Stat 301 – Lecture 5
Two Independent Samples
Question
In
2000, did men and
women differ in terms of
their body mass index?
1
Populations
random
selection
2. Male
Inference
1. Female
random
selection
Samples
2
Body Mass Index
Females
Males
n1  50
n2  50
Y1  27.484
Y2  26.868
s1  7.860
s2  7.215
s p  7.544
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Stat 301 – Lecture 5
95% Confidence Interval
Y  Y   t s
*
1
2
p
1 1

n1 n2
t * from t - table with
df  n1  n2  2
4
95% Confidence Interval
Y  Y   t s
*
1
2
p
1 1

n1 n2
27.484  26.868  1.98457.544
0.616  1.98451.509 
1
1

50 50
0.616  2.995
 2.38 to 3.61
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Interpretation
We are 95% confident that the
difference in population mean
BMI for women compared to
men is between –2.38 and 3.61.
Women could have a mean BMI
as much as 2.38 lower than
men or as much as 3.61 higher
than men.
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2
Stat 301 – Lecture 5
Difference?
Because zero is in the
confidence interval, there
could be no difference in
population mean BMI’s for
women compared to men.
This agrees with the test of
hypothesis.
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Two-sample model
Y  i  
•Y represents a value of the variable
of interest

• i represents the ith population mean
• represents the random error
associated with an observation
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Conditions
The random error term,  , is
 Independent
 Identically
distributed
 Normally distributed with
standard deviation, 
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Stat 301 – Lecture 5
Residuals
Estimate of error
(Observation – Fit)
Residual
ˆ  Y  Yi
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Checking Conditions
Independence.
 Hard
to check this but the fact
that we obtained the data
through separate random
samples of women and men
assures us that the statistical
methods should work.
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Checking Conditions
Identically distributed.
 Check
using an outlier box plot.
Unusual points may come from
a different distribution
 Check using a histogram. Bimodal shape could indicate two
different distributions.
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Stat 301 – Lecture 5
Checking Conditions
Normally distributed.
Check
with a histogram.
Symmetric and mounded in
the middle.
Check with a normal
quantile plot. Points falling
close to a diagonal line.
13
Distributions
3
.99
2
.95
.90
1
.75
0
.50
Normal Quantile Plot
BMI centered by Gender
.25
-1
.10
.05
-2
.01
-3
30
20
15
Count
25
10
5
-20
-15
-10
-5
0
5
10
15
20
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BMI Residuals
 Histogram is skewed left and
mounded to the right of zero.
 Box plot is fairly symmetric with two
potential outliers on the high side.
 Normal quantile plot has points
following the diagonal line for the
first part but then wiggles around
for larger values.
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Stat 301 – Lecture 5
Equal Variance?
 All of the error terms are
supposed to be from the
same distribution with a
single standard deviation, σ.
 Display the residuals for
each group, male and
female.
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17
Equal Variance?
 Both males and females show
about the same variability.
The sample standard
deviations are very close.
 The equal variance condition
is satisfied.
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Stat 301 – Lecture 5
BMI Residuals
The identically distributed
and normally distributed
error conditions necessary
for statistical inference may
not be met for these data.
19
Consequences
The P-value for the test may not
be correct.
 Even so, there is not much of a
difference between women and
men, and I would not change
my conclusion from the test of
hypothesis.
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Consequences
The stated confidence level
may not give the true coverage
rate.
 I would still use the confidence
interval but recognize that the
true coverage rate is probably
less than 95%.
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