Consider these research statements:

(1) “We are 90 percent confident that the population mean is between 5 and 15.”

(2) “90 percent of the confidence intervals formed in this way will contain the population mean, and this particular interval extends from a lower limit of 5 to an upper limit of 15.”

(3) "There is a 90 percent chance that the population mean is between 5 and 15."

Are these statements saying the same thing?

I think statement (3) is most correct, because it phrases in the correct language, but can someone help in explaining why precisely? Why are (1) and (2) bad candidates?

  • 1
    $\begingroup$ For frequentist confidence intervals, (2) is the recommended statement while (3) is wrong: the population mean is fixed (albeit unknown) and so has a probability of $0$ or $1$ of being between $5$ and $15$. (1) is unhelpfully ambiguous as to whether (2) or (3) is intended as the meaning and whether it is the meaning hearers/readers receive. If you prefer statements like (3) then use Bayesian credible intervals and techniques. $\endgroup$ – Henry Oct 18 '16 at 17:48
  • $\begingroup$ @Henry, I'm trying to choose one. Can you explain with a formal answer as to why which one is best? $\endgroup$ – mary Oct 18 '16 at 18:24
  • $\begingroup$ I could repeat my comment as an answer if you wish, but I suspect it is not quite what you are looking for $\endgroup$ – Henry Oct 18 '16 at 18:33
  • $\begingroup$ @Henry, it isn't what I'm looking for. need some help seeing why the other statements dont work and what's missing in them $\endgroup$ – mary Oct 18 '16 at 19:27

The distinguishing feature between statement $2$ versus $1$ and $3$, is that in statement $2$, care is taken to avoid implying that the parameter is somehow variable or subject to randomness. In the frequentist paradigm, a parameter is a fixed unknown quantity, and anything we calculate from a sample is a realization of one or more random variables.

In other words, the confidence interval is not the fixed quantity: it will vary from experiment to experiment, from sample to sample. Thus, the frequentist will say, for example, that 95% of such intervals will contain the true value of the parameter. The frequentist does not say that there is a 95% probability that the parameter lies in the constructed interval, because this would imply that the value of the parameter varies from sample to sample--which it does not.

The frequentist viewpoint is actually rather intuitive; for instance, suppose we are given a six-sided die numbered $1$ to $6$, but are not told whether it is fair or loaded; moreover, if it were loaded, we are also not told in what way it is loaded--does it give more ones, or fives, for instance? But the intrinsic physical properties of the die is unchanging: it is entirely natural to postulate that, even though we do not know exactly how the die behaves, we can reason that it behaves consistently according to some probability model, and that the parameters that uniquely determine this model will not change from roll to roll, because this would suggest that the die would change in a material or physical sense.

So, if we are interested in the proportion $p$ of sixes that this die rolls, then it is reasonable to assume that $p$ is a fixed but unknown property of the die. If you roll the die ten times and get $3$ sixes, then under the frequentist viewpoint, this presumably gives you some information about $p$. But if I roll the same die ten times and I get $5$ sixes, my inference about $p$ may differ from yours, in particular, if neither of us knows the outcome of the other's experiment. These experiments are realizations of a random process. If you then roll the die $100$ times and get $43$ sixes, you could make a case for saying that you now have more information about $p$ than I have got with my ten rolls.

But in any case, we can see that no matter how many rolls of the die you might observe, there is no way to precisely infer with absolute certainty the true value of $p$. The frequentist would say you could find progressively more narrow confidence intervals, but no matter how small the width of such an interval, there will always remain a possibility that the sample observed from the experiment resulted in the calculation of an interval that fails to contain the true $p$--but as $p$ is forever unknown to us, we don't know when such an erroneous inference may occur.


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