Here's what I got for the confidence interval:
Let n = sample size, K = number of successes, p = sample proportion (=K/n), pi = true
proportion.
If n = 1250 and K = 1 (p = 1/1250), we can be 95% sure that pi > about 0.000041
(small-sample one-sided 95% confidence interval using the binomial distribution). In
particular, pi = 0 is rejected.
Here are a couple of hypothesis tests to verify this:
H0: pi = 0
H1: pi > 0
p-value = P(K >= 1) assuming H0 is true
= 0
So pi = 0 is rejected (and it always will be if there are a positive number of
successes.)
H0: pi = 0.000041
H1: pi > 0.000041
p-value = P(K >= 1) assuming H0 is true
= 0.04996
pi = 0.000041 is the limit of the confidence interval.
(using BINOMDIST from Excel)
Rodney
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Rodney Carr
School of Management Information Systems
Deakin University
PO Box 423
Warrnambool VIC 3280
Australia
email: [EMAIL PROTECTED] phone: + 61 3 5563 3458
mobile: 0417 307 692 fax: + 61 3 5563 3320
www: http://www.man.deakin.edu.au/rodneyc
-----Original Message-----
From: Donald Burrill [SMTP:[EMAIL PROTECTED]]
Sent: Wednesday, June 21, 2000 5:28 PM
To: Dale Berger
Cc: [EMAIL PROTECTED]; [EMAIL PROTECTED]
Subject: Re: Rates and proportions
On Tue, 20 Jun 2000, Dale Berger wrote:
> If we observe one escape out of 1250 inmates, why can't we reliably
> rule out zero as the population escape rate?
Because k = 1 (for n = 1250) is not significantly different from k = 0.
> The normal approximation to the binomial may not be appropriate here.
No, I don't expect it is. So use the binomial distribution.
That's supposing that one wants a statistical argument. If a purely
logical argument suffices, it is indeed the case that a counterexample
demonstrates the falsity of a proposition. But it may still be not
unreasonable to ask, with what probability may one observe one (or more)
escapes outof n=1250 (or whatever n actually applies), if the true
probability of an escape is <some suitably small non-zero value>?
(I specify non-zero only because it's difficult to carry out some
computations when p=0 exactly.)
And it is certainly reasonable to ask what confidence interval on p is
associated with k = 1.
-- Don.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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