# Two proportions One sample t Two sample t

```One proportion
Two proportions
(from independent samples or a randomized
experiment)
One sample t
(including
Matched Pairs)
Two sample t
(from independent samples
or a randomized experiment)
Population(s)
p
p1 , p 2

1 ,  2
Mean of the
sampling
distribution
p
p1  p2

1   2
Sample(s)
p̂
pˆ 1 , pˆ 2
x
x1 , x2
Standard
Deviation of the
sampling dist’n
of the statistic
Estimate, called
the Standard
Error
SE 
pˆ (1  pˆ )
n
Check for SRS
design of study
Check for Large-sample:
normality for a successes  15
Confidence failures  15
Interval
Plus-four: See list.
Check for
normality for a
Significance Test
Critical Value
npo  10
n(1  po )  10
See conditions list.
z*
depends only on
confidence level

p1 (1  p1 ) p2 (1  p2 )

n1
n2
p(1  p)
n
SE 
n
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
Two independent SRSs or random assignment
to groups.
Large-sample: In each sample
successes  10
failures  10
 12
SE 
n1
s
n

2
SE 
 22
n2
2
s1
s
 2
n1
n2
SRS
Two independent SRSs or
random assignment to groups.
See conditions list. Same
conditions for confidence
interval and significance test.
See conditions list. Same
conditions for confidence
interval and significance test.
See conditions list. Same
conditions for confidence
interval and significance test.
See conditions list. Same
conditions for confidence
interval and significance test.
t*
depends on confidence level
and degrees of freedom
t*
depends on confidence level
and degrees of freedom
Plus-four: See conditions list.
In each sample:
successes  5
failures  5
See conditions list.
z*
depends only on confidence level
One proportion
Margin of Error
Confidence
Interval
E  z *
pˆ (1  pˆ )
n
pˆ (1  pˆ )
pˆ  z * 
n
z
Two proportions
(from independent samples or a randomized
experiment)
z *
pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )

n1
n2
pˆ (1  pˆ1 ) pˆ 2 (1  pˆ 2 )
( pˆ1  pˆ 2 )  z *  1

n1
n2
pˆ  po
One sample t
(including
Matched Pairs)
t *
s
x  t *
n
t
z
( pˆ1  pˆ 2 )  0
pˆ (1  pˆ ) pˆ (1  pˆ )

n1
n2
Where we use the pooled proportion
pˆ 
total successes
n1  n2
We use Normal table to find an exact P-value
or t-table to find an interval containing the
P-value
s12 s2 2

n1 n2
( x1  x2 )  t * 
s12 s2 2

n1 n2
HO : 1  2
HO : p1  p2
then
t *
If
If
po (1  po )
n
Where po is the
Test statistic hypothesized
population
proportion;
We use Normal
Table to find an
exact P-value or
t-table to find an
interval containing
the P-value
s
n
Two sample t
(from independent samples
or a randomized experiment)
x  o
s
n
Where o is the
hypothesized population
mean ; we use t-table to find
an interval containing the
P-value
then
t
( x1  x2 )  0
s12 s2 2

n1 n2
Software gives us the exact
degrees of freedom and Pvalue.
By hand, use the smaller
sample size minus 1 and
an interval containing the
P-value.
```