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Inference for 2 Samples (2 Independent Proportions (Difference in…
Inference for 2 Samples
Confidence Interval
General Form
Sample Statistic +- (Multiplier) (Standard Error)
Point Estimate +- (Multiplier) (Standard Error)
Test Statistic
General Form
Test Statistic=(Sample Statistic-Null Parameter)/Standard Error
5 Step Hypothesis Testing Procedure
Check assumptions & write hypotheses
Calculate the test statistic
Determine the
p
value
Make a decision
State a "real world" conclusion
2 Independent
Proportions
(Difference in
Proportions
)
Assumptions:
n1p1
>=10,
n1
(1-
p1
)>=10,
n2p2
>=10, &
n2
(1-
p2
)>-10
If assumptions is met
Use
Normal
Distribution &
z
Test Statistic
If assumption is
NOT
met
Use either
bootstrapping
randomization
Fisher's exact method
Standard deviation (standard error):
SE0
=(((
p1
(1-
p1
))/
n1
)+((
p2
(1-
p2
))/
n2
))^1/2
SE0
=((
p
(1-
p
)((1/
n1
)+(1/
n2
)))^1/2
Assumption:
np
>=10 &
n
(1-
p
)>=10
If assumption is met
Sample Statistic:
p1
-
p2
p1
-
p2
+-
z
(((
p1
(1 -
p1
))/
n1
) + ((
p2
(1 -
p2
))/
n2
))^1/2
Multiplier:
z
= (
p1
-
p2
)/
SE0
Standard Error:
(((
p1
(1 -
p1
))/
n1
) + ((
p2
(1 -
p2
))/
n2
))^1/2
Test Statistic:
z = (
p1n1
+
p2n2
)/(
n1
+
n2
)
p
Value:
The percentage of a
normal
distribution that is further from the mean than the
test statistic
Decide
between
null
&
alternative
If
p
<=
a
Reject
the
null
If
p
>
a
Fail
to reject the
null
"Real World" Conclusion
How to do it in
Minitab Express
Go to
Statistics
/
2-Sample Inference
/
Proportions
Change "Both samples are in one column" to
Summarized data
"Number of
events
" =
p
Select "
Options
"
"Test method:" = "
Estimate the proportions separately
"
"Confidence level:" =
95%
"Number of
trials
" =
n
How to do it in
Minitab Express
Find
p
value
Go to
Statistics
/
Probability Distribution
/
Distribution Plot
Display Probability
Under "Shade the area corresponding to the following:" select "
A specified x value
"
"X value:" =
z
Select the appropriate tail
Find
z
&
p
value
Go to
Statistics
/
2-Sample Inference
/
Proportions
Change "Both samples are in one column" to
Summarized data
"Number of
trials
" =
n
Select "
Options
"
"Test method:" = "
Use the pooled estimate of the proportion
"
"Number of
events
" =
p
2 Independent Means
Is
n
for both samples at least 30?
If so, then use
t
distribution
Point Estimate:
x1
-
x2
CI:
(
x1
-
x2
) +-
t
(((
s1
^2)/
n1
) + ((
s2
^2)/
n2
))^1/2
Standard Error:
(((
s1
^2)/
n1
) + ((
s2
^2)/
n2
))^1/2
Multiplier:
t
can be found with software.
df
= smallest
n
- 1
If not, then use
bootstrapping
or
randomization
Is
n
for both samples at least 30?
If not, then use
bootstrapping
or
randomization
If so, then use
t
distribution
Test Statistic:
t
= (
x1
-
x2
)/(((
s1
^2)/
n1
) + ((
s2
^2)/
n2
))^1/2
p
Value:
The percentage of a
normal
distribution that is further from the mean than the
test statistic
How to do it in
Minitab Express
How to find
t
Multiplier
Go to
Statistics
/
Probability Distribution
/
Distribution Plot
Display Probability
Under "Distribution" change "Normal" to "
t
"
Enter
df
Select "
Equal tails
"
"Probability:" = 100 -
CI
Find
CI
Go to
Statistics
/
2-Sample Inference
/
t
Change "Both samples are in one column" to
Summarized data
"Sample size:" =
n
"Sample mean:" =
x
Select "
Options
"
"Confidence level:" =
95%
Make sure "Assume equal variances" is
NOT
checked
"Sample standard deviation:" =
(((
s1
^2)/
n1
) + ((
s2
^2)/
n2
))^1/2
Go to
Statistics
/
Probability Distribution
/
Distribution Plot
Display Probability
Under "Distribution" change "Normal" to "
t
"
Enter
df
Under "Shade the area corresponding to the following:" select "
A specified x value
"
Select the appropriate tail
"X value:" =
t