Chapter 14
Hypothesis Tests:
Two Independent Samples
2
Chapter 14 Two Independent
Samples
•Major Points
• An example
• Distribution of differences between
means
• Heterogeneity of Variance
• Nonnormality
• Confidence limits
Cont.
3
Chapter 14 Two Independent
Samples
•Major Points--cont
• SPSS printout
• Review questions
•Violent Videos Again
Chapter 14 Two Independent
Samples
• Bushman (1998) Violent videos and
aggressive behavior
• Doing the study Bushman’s way—
almost (previously we looked at a
one group example, comparing
group mean to population mean)
• Bushman actually had two
independent groups
• Violent video versus non-violent
video
• We want to compare mean number
of aggressive associations between
groups
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5
Chapter 14 Two Independent
Samples
•The Data
Condition
Mean St. Dev. Variance
n
Violent video
7.10
4.40
19.36
100
NonViolent
video
5.65
3.20
10.24
100
Chapter 14 Two Independent
Samples
6
•Analysis
• These are Bushman’s data, though he
had more dimensions.
• We still have means of 7.10 and 5.65,
but both of these are sample means.
• We want to test differences between
sample means.
• Not between a sample and a
population mean
• Not between related samples
• We call these Independent samples
Cont.
7
Chapter 14 Two Independent
Samples
•Analysis--cont.
• Q: How are sample means distributed if
H0 is true?
• Need sampling distribution of differences
between means
• Same idea as before, except statistic is
(mean1 - mean2)
• Distribution of Differences between
means over repeated samplings
X
8
Chapter 14 Two Independent
Samples
•Sampling Distribution of Mean
Differences
• Mean of sampling distribution = 1 - 2
(remember that the mean of the sampling
distribution should equal the population
value, here the pop. mean difference)
• Standard deviation of sampling
distribution (standard error of mean
differences) =
sX 1 X 2 
s12 s22

n1 n2
Note. The variance of the differences between 2 samples is equal to the
sum of their variances
Cont.
Chapter 14 Two Independent
Samples
•Sampling Distribution--cont.
• Distribution approaches normal as n
increases.
• Later we will modify this to “pool”
variances.
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10
Chapter 14 Two Independent
Samples
•Analysis--cont.
• Same basic formula as before, but with
accommodation to 2 groups.
X1  X 2
X1  X 2
t

sX 1 X 2
s12 s22

n1 n2
• Note parallels with earlier t, p 291
11
Chapter 14 Two Independent
Samples
•Our Data
X1  X 2
7.10  5.65
t

2
2
19.36 10.24
s1 s2


100 100
n1 n2
1.45 1.45


 2.66
.296 .544
12
Chapter 14 Two Independent
Samples
•Degrees of Freedom
• Need to know the df to look up the critical
t value for rejecting null
• Each group has 100 subjects.
• Each group has n - 1 = 100 - 1 = 99 df
• Total df = n1 - 1 + n2 - 1 = n1 + n2 - 2
100 + 100 - 2 = 198 df
• t.025(198) = +1.97 (approx.)
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Chapter 14 Two Independent
Samples
•Conclusions
• Since 2.66 > 1.97, reject H0.
• Conclude that those who watch violent
videos produce more aggressive
associates than those who watch
nonviolent videos.
Chapter 14 Two Independent
Samples
•Assumptions
• Two major assumptions
• Both groups are sampled from
populations with the same
variance
• “homogeneity of variance”
• Both groups are sampled from
normal populations
• Assumption of normality
• Frequently violated with little
harm.
• Sample sizes greater than
30, Central Limit theorem
says approaches normal
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Chapter 14 Two Independent
Samples
15
•Pooling Variances
• If we assume both population
variances equal, then a
weighted average of sample
variances would be better
estimate (weighted by each
2 or n size).
2
samples
df,
(n  1) s  n  1s
s 
2
p
1
1
2
2
n1  n2  2
99(19.36)  9910.24 29304


 14.8
100  100  2
198
Cont.
Chapter 14 Two Independent
Samples
•Pooling Variances--cont.
• Substitute sp2 in place of separate
variances in formula for t.
• Will not change result if sample sizes
equal
• Do not pool if one variance more than 4
times the other AND unequal group sizes,
or if the test of unequal variances
(Levene’s test) is significant.
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Chapter 14 Two Independent
Samples
•Heterogeneous Variances
• Refers to case of unequal population
variances (one variance is more than
4 times the other or test is significant)
• We don’t pool the sample variances.
• We adjust df and look t up in tables for
adjusted df, this is a more
conservative test (need larger t to be
significant).
• One option is to use the Minimum df =
smaller group n - 1.
• Most software calculates optimal
df, slightly larger than smaller n-1.
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18
Chapter 14 Two Independent
Samples
•Confidence Limits on the mean difference


CI .95  X 1  X 2  t .025 s X 1  X 2
 (7.1  5.65)  1.97  .544
 1.45  1.97  .544  1.45  1.07
 0.38  diff  2.53
Cont.
Chapter 14 Two Independent
Samples
•Confidence Limits--cont.
• p = .95 that interval formed in this way
includes the true value of 1 - 2.
• Not probability that 1 - 2 falls in the
interval, but probability that interval
includes 1 - 2.
• Reference is to “interval formed in this
way,” not “this interval.” Minor
difference in wording.
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20
Chapter 14 Two Independent
Samples
•Computer Example
• SPSS reproduces the t value and the
confidence limits.
• All other statistics are the same as well.
• Note different df depending on
homogeneity of variance.
• Don’t pool if heterogeneity
• and modify degrees of freedom
Chapter 14 Two Independent
Compare
Samples the two
means
Group Statistics
AGGRESS
Mean
Std. Error
Mean
N
1.00
100
7.1000
4.4000
.4400
2.00
100
5.6500
3.2000
.3200
Quick check for
normality, does mean +
and – 2 SD equal range?
If not, probably skewed,
non-normal
Independent Samples Test
Levene’s test is decision
making point, null is that
there is no diff in
variances, if it is sig (p <
.05) assumption violoated,
move down
AGGRESS
Std.
Deviation
GROUP
Each
group
has 100
people
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Estimate of
sampling
error, SD of
sampling
disctribution
Equal
variances
assumed
Equal
variances not
assumed
Levene's
T est for
Equality of
Variances
95% confident that the
diff in group means in
population falls
between – and -, notice
that 0 is not in interval
This is the test of
difference
between group
means, Rule of
thumb, t > 2
reject
t-test for Equality of Means
Mean
gr1 –
mean
gr2
95% Confidence
Interval of the
Difference
Std.
Error
Differe
nce
Lower
Upper
Sig.
(2-tailed)
Mean
Differen
ce
198
.008
1.4500
.5441
.3771
2.5229
181
.008
1.4500
.5441
.3765
2.5235
F
Sig.
t
df
8.8
.003
2.665
2.665
Optimal df
computed, not
pooled
Sig or p </= alpha
.05, reject null, there
was a stat sig diff
SE Diff
in group
means
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Chapter 14 Two Independent
Samples
Interpretation
• Have the assumptions been met?
– Normality: histograms or mean + and – 2 SD’s
– Equal variances: Interpret Levene’s test
• Is the test of Ho statistically significant?
– T and sig level
• What is the effect size? Is the mean difference
large? Is is about a 1 sd difference, 2 sd’s?
How LARGE is it?
– Diff in means/pooled SD (quick way is to average SD
for 2 groups)
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Chapter 14 Two Independent
Samples
•Review Questions
• Why do you suppose it makes a difference
whether we compare a sample and a
population mean or two sample means?
• What do we know about the sampling
distribution of differences between
means?
Cont.
Chapter 14 Two Independent
Samples
24
•Review Questions--cont.
• What do we mean by “the standard error
of differences between means?”
• How do we calculate the degrees of
freedom with two groups?
• What are the underlying assumptions?
• What does it mean to “pool” the
variances?
Cont.
Chapter 14 Two Independent
Samples
•Review Questions--cont.
• When would we not pool the variances?
• What is the difference between how we
calculate confidence limits with one or
with two means?
• What does the text mean by modifying the
df when we have heterogeneity of
variance?
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