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Confidence Interval
Let's say I have an event for which I do not know the probability p of happening. However I have sample data consisting of 14 points for which this event never happened.What is the confidence interval at 95% for this event ? (make any sensible assumption you need)
Last edited by bluetrin on May 9th, 2013, 10:00 pm, edited 1 time in total.
Confidence Interval
I am afraid, my solution was too easy for you 95% confidence interval:[$](1-p)^n=0.05[$][$]n\cdot ln(1-p)=ln(0.05)[$]for small p:[$]n \cdot -p=ln(0.05)[$][$]n\cdot -p=ln(0.05)[$][$]p=-\frac{ln(0.05)}{n}[$]Since ln(0.05) is very close to -3, this leads to the rule of 3 for an interval of 95%: i.e. 3/n gives you a good approximation (link).You are probably right with Jeffreys' or Bayes'.
Last edited by bluetrin on May 12th, 2013, 10:00 pm, edited 1 time in total.
Confidence Interval
Consider only the upper bound, i.e. the lower bound is 0. Denote upper bound by u, and let X be Bin(14, p)Now put H0: p > u, H1: p < uP(Obtain our data or worse under H0) = P(X = 0 | X bin(14, u) = (1 - u)^14 >= 0.05 [If we do not wish to reject] => u = 1 - 0.05^(1/14).That is, we reject H0 at 95% for values of u less than 1 - 0.05^(1/14) ~= 0.1926.While there is now lower bound on p, if it were larger than 0.1926 then our observed data would be more unusual than 1/20.I.e. our confidence interval is [0, 0.1926]