Database Reference
In-Depth Information
Taking
u
2
¼ U
p
, we got the following approach for the solution of the inequality:
u
2
¼ t
2
a
2
z
2
2
abz þ b
2
α
u
2
¼
2
2
z
2
þ β
a
2
z
2
2
abz þ b
2
u
2
2
z
2
u
2
2
α
β
0
¼
2
α
2
z
2
þ β
0
¼ z
2
a
2
u
2
2
Þ z
2
aðÞþb
2
u
2
2
ð
α
β
b
2
u
2
2
2
ab
β
0
¼ z
2
z
2
þ
2
:
a
2
u
2
α
a
2
u
2
α
The solution formula for quadratic equations
r
v
2
4
w
v
2
z
u
,
o
¼
ð
11
5
Þ
:
with
b
2
u
2
2
2
ab
β
v ¼
2
,
w ¼
2
,
ð
11
6
Þ
:
a
2
u
2
a
2
u
2
α
α
leads to the desired confidence interval:
z
u
d þ
1
z
o
z
u
1
d z
o
1
ð
11
7
Þ
:
:
We now turn to the implementation. First, we need to determine the values
a
,
b
,
. Here
EX
A
is “value per visit” of the control group and
EX
B
the similar
value for the recommendation group. For the calculation of the confidence interval
for the average order revenue, similarly “avg. order value” has to be used.
A confidence interval “CRO” can also be determined.
The variance
D
2
X
A
of the control group and
D
2
X
B
of the recommendation group
can be calculated via the quadratic sums of the order revenues:
α
,
β
D
2
X
A
¼ EX
A
EX
A
2
ð
Þ
:
2
D
2
X
B
¼ EX
B
EX
B
ð
Þ
The values
n
A
and
n
B
depend on the target quantity “visits” or “orders.”
The quantile of the normal distribution
U
p
, and hence
u
, depends on the desired
confidence level and is a constant:
• 90 % confidence interval:
U
p
¼ U
0,95
¼
1.6449
• 95 % confidence interval:
U
p
¼ U
0,975
¼
1.9600
• 99 % confidence interval:
U
p
¼ U
0,995
¼
2.5758
Therewith, using (
11.6
), we can compute
p
,
q
, and by (
11.5
) and (
11.7
), we get
the desired interval.
