Xiao Li wrote:
>
> Let N be the population size. Let p denote the proportion of this
> population with a certain characteristic (ie. has diabetes, divorced,
> homosexual, etc.) Let's say we are trying to estimate p using a
> sample of size n. So we collect our sample and calculate p^ to be the
> proportion of elements in our sample with our characteristic of
> interest. Then we calculate the standard of deviation of p^ (also
> known as the standard error) because it gives us an idea of how close
> our sample proportion is to the real proportion.
>
> Now, my statistics textbook says the standard of deviation is given by
> SD(p^) = root[(p^(1-p^))/n] when sampling with replacement or when
> the sample size is significantly smaller than the population size.
>
> My question is why isn't there a N variable in our equation for
> SD(p^)? According to this formula, if we are trying to figure out the
> proportion of people with diabetes in Berkeley, California with sample
> size n, we would get the same standard of error as if we are trying to
> figure out the proportion of people with diabetes in the world with
> sample size n. (In the former, the N is a lot smaller than in the
> later).
That's right - assuming each sample to be random from the population
in question, and the population not to be so small that removing your
sample depletes it significantly.
Suppose I show you a barrel that looks as if it would contain about a
million jellybeans, and you grab a handful (say, fifty) and you count
them and find that half are green. You'd be able to estimate that half
the jellybeans in the barrel are green, and you'd have some idea of how
good that estimate was - say between about 35% and 65% with about 95%
confidence.
Now suppose I tell you - surprise - the barrel has a false bottom and
only contains 10,000 jellybeans, 1% of the number you'd assumed. Does it
seem reasonable that your estimate has just got a lot better? (People
making this mistake usually assume that a small population can be
estimated better, not worse.)
Or suppose the barrel is connected to a huge underground warehouse
contianing a billion well-mixed jellybeans. Do you suddenly lose
confidence in your estimate?
-Robert Dawson
.
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