> N0<2000; t<200; k<400
> alpha<c(0.05,0.02,0.01)
> ralpha<qhyper(1alpha,t,N0t,k)
> data.frame(alpha,ralpha)
alpha ralpha
1
0.05
49
2
0.02
51
3
0.01
53
For example, we must capture al least 49 that were tagged in order to reject
H
0
at the
↵
= 0
.
05
level. In this case
the estimate for
N
is
ˆ
N
= [
kt/r
↵
] = 1632
. As anticipated,
r
↵
increases and the critical regions shrinks as the value
of
↵
decreases.
Using the level
r
↵
determined using the value
N
0
for
N
, we see that the power function
⇡
(
N
) =
P
N
{
R
≥
r
↵
}
.
R
is a hypergeometric random variable with mass function
f
R
(
r
) =
P
N
{
R
=
r
}
=
(
t
r
)(
N

t
k

r
)
(
N
k
)
.
The plot for the case
↵
= 0
.
05
is given using the
R
commands
> N<c(1300:2100)
> pi<1phyper(49,t,Nt,k)
> plot(N,pi,type="l",ylim=c(0,1))
We can increase power by increasing the size of
k
, the number the value in the second capture. This increases the
value of
r
↵
. For
↵
= 0
.
05
, we have the table.
> k<c(400,600,800)
> N0<2000
> ralpha<qhyper(0.95,t,N0t,k)
> data.frame(k,ralpha)
k ralpha
1 400
49
2 600
70
3 800
91
We show the impact on power
⇡
(
N
)
of both significance level
↵
and the number in the recapture
k
in Figure 18.3.
Exercise 18.4.
Determine the type II error rate for
N
= 1600
with
•
k
= 400
and
↵
= 0
.
05
,
0
.
02
,
and 0.01, and
•
↵
= 0
.
05
and
k
= 400
,
600
,
and 800.
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