The Sampling Distribution of aStatistic

Recall that a statistic is simply a number which we somehow attach to a sample of some population. Here are examples of simple minded statistics:

• The largest number in the sample.• The smallest number in the sample.• The range of the sample.• The midpoint of the range.• The median of the sample.• The average of the sample.

Which statistic do we use? Obviously …

… that depends onwhich parameter of the population we want to estimate! (duh !) For instance we could use:

• The largest number in the sample to guessthe maximum of the population.

• The smallest to guess the minimum of the population..

• The range of the sample to guessthe spread of the population.

• The range/8 to guessthe standard deviation of the population

• The average of the sample to guessthe mean of the population.

Let’s do an example. Our population consists of:

1,000 beanbags, some weighing

• zero ounces (filled with air) some weighing• two ounces (filled with peas) and some weighing• seven ounces (filled with whatever.)

So we have a population that consists of 1000 numbers, some 0’s, some 2’s and some 7’s.We would like to guess the mean of the population and maybe the standard deviation,but we have enough resources to sample only three members of the population.(Beanbags aren’t cheap, the bag is made of gold) Let’s start by listing all the possible samples of three entries we can get:We list each with the resulting sample average:

• So, if our sample is 2, 7, 7 we would guess 5.33, but if it is 2, 0, 7 we would guess 3.

You can see that our guess can be anyone of these numbers:

These numbers are just the values of a random variable (they vary at random!), and if we knew the probability distribution we could make some progress.Progress starts with naming things (yourself, this land I claim in the name of …., the Fighting Irish, etc.), so let’s name a few things.

• The Random Variable above is called the sample mean.

• The probability distribution of the sample mean is called the

sampling distribution of the mean.

Let’s return to our example. The values of the RV “sample mean” are:

So we need to fill the blanks in the following table:

Some blanks we can fill, sort of:

where

Fk = (# k’s)/1000 = p(k), k = 0, 2, 7

But unless we know F0 , F2 , F7 we areapparently stuck!Not quite.

Basic Assumption aboutSampling Procedure

When we filled the three blanks in the previous slide we tacitly assumed that

• Probabilities stay the same in each pick

and

• Probabilities multiply

In the next slide we rephrase the two statements above as follows:

Basic Assumption aboutSampling Procedure (cont’d)

• Definition. A sample of size Nconsists of N entries picked from the population of interest in such a way that

1. Each pick is independent of all the others.

2. Each pick comes from the same population of interest.(We usually assume that random sampling achieves 1 above, 2 requires care in defining the procedure.)

Let’s invent some numbers for F0 , F2 , and F7 .

F0 = 0.1

F2 = 0.6

F7 = 0.3

In the next slide we show, for each of the 27 possible 3-samples,

the average and the probability of fishing that particular sample from our population.

• Note how our two basic assumptions that

Probabilities stay the same

and

Probabilities Multiply

allow us to compute the probabilities of each sample. Now we can fill the probability distribution table for the sample mean,

we just add probabilities

The Wonderful Secret

Regardless of how large or small n is, the

expected value of the sample mean

is exactly the mean of the population!

In other words

Now do the following exercise:

There are 16,000 undergraduates at Podunk U., 5,000 freshmen, 4,000 sophomores, 4,000 juniors and 3,000 seniors.

1. Pick a random sample of size 2 from the undergraduates of Podunk U. and average their number of years of enrollment.

2. Display the Probability Distribution table of the sample mean.

(There are 16 possible distinct samples.)

Verify the previous statement.