Dear DCstats and ERosa Starting a new section here because the preceding is hopelessly indented DCstats and ERosa you are mistaken in your thinking Jdannan and Avenue are quite correct I know this might be very difficult for you to accept You claim to have references supporting your interpretation However I think you are misinterpreting the statements from the textbooks you are reading I can t speak for your petroleum industry statistican friend My guess is that you conveyed your misinterpretation of the situation to him her I have a ph d in mathematics and after reading this talk page and thinking I was almost losing my mind I consulted with several other ph d mathematician and statistician friends of mine who confirmed that Jdannan and Avenue are indeed correct And the interpretation given in the article is basically correct as well In fact it s the interpretation I ve seen given in every stats book I ve looked at Some references since these seem to be very important to people Warning There is much confusion about how to interpret a confidence interval A confidence interval is not a probability statement about since is a fixed quantity not a random variable Larry Wasserman All of Statistics A Concise Course in Statistical Inference p 92 CAUTION A 95 confidence interval does not mean that there is a 95 probability that the interval contains Remember probability describes the likelihood of undetermined events Therefore it does not make sense to talk about the probability that the interval contains since the population mean is a fixed value Think of it this way Suppose I flip a coin and obtain a head If I ask you to determine the probability that the flip resulted in a head it would not be 0 5 because the outcome has already been determined Instead the probability is 0 or 1 Confidence intervals work the same way Because is already determined we do not say that there is a 95 probability that the interval contains Michael Sullivan III Statistics Informed Decisions Using Data p 453 The idea of interval estimation is complicated an example is in order Suppose that for each x is a real random variable normally distributed about with unit variance then as is very easy to see with the aid of a table of the normal distribution if M x is taken to be the interval x 1 9600 x 1 9600 then 1 P in M x where is constant and almost equal to 0 95 It is usually thought necessary to warn the novice that such an equation as 1 does not concern the probability that a random variable lies in a fixed set M x Of course is given and therefore not random in the context at hand and given is the probability that M x which is a contraction of x has as its value an interval that contains Leonard J Savage The Foundations of Statistics p 260 Your misunderstanding seems to be based on an inadequate grasp of a few facts The population mean is a fixed constant not a random variable The endpoints of the confidence interval are random variables not fixed constants A statement about the level of confidence of a confidence interval is a probabilistic statement about the endpoints of the interval considered as random variables The issue about whether is known or not is a complete red herring What matters is that is fixed not whether it s known This also has absolutely nothing to do with probabilistic interpretations of quantum physics good lord Let me go through an example which has the most simplifying assumptions Suppose we have a normally distributed random variable X on a population with mean 100 and standard deviation 16 and suppose we select samples of size n 100 Then the sample mean X bar is a sampling distribution defined on the space of all possible samples of size 100 and it has mean 100 and standard deviation sqrt n 16 sqrt 100 16 10 1 6 A 95 confidence interval for based on this sampling is then given by X bar z 2 sqrt n X bar z 2 sqrt n X bar 1 96 1 6 X bar 1 96 1 6 X bar 3 136 X bar 3 136 Now note the following facts The confidence interval is defined to be a random interval i e the endpoints of the confidence interval are random variables defined in terms of the sample mean X bar We know what the population mean is and yet we wer