In a sample of 580 voters,
277 are for Kitzhaber, 240 for Dudley. [We
assume the sample is an SRS for this example, which may not be entirely true.]
[This is for the 2010 gubernatorial election in Oregon; the poll results were released 4 days before election day.]
If this were a SRS (perhaps
true), the corresponding 95% confidence interval for Kitzhaber support would be
47.8% +/- 4.1% OR 43.7% to 51.9%.
Popular wrong answer #1: “This means we are 95% confident that the percentage of voters who favor Kitzhaber is in the range 43.7%
to 51.9%.”
[emphasis added; scroll down for my response]
Sorry, but this response has
essentially no content. Just as "Depreciated assets are assets
that have been depreciated" or "A publicly-financed election is an
election that is publicly financed" are statements that convey no
information, saying that "A 95% confidence interval is an interval for
which we are 95% confident that it is correct" doesn't tell us anything
we didn't already know. Sorry, zero points.
Popular wrong answer #2: “This means there’s a 95%
chance that the percentage of voters who favor Kitzhaber is in the range 43.7%
to 51.9%.”
[Scroll down for my response]
No, sorry, the percentage for
Kitzhaber is either
in the range 43.7% to 51.9% (a 100% chance) or out of that range (a 0%
chance). While samples vary from poll to poll, the percentage for
Kitzhaber, at the point in time these results were published, does not
vary, and is not subject to chance. Sorry, zero points.
Popular wrong answer #3: “This means that if we took
repeated samples, 95% of them would have Kitzhaber support percentages
in the
range 43.7% to 51.9%.”
[Scroll down for my response]
No, 95% of samples will have Kitzhaber support percentages in the range "Pop. pct. ± 2SEpct."
That’s not the same as 43.7% to 51.9%, which is just the confidence interval derived one of the
samples. This sample and its confidence interval is not privileged to dictate
what other
samples/intervals must do, only Pop. Pct ± 2SEpct is so privileged - and Pop. Pct is the very thing we don't know. This answer gets about 40%-50% credit.
“If we took repeated samples,
each of them would give us a different confidence interval. 95%
of those intervals would enclose the true
population percentage of support for Kitzhaber, but we can’t know if
this
interval (or any one of the intervals) is one of those 95% that are
‘good’ or if it’s one of the 5% ‘bad’
intervals that misses the true population percentage. All we can
say is this process 'works' 95% of the time, but we don't know which
times it is working.”
“There’s a 95% chance of
getting a sample1 whose corresponding confidence interval encloses the true
population percentage of support for Kitzhaber.
However, once the sample’s drawn, it’s no longer a matter of
chance. Our sample either gives us a ‘good interval’
or a ‘bad one’ which misses the true population percentage, and we can’t know
which kind we have. All we can say is this process 'works' 95% of the time, but we don't know which times it is working.”
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