This argument relies on the symmetry of the normal and t distributions - what happens with a confidence interval for a variance?
In any case, it still misses my fundamental question - interpreting the interval explicitly in terms of probability.
Regards,
Alan
On Monday, November 18, 2002, at 08:31 AM, Dennis Roberts wrote:
for what it's worth ...
i start with a typical sampling distribution of means ... talk about how many X bars fall certain distances from mu ...
then, i say what if you took a sample ... got a sample mean that happens to be identical to the mu ... and, went 1 standard error on each side of it .... does that interval capture mu? yes of course ...
then i keep moving the X bar out further and further out from mu ... each time going 1 standard error on either side of X the next X bar .. and asking ... does the interval contain mu?
i keep doing that until X bar coincides with a distance of 1 standard error away from mu ...
doing this on either side of mu will produce for you intervals that all capture mu ...
then, if you go slightly further from mu than 1 standard error ... and continue to go 1 standard error on either side of your X bar ... now the intervals DO NOT include mu ...
so, using this approach ... about 68% of 1 standard error bands around all possible X bars ... will capture mu ... and the remaining 32% won't ...
instead of 1 standard error ... try it with 2 ... to show that 95% of the intervals capture mu
i have this piece of cardboard ... about 2 feet long and 6 inches high ... with 2 standard errors marked off from a midpoint ... that i label X bar .... and, after i draw a normal distribution on the board ... with baseline .... i slide this home made "instructional aid" horizontal back and forth .... illustrating the point
i try to show that ... there is an equivalency between saying:
1. 68% of the sample means fall 1 standard error from the mu value
and
2. 68% of all 1 standard error bands around all possible sample X bars will capture or include mu
i try to let them see that IF you would like to know something about what the mu value might be ... an interval around the sample X bar ... given the confidence you want to have with this .... can assist you in that quest
of course, i also talk about the tradeoff between confidence and precision
i am not sure why you are displeased with text book renditions ... is it because they seem to interpret them incorrectly? don't give them sufficient play? abuse their use? or what?
..
..
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
.. http://jse.stat.ncsu.edu/ .
=================================================================
. . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
