On Sat, 21 Apr 2012 08:01:57 -0700, Jim Clark wrote: [snip] >What I do know is that if you select a sample of N observations with mean M and >standard deviation S out of a population with mean MU and standard deviation >SIGMA, then: > >1. M will fall within MU +/- z(alpha/2)*SIGMA/sqrt(N) with probability = 1 - >alpha (hypothesis testing), and
I believe this may be Fisher's position with respect to a one sample test. >2. Equivalently, MU will fall within M +/- z(alpha/2)*SIGMA/sqrt(N) with >probability = 1 - alpha (confidence intervals). If you have only one CI, your second point is wrong -- this is what Neyman was emphasizing when he said that for a given CI, it either contained the population parameter (Prob= 1.00) or it didn't (Prob= 0.00). For a more involved explanation, consider the following quote: | Note that when we compute a 95% confidence interval for a particular |sample, we have only one interval. Strictly speaking, that particular interval |does not mean that the probability that the population mean lies within that |interval is 0.95. For that statement to be true, it would have to be the case |that the population mean is a random variable, like the heads and tails in a |coin are random variables, and 1 through 6 on a die are random variables. | |The population mean is a single point value that cannot have a multitude |of possible values and is therefore not a random variable. If we relax this |assumption, that the population mean is a point value, and assume instead |that ‘the’ population mean is in reality a range of possible values (each value |having different probabilities of being the population mean), then we could |say that any one 95% confidence interval represents the range within which |the population mean lies with probability 0.95. See the book by Gelman and |Hill (2007) for more detail on this approach. | |It’s worth repeating the above point about confidence intervals. The meaning |of the confidence interval depends crucially on hypothetical repeated samples: |the confidence intervals computed in 95% of these repeated samples will |contain the population mean. In essence, the confidence interval from a single |sample is a random variable just like heads and tails in a coin toss, or the |numbers 1 through 6 in a die, are random variables. Just as a fair coin has |a 0.5 chance of yielding a heads, and just as a fair die has a 1/6 chance of |landing a 1 or 2 etc., a confidence interval in repeated sampling has a 0.95 |chance of containing the population mean. (p60-61) Ref: The Sampling Distribution of the Sample Mean Shravan Vasishth and Michael Broe 2011, The Foundations of Statistics: A Simulation-based Approach, Pages 43-80 Fisher did not believe in confidence intervals because he thought it absurd having to assume some number of future replications to get the CI. Quoting Chirstensen from my previous post: |one can use Fisherian testing to arrive at "confidence regions" |that do not involve either fiducial inference or repeated sampling. |A (1 - alpha) confidence region can be defined simply as a |collection of parameter values that would not be rejected by a |Fisherian alpha level test, that is, a collection of parameter values |that are consistent with the data as judged by an alpha level test. |This definition involves no long run frequency interpretation of |"confidence." It makes no reference to what proportion of |hypothetical confidence regions would include the true parameter. Your statement 2 above seems to me to be more consistent with Fisher's view (though, technically, this is not a confidence interval in the Neyman sense). -Mike Palij New York University [email protected] --- You are currently subscribed to tips as: [email protected]. To unsubscribe click here: http://fsulist.frostburg.edu/u?id=13090.68da6e6e5325aa33287ff385b70df5d5&n=T&l=tips&o=17415 or send a blank email to leave-17415-13090.68da6e6e5325aa33287ff385b70df...@fsulist.frostburg.edu
