Quote:
Originally Posted by fourfades
So make the "technical" definition apply to real life. Thanks
Quote:
Originally Posted by statmanhal
The question I have is how would you interpret or use that information?
One particular CI will just give you an "idea" of where the true parameter lies.
Maybe it'll help if I explain the idea behind CI. In probability theory (not statistics), if you take the average of n independent identically distributed (iid) random variables w/ mean u and standard deviation a, the average is a random variable, and it converges to the normal distribution as n tends to infinity by Central Limit Theorem, and its mean is u and standard deviation is a/sqrt(n) by simple calculation.
So let X = (1/n)(Z1 + ... + Zn), the average of all Z's, where each Z is iid w/ mean u and sd a. X converges to Normal (u, a/sqrt(n)) as n goes to infinity. So we can say:
P(u-(1.96a/sqrt(n))< X < u+1.96a/sqrt(n))) = 0.95. as n tends to infinity.
which using math, is saying the exact same statement as:
P(X-(1.96a/sqrt(n)) < u < X+(1.96a/sqrt(a))) = 0.95
So now back to statistics. Notice, that above equation looks very similar to a confidence interval, in fact, the above equation is the idea behind confidence intervals. Also notice, the above equation states that the probability u is in the interval (X-(1.96a/sqrt(n), X+(1.96a/sqrt(n)) is 95%, which is a true statement, and it's what I've been saying (again remember, X is random before we gather data).
Now we gather data, and find that X=x (small x, our sample mean, is a #, it's not random). We don't know a, but we use the "bootstrap method" and guess a is the sd of samples. Now we plug in x in above interval and call it our confidence interval, which now is:
(x-(1.96a/sqrt(n)),x+(1.96a/sqrt(a)))
Does this mean u is in the above interval w/ 95% probability? No! There is nothing probabilistic in the above interval once we replace X w/ x. Everything in the above statement are #s, not random variables. All we can say is that 95% of CI's (over an infinite number of CI's) created in this fashion will contain the true mean. Unfortunately, you don't know which one w/ absolute certainty.
Last edited by :::grimReaper:::; 08-13-2010 at 05:26 AM.