Re: Interpreting confidence intervals
- From: Chip Eastham <hardmath@xxxxxxxxx>
- Date: Sun, 5 Jul 2009 20:58:36 -0700 (PDT)
On Jul 5, 11:15 pm, hierholzer <hierhol...@xxxxxxxxxx> wrote:
I think my professor is giving misleading information
with respect to the interpretation of confidence
intervals. For example, we take one sample and
from that give the following interpretation(assume
a 95% CI):
"I'm 95% confident that the pop. mean for
x is anywhere from .245 below to .785
above that of y"
Why do I think this is wrong? My book
says your endpoints are random variables
and so your interval is a random interval(
I agree with that). With an infinite number
of random samples we can compute a 95%
CI for each, and 95% will contain true pop.
mean.
It goes on to say that in practice we deal with
one random sample and calculate one confidence
interval. Because the interval may contain
mu or not, it is unreasonable to attach a
probability level to this specific event.
It continues by stating:
"The appropriate state is the observed interval
[l, u] brackets the true value of mu with confidence
100(1-a). This statement has a frequency
interpretation; that is, we don't know if the
statement is true for this specific example,
but the method used to obtain the interval
[l, u] yields correct statements 100(1-a)% of
the time."
Given the above, then:
"I'm 95% confident that the pop. mean for
x is anywhere from .245 below to .785
above that of y"
Should read:
"There is a confidence of 95
that the pop. mean for x is anywhere
from .245 below to .785 above that of y"
Or
"The method used to obtain a given
interval, in this particular case: (-.245, .785),
will capture the pop. mean 95% of the time".
Thoughts? Recommendations on wording
it differently?
I don't really see the distinction between
your professor's meaning and that of your
final version. Perhaps yours is more
informative, since it emphasizes the "method
used to obtain a given interval" (which
in this case depends on sampling), but
ultimately we make our way to a useful
way of understanding how we are 95%
confident that the population mean falls
into a particular interval.
It is true that this is not the only
interval with that property, but I don't
find the original wording unduly faulty
by not going into that level of detail.
Indeed your wording only hints at the
fact that other methods may give other
intervals, also with the property of
bracketing the population mean (at least)
95% of the time.
regards, chip
.
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