Differences Between
Group Means
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Dr. Michael R. Hyman, NMSU
Differences between Groups
when Comparing Means
• Interval or ratio scaled variables
• t-test
– When groups are small
– When population standard deviation is
unknown
• z-test
– When groups are large
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Null Hypothesis about Mean
Differences between Groups
 
1
2
OR
  0
1
2
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t-Test for Difference of Means
1   2
t
S X1  X 2
X1 = mean for Group 1
X2 = mean for Group 2
SX1-X2 = the pooled or combined standard error
of difference between means
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Pooled Estimate of the Standard
Error
t-test for the Difference of Means
S X1  X 2
 n1  1S12  ( n2  1) S 22 )  1
1 
  
 
n1  n2  2

 n1 n2 
S12 = the variance of Group 1
S22 = the variance of Group 2
n1 = the sample size of Group 1
n2 = the sample size of Group 2
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t-Test for Difference of Means
Example
 202.1  132.6
 
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
2
S X1 X 2
2
 1 1 
  
 21 14 

 .797
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16.5  12.2
4 .3
t

.797
.797
 5.395
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Comparing Two Groups when
Comparing Proportions
• Percentage Comparisons
• Sample Proportion - P
• Population Proportion - 
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Differences between Two Groups
when Comparing Proportions
The hypothesis is:
Ho: 1  2
may be restated as:
Ho: 1  2  0
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Z-Test for Differences of
Proportions
Z 
 p1 
p 2    1   2 
S p1  p 2
p1 = sample portion of successes in Group 1
p2 = sample portion of successes in Group 2
(p1 - p1) = hypothesized population proportion 1 minus hypothesized
population proportion 1 minus
Sp1-p2 = pooled estimate of the standard errors of difference of
proportions
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Z-Test for Differences of
Proportions: Standard Deviation
S p1  p2 
pp
=
p
q
=
n1 =
n2 =
1
1 

pq 

 n1 n2 
pooled estimate of proportion of success in a sample of both
groups
(1- p
p) or a pooled estimate of proportion of failures in a sample
of both groups
sample size for group 1
sample size for group 2
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Z-Test for Differences of
Proportions: Example
S p1  p2
1 
 1
 .375 .625 


 100 100 
 .068
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Z-Test for Differences of
Proportions
n1 p1  n2 p2
p
n1  n2
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Z-Test for Differences of
Proportions: Example

100 .35  100 .4 
p
100  100
 .375
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Hypothesis Test of a Proportion
 is the population proportion
p is the sample proportion
 is estimated with p
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Hypothesis Test of a Proportion
H0 :   . 5
H1 :   . 5
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Sp 
0.60.4
100
 .0024
.24

100
 .04899
20
.6  .5
p 

Zobs 
.04899
Sp
.1
 2.04

.04899
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Hypothesis Test of a Proportion:
Another Example
n  1,200
p  .20
Sp 
pq
n
Sp 
(.2)(.8)
1200
Sp 
.16
1200
Sp  .000133
Sp  . 0115
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Hypothesis Test of a Proportion:
Another Example
Z
p
Sp
.20  .15
.0115
.05
Z
.0115
Z  4.348
The Z value exceeds 1.96, so the null hypothesis should be rejected at the .05 level.
Indeed it is significantt beyond the .001
Z
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