1
Formula Sheet for the Final Exam
.
If X has a Binomial Distribution with probability of success p and n number of trials, then:
1.
𝑃(? = ?) = (
?
?
) ?
?
?
?−?
, ? = 0, 1, 2, ... , ?,
2.
The mean is
𝐸(?) = ?. ?
3.
The variance
𝜎2 = ?. ?. ?
where
? = (1 − ?)
4.
Standard deviation
q
p
n
.
.
Least Square Line:
__
1
__
0
1
1
0
^
,
x
b
y
b
and
S
S
r
b
where
x
b
b
y
x
y
,
Any random variable
X with normal distribution can be:
Standardized to obtain
X
Z
, with new mean = 0 and Variance = 1.
For the population proportion
p
, confidence interval is
ˆ
p
/2
ˆ ˆ
/
z
pq n
. The sample size
n
is large when
?𝜋 > 10 ??? ?(1 − 𝜋) > 10.
1
0.80
0.85
0.90
0.95
0.99
/ 2
z
1.28
1.44
1.645
1.96
2.575
In testing
0
H
:
p
=
𝜋
0
, the test statistic is
? =
?̂ − 𝜋
0
√
𝜋
0
(1 − 𝜋
0
)
?
, ?ℎ??? ?̂ =
?
?
The mean of the sampling distribution of the sample mean is
𝜇
?
̅
= 𝜇
The standard error of the sampling mean
?
̅
is
𝜎
√?
Testing the difference between two proportions, Under the null hypothesis:
?
0
∶ 𝜋
1
− π
2
= 0
The pooled proportion is given by:
2
1
2
1
ˆ
n
n
x
x
p
Pooled
The test statistic is
2
1
2
1
2
1
2
1
0
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
)
ˆ
ˆ
(
ˆ
ˆ
n
q
p
n
q
p
p
p
p
p
SE
p
p
z
Pooled
Pooled
Pooled
Pooled
Pooled
2
2
2
1
1
1
2
2
1
ˆ
1
ˆ
ˆ
1
ˆ
ˆ
ˆ
Pr
%
100
)
1
(
n
p
p
n
p
p
z
p
p
oportions
Population
Two
between
Difference
The
for
Interval
Confidence
2
2
/
2
2
/
)
25
(.
then,
,
p
about
nothing
know
you
If
)
ˆ
1
(
ˆ
then
,
experience
pass
from
p
of
estimate
good
a
know
you
If
E
z
n
E
z
p
p
n
n
p
p
z
p
oportion
Population
for
Interval
Confidence
)
ˆ
1
(
ˆ
ˆ
:
Pr
%
100
)
1
(
2