Re: Simple confidence interval
- From: Jack Tomsky <jtomsky@xxxxxxxxxxxxx>
- Date: Fri, 16 Mar 2007 11:14:45 EDT
Suppose I have some process with an unknown, but
presumably random,
binary outcome (0 or 1). Every time I run a trial it
either succeeds
or fails.
Say I run 10 trials and I get 7 successes. I believe
.7 is the bnest
estimate of the actual probability of a success.
Is there a way to calculate a probability range (such
as .62-.74) that
I can say has a 90% (or 95%) probability of
containing the true
probability?
Can someone help me with the formula (or the method
and I'll look up
the formula) to calculate that range given:
1. The number of trials (10 in my example),
2. The number of successes (7 in my example), and
3. The "confidence level", if that's the right term
(.9 in my
example)?
Thanks for any help.
--
You can obtain confidence intervals for the binomial parameter by using the relationship between the binomial distribution and the beta distribution and the relationship between the beta distribution and the F distribution.
Let x be the observed number of successes, n be the number of trials, and 1-alpha be the confidence level.
The 1-alpha level confidence interval for p is (pL,pU), where
pL = x/[x+(n-x+1)*F(1-alpha/2; 2(n-x+1),2x)]
pU = (x+1)/[x+1+(n-1)*F(alpha/2; 2(n-x),2(x+1)]
If x=0, then pL = 0.
If x = n, then pU = 1.
Jack
.
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