samples → quantitative response → interest in mean difference
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One sample
SRS from one population
Comparisons made to an external reference population
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Paired Sample
Two samples with each observation in sample 1 matched to a
unique observation in sample 2
Just like a one-sample problem except inferences directed
toward within-pair differences DELTA
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Independent sample inference
Independent samples from two populations
No matching or pairing
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What type of sampling method?
1. Measure vitamin content in loaves of bread and see
2.
3.
if the average meets national standards.
Compare vitamin content of bread immediately
after baking versus 3 days later (same loaves are
used on day one and 3 days later)
Compare vitamin content of bread immediately
after baking versus loaves that have been on shelf
for 3 days
1 = single sample
2 = paired samples
3 = independent samples
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Illustrative example: independent
samples
Goal: compare response variable in two groups
Fasting cholesterol (mg/dl)
Group 1 (type A personality):
233, 291, 312, 250, 246, 197, 268,
224, 239, 239, 254, 276, 234, 181,
248, 252, 202, 218, 212, 325
Group 2 (type B personality)
344, 185, 263, 246, 224, 212, 188,
250, 148, 169, 226, 175, 242, 252,
153, 183, 137, 202, 194, 213
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Data setup for independent samples
Two columns
Response
variable in one
column
Explanatory
variable in other
column
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Side-by-side boxplots
Compare locations, spreads, and shapes
400
Interpretation:
21
20
(1) Different locations
(group 1 > group 2)
300
(2) Different spreads
(group 1 < group 2)
200
100
N=
20
20
1
2
(3) Shape: fairly
symmetrical (but
both with outside
values)
GROUP
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Summary statistics by group
If no major departures from Normality, report means and standard
deviations (and sample sizes)
Group
n
mean
std dev
1
2
20
20
245.05
210.30
36.64
48.34
Take time to look at your results.
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Notation for independent samples
Parameters (population)
Group 1
N1
µ1
σ1
Group 2
N2
µ2
σ2
Group 1
n1
s1
Group 2
n2
x1
x2
Statistics (sample)
s2
x1 x2 is the sample mean difference
x1 x2 estimates 1 2
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Sampling distribution of mean difference
The sampling distribution of the mean difference is key to inference
x1 x2 ~ N ( 1 2 , SE x1 x2 )
{FIGURE DRAWN ON BOARD}
The SDM difference tends to be Normal with expectation μ1 − μ2
and standard deviation SE; (SE discussed next slide)
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Pooled Standard Error
Illustrative data (summary statistics)
Group
ni
si
xbari
1
20
36.64
245.05
2
20
48.34
210.30
df1 n1 1 20 1 19
s
2
pooled
df 2 n2 1 20 1 19
df df1 df 2 19 19 38
(df1 )( s12 ) (df 2 )( s22 )
df
(19)(36.64 2 ) (19)( 48.34 2 )
38
1839.623
1 1
1
1
SEx1 x2 s 2pooled 1839.623 13.56
20 20
n1 n2
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Confidence interval for µ1 – µ2
(1−αlpha)100% confidence interval for µ1 – µ2
( x1 x2 ) t df ,1 SE x1 x2
2
Illustrative example (Cholesterol in type A and B men)
( x1 x2 ) (t n 1,.975 )( SE x1 x2 )
(245.05 210.30) (2.02)(13.56)
34.75 27.39
(7.36, 62.14)
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Comparison of CI formulas
(point estimate) (t*)( SE )
Type of
sample
single
paired
independent
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point
estimate
df for t*
n 1
x
nd 1
xd
x1 x2
(n1 1) (n2 1)
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SE
n
delta
n
1 1
s 2pooled
n1 n2
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Independent t test
A. H0: µ1 = µ2
vs.
H1: µ1 > µ2 or
H1: µ1 < µ2 or
H1: µ1 µ2
tstat
( x1 x2 )
SE x1 x2
with df df1 df 2
B. Independent t
statistic
C. P-value – use t
table or software
utility to convert
tstat to P- value
D. Significance level
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Pooled t statistic
(n1 1) (n2 2)
Illustrative example
SE x1 x2 13.56
x1 x2 245.05 210.30 34.75
df 19 19 38
tstat
x1 x2 34.75
2.56
SE x1 x2 13.56
One - sided P between 0.01 and 0.005
Two - sided P between 0.01 and 0.02
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SPSS output
These are the pooled (equal variance) statistics calculated in HS 167
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Conditions necessary for t procedures
Validity assumptions
good information (no information bias)
good sample (“no selection bias”)
good comparison (“no confounding” – no lurking
variables)
Distributional assumptions
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Sampling independence
Normality
Equal variance
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Sample size requirements for
confidence intervals
1.96
n
d
2
This will restrict the margin of error to no bigger than plus or minus d
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Sample size requirement for CI
Suppose, you have a variable with = 15
4 152
For d 5, use n
36
2
5
4 152
For d 2.5, use n
144
2
2.5
4 152
For d 1, use n 2 900
1
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Sample size
requirements
increases when you
need precision
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Sample size for significance test
Goal: to conduct a significance test with adequate
power to detect “a difference worth detecting”
The difference worth detecting is a difference
difference worth finding.
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In a study of an anti-hypertensives
for instance, a drop of 10 mm Hg might be worth
detecting, while a drop of 1 mm Hg might not be
worth detecting.
In a study on weight loss, a drop of 5 pounds might
be meaningful in a population of runway models, but
may be meaningless in a morbidly obese population.
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Determinants of sample size
requirements
“Difference worth detecting” ()
Standard deviation of data ()
Type I error rate ()
We consider only .05 two-sided
Power of test (we consider on 80%
power)
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Sample size requirements for test
Approx. sample size needed for 80%
power at alpha = .05 (two-sided) to
detect a difference of Δ:
16
n
1
2
2
Illustrative example: Suppose Δ = 25 and
= 45 …
2
16 45
n
1 52.8 53
2
25
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