I think you are confusing the idea of a sample with the source of a
binomial random variable. The binomial model applies when some action is
repeated a specified number of times, n; when we are interested in the
occurrence or not of some outcome; when the probability of that outcome
is the same for all repetitions of the action (ie all trials); when
trials are independent; and when the variable of interest is the number
of times the outcome of interest occurs.
Dead simple example: toss a coin 10 times; assume the 10 tosses are
independent; we want the number of heads; assume the probability of a
head on each toss is 0.5 (or whatever). The variable X = number of heads
out of 10 tosses is binomially distributed - more precisely, the
binomial model is a (very) good model for this situation.
A sampling distribution is just a probability distribution which occurs
as a result of sampling. In the present context, we might take a sample
of values of X. A sample of size 20, for example, would mean repeating
the whole shebang 20 times - each time you toss the coin 10 times and
record the number of heads. Now suppose we want to measure some
characteristic of this sample - for example, the mean value of X, or the
proportion of times the value of X is greater than 5, or .......... This
measure is a statistic of the sample. It is clearly also a random
variable since it varies over the samples taken. The probability model
which describes how the statistic varies over the population of all
possible samples of that size is called the sampling distribution for
that statistic.
So a sampling distribution is just an ordinary probability distribution,
in the particular case where the population is a population of samples.
If you take samples of size 1, and the statistic you record for that
sample is simply the value of X, you have the 'parent' distribution - so
the latter is just one of the sampling distributions you can have for a
particular situation.
With the binomial there is a complication - if you have a particular
characteristic in a population, and you take a simple random sample from
that population, and measure the number of times the characteristic
occurs in the sample, the binomial model describes this. Exactly, if the
sampling is with replacement, approximately if without replacement.
To answer your first question - ANY binomial model, whatever its origin,
is approximately normal for large enough n. This has nothing to do with
sampling (except that the application may be in sampling, as in the
previous paragraph).
A bit long winded - sorry!
Alan
James Ankeny wrote:
>
> Hello,
> I have a question regarding the so-called normal approx. to the binomial
> distribution. According to most textbooks I have looked at (these are
> undergraduate stats books), there is some talk of how a binomial random
> variable is approximately normal for large n, and may be approximated by the
> normal distribution. My question is, are they saying that the sampling
> distribution of a binomial rv is approximately normal for large n?
> Typically, a binomial rv is not thought of as a statistic, at least in these
> books, but this is the only way that the approximation makes sense to me.
> Perhaps, the sampling distribution of a binomial rv may be normal, kind of
> like the sampling distribution of x-bar may be normal? This way, one could
> calculate a statistic from a sample, like the number of successes, and form
> a confidence interval. Please tell me if this is way off, but when they say
> that a binomial rv may be normal for large n, it seems like this would only
> be true if they were talking about a sampling distribution where repeated
> samples are selected and the number of successes calculated.
>
> _______________________________________________________
> Send a cool gift with your E-Card
> http://www.bluemountain.com/giftcenter/
>
> =================================================================
> Instructions for joining and leaving this list and remarks about
> the problem of INAPPROPRIATE MESSAGES are available at
> http://jse.stat.ncsu.edu/
> =================================================================
--
Alan McLean ([EMAIL PROTECTED])
Department of Econometrics and Business Statistics
Monash University, Caulfield Campus, Melbourne
Tel: +61 03 9903 2102 Fax: +61 03 9903 2007
=================================================================
Instructions for joining and leaving this list and remarks about
the problem of INAPPROPRIATE MESSAGES are available at
http://jse.stat.ncsu.edu/
=================================================================
