# The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true population parameter 90% of the time."[1] Note that this does not refer to repeated measurement of the same sample, but repeated sampling.

According to frequentists, why can't probabilistic statements be made about population paramemters?
The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true population parameter 90% of the time."[1] Note that this does not refer to repeated measurement of the same sample, but repeated sampling.
And:
The confidence interval can be expressed in terms of a single sample: "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." Note this is a probability statement about the confidence interval, not the population parameter.
And:
A 95% confidence interval does not mean that for a given realised interval calculated from sample data there is a 95% probability the population parameter lies within the interval, nor that there is a 95% probability that the interval covers the population parameter.[11] Once an experiment is done and an interval calculated, this interval either covers the parameter value or it does not; it is no longer a matter of probability.
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Step 1
Suppose that you want to model the random behaviour of a certain population. Then you have to associate to the population a density function f (that is, you choose a "normal distribution", "exponential distribution", etc.), and a parametre $\theta$ (that is, if for example your density is a normal, then $\theta$ can be the population mean or the variance, etc.).
Suppose that you have decided which f you want, that is, the distribution for your population. The goal now is to estimate $\theta$. In frequentist statistics, $\theta$ is an unknown contant to be discovered. That is why we speak about confidence and not about probability.
Example: imagine I want to model the height of the people in England. I associate to it the normal distribution, so f is the density function of a normal. Now I want to estimate $\mu =\text{population mean}$. One takes a sample ${X}_{1},\dots ,{X}_{n}$ of heights and uses the fact that
$\frac{\overline{{X}_{n}}-\mu }{{s}_{n}/\sqrt{n}}\sim {t}_{n-1}.$
One computes a and b so that
$P\left(a<\frac{\overline{{X}_{n}}-\mu }{{s}_{n}/\sqrt{n}}
that is,
$P\left(\overline{{X}_{n}}-a\cdot {s}_{n}/\sqrt{n}<\mu <\overline{{X}_{n}}-b\cdot {s}_{n}/\sqrt{n}\right)=0.95.$
Step 2
Here it makes sense to speak about probability because $\overline{{X}_{n}}$ is a random variable. Now, what you do is to substitute $\overline{{X}_{n}}$ (random variable) by the sample mean $\overline{{x}_{n}}$ (constant value), and your confidence interval is
$I=\left[\overline{{x}_{n}}-a\cdot {s}_{n}/\sqrt{n},\overline{{x}_{n}}+b\cdot {s}_{n}/\sqrt{n}\right].$
The parametre $\mu$ is a constant, so either it belongs to I or not (you do not have probability here). But you have a lot of confidence that it will belong to I.
Remark: opposite to frequentist statistics, one may use bayesian statistics, which assumes that the parametre $\theta$ is a random variable, with a probability distribution to be discovered. In this case one speaks about credible regions (probabilities) and not confidence intervals (confidence).