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Here the objective is to take samples from a population and construct an interval [l,u] for the unknown value of the parameter being estimated.  Typically, there is a confidence coefficient associated with the interval, e.g. one may construct a 99% confidence interval or a 96% confidence interval, or in general, a 100*(1-a)% interval where a is a small fraction (typically less than or equal to 0.1)

One should be very clear on how to interpret such an interval.  Given a 100*(1-a)% confidence interval [l,u] for the value of some parameter q, it is NOT true that "there is a probability of (1-a)  that the value of q lies within this interval."  In fact, there IS no such probability since the value either lies within the interval (in which case the probability is equal to 1) or it does not lie within the interval (in which case the probability is equal to 0)!

A better way to interpret the interval is that there is a probability of (1-a) that the interval covers the (unknown) value of q.  More precisely, suppose we were to follow the correct sampling procedure used to construct the interval and we were to repeat this procedure a very large number of times, constructing a confidence interval each time. Note that these intervals will in general be different each time since they are based upon different random samples that will in general be different from each other.  If we were to follow this procedure, then we would expect about 100*(1-a)% of the intervals to contain the true value of q.

Example:  For the simplest case consider a random sample of size n drawn from a Normal population N(m,s²), whose mean m is unknown but S.D. s  is known.  Suppose the  sample mean is .  Then a 100(1-a)% Confidence Interval for m is given by .