How do I handle a proportion CI when my sample p=1?
It is an accuracy test. The equipment is very
y accurate. Time constraints are a driver in test
design. At lower sample sizes often the accuracy
will be 100%. I don't infer that the population will
be 100%, but how is CI calculated when p(1-p)=0?

How do I handle a proportion CI when my sample p=1?
It is an accuracy test. The equipment is very
y accurate. Time constraints are a driver in test
design. At lower sample sizes often the accuracy
will be 100%. I don't infer that the population will
be 100%, but how is CI calculated when p(1-p)=0?

You obviously mean the case where X = 0 or X = N.
The assumption is that X ~ Bin(p,N) with unknown parameter p.

If X = N, the 1-alpha confidence interval for p is

(alpha)^(1/N) < = p <= 1.

If X = 0, the 1-alpha confidence interval for p is

0 <= p <= 1 - (alpha)^(1/N).

On Jul 7, 2:26 pm, Jack Tomsky
jtom... at (no spam) ix.netcom.com> wrote:
How do I handle a proportion CI when my sample
p=1?
It is an accuracy test. The equipment is very
y accurate. Time constraints are a driver in test
design. At lower sample sizes often the accuracy
will be 100%. I don't infer that the population
will
be 100%, but how is CI calculated when p(1-p)=0?

You obviously mean the case where X = 0 or X = N.
The assumption is that X ~ Bin(p,N) with unknown
parameter p.

If X = N, the 1-alpha confidence interval for p is

(alpha)^(1/N) < = p <= 1.

That interval corresponds to a hypothesis test that
puts all the
rejection area in the upper tail of the sampling
distribution
and never rejects if the sample X is too small (i.e.,
if the
hypothesized value of p is too high). To justify
using that
interval when X = N, you would have to want the
interval to be of
the form lowerbound <= p <= 1 no matter what the
value of X was.


If X = 0, the 1-alpha confidence interval for p is

0 <= p <= 1 - (alpha)^(1/N).

Similarly, that interval corresponds to a hypothesis
test that puts
all the rejection area in the lower tail of the
sampling distribution
and never rejects if the sample X is too big (i.e.,
if the
hypothesized value of p is too low). To justify using
that
interval when X = 0, you would have to want the
interval to be of
the form 0 <= p <= upperbound no matter what the
value of X was.

On the other hand, if you want the interval to be of
the form
0 < lowerbound <= p <= upperbound < 1 when X is
is not 0 or N,
then you should change alpha to alpha/2 in the
expressions for
the CI when X is extreme:
X = N ==> (alpha/2)^(1/N) < = p <= 1,
X = 0 ==> 0 <= p <= 1 - (alpha/2)^(1/N).