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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According to frequentists, why can&#039;t probabilistic statements be made about population paramemters?The confidence interval can be expressed in terms of samples (or repeated samples): &quot;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.&quot;[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: &quot;There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter.&quot; 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.

Eva Cochran · Accepted Answer

Step 1Suppose 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 &quot;normal distribution&quot;, &quot;exponential distribution&quot;, etc.), and a parametre   θ (that is, if for example your density is a normal, then   θ 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   θ. In frequentist statistics,   θ 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   μ  =  population mean. One takes a sample       X    1    ,  …  ,      X    n   of heights and uses the fact that                                          X            n                    ¯                    −      μ                      s        n                    /                    n              ∼      t          n      −      1        .One computes a and b so that  P      (    a    &amp;lt;                                                      X              n                        ¯                          −        μ                              s          n                          /                          n                      &amp;lt;    b    )    =  0.95  ,that is,  P      (                            X          n                ¯              −    a    ⋅          s      n              /              n        &amp;lt;    μ    &amp;lt;                            X          n                ¯              −    b    ⋅          s      n              /              n        )    =  0.95.Step 2Here it makes sense to speak about probability because                     X        n            ¯       is a random variable. Now, what you do is to substitute                     X        n            ¯       (random variable) by the sample mean                     x        n            ¯       (constant value), and your confidence interval is  I  =  [                    x        n            ¯        −  a  ⋅      s    n        /        n    ,                    x        n            ¯        +  b  ⋅      s    n        /        n    ]  .The parametre   μ 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   θ 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).

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