How we interpret confidence intervals.For the formula X ¯ ± z α 2 σ n , we substitute x ¯ o b s for X ¯ , but isn't that observed sample mean x ¯ o b s only for one particular sample?Also why is it incorrect to interpret a confidence interval as the probability that the actual μ lies in a interval (a,b) is ( 1 − α ) 100 %

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How we interpret confidence intervals.For the formula             X      ¯        ±      z                            α          2                                σ              n            , we substitute                     x        ¯                    o      b      s       for             X      ¯      , but isn&#039;t that observed sample mean                     x        ¯                    o      b      s       only for one particular sample?Also why is it incorrect to interpret a confidence interval as the probability that the actual   μ lies in a interval (a,b) is   (  1  −  α  )  100  %

cercimw · Accepted Answer

Step 1The reason why it is incorrect to interpret a confidence interval as &quot;the probability that the actual μ lies in a interval (a,b) is   (  1  −      α    )  100%&quot; is becaue   μ is fixed but unknown, not a random variable.Step 2The actual probability would be 1 or 0 depending on whether   a  &amp;lt;  μ  &amp;lt;  b or not.It is the interval itself which is variable, since it depends on the actual sample obtained, so we refer to &quot;the probability that the interval contains   μ&quot; rather than the &quot;probability   μ lies in the interval&quot;.

shaunistayb1 · Accepted Answer

Step 1There is something with a probability of   (  1  −  α  ) or more, but that something has a more complex description than &quot;is the estimate of   μ correct&quot;.The something is: &quot;given the probability assumptions (such as normal distributions for the quantities of interest), the probability is at least that high that the estimation procedure we followed, applied to data generated in a way that satisfies the assumptions, will give an interval containing the correct answer&quot;.Step 2In the standard accounts of probability, &quot;is the estimate of   μ correct&quot; is not a random event to which a probability can be assigned, except in the trivial sense that the probability being 0 or 1 is another way of saying whether the estimate is correct or not.

How we interpret confidence intervals. For the formula barX pm z_{alpha/2} sigma/sqrtn, we substitute barx_{obs} for barX, but isn't that observed sample mean barx_{obs} only for one particular sample?

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