In practice in applying statistical techniques, we are interested in random variables defined on the population under study. In the examples mentioned in the preceding page, we might be interested in:

Example 1: The difference in
blood
pressure with and without taking a certain drug.

Example 2: The number of heart bypass surgeries performed in a particular year, or the number of such surgeries that are successful, or the number in which the patient has complications of surgery, etc.

Example 3: The number that comes up on the die.

If we
take a sample of units from the population, we have a corresponding
sample of values of the random
variable. Example 2: The number of heart bypass surgeries performed in a particular year, or the number of such surgeries that are successful, or the number in which the patient has complications of surgery, etc.

Example 3: The number that comes up on the die.

For example, if the random variable (let's call it Y) is difference in blood pressure with and without taking the drug, then the sample will consist of values we can call

y_{1}, y_{2},
...,
y_{n},

where n = number of people in
the
sample from the population

and whereThe people in the sample are
listed as person 1, person 2, etc.

y_{1} = the
difference in
blood pressures (that is, the value of Y) for the first person in the
sample,

y_{2} = the difference in blood pressures (that is,
the value of Y) for the second person in the sample

etc.

y

etc.

»» If the next paragraph is too complex or mathematical for you, just skip to The Bottom Line below.

Abstractly, we can think of this situation as describing n random variables Y

Y_{1} is defined as
the
value of Y for the first person in a sample of the population;

Y_{1} is defined as
the
value of Y for the first person in a sample of the population;

etc.

The difference between using the small y's and the large Y's
is
that when we use the small y's we
are thinking of a fixed sample of size n from the population, but when
we use the large Y's, we are thinking of letting the sample vary
(but always with size n).etc.

Precise definition of simple random sample of a random variable:

"The sample y_{1},
y_{2}, ..., y_{n} is a simple random sample" means that
the associated random variables Y_{1},
Y_{2},
..., Y_{n }are
independent.

Intuitively, "independent" means that the values of any subset of the
random variables YConnection with the initial definition of simple random sample

Recall Example 3 above:

We are tossing a die;
the
number that comes up on the die is our random variable Y.

If we use the Moore and McCabe's definition of simple random sample, our population is all possible tosses of the die. Our simple random sample is n different tosses. The different tosses of the die are independent events, which means that in the precise definition above, the random variables Y

Compare this with example 2:

Our population is all
hospitals in
the U.S. that perform heart bypass surgery.

If we use Moore and McCabe's
definition of simple random sample of size n, we end up
with n distinct
hospitals. This means that when we have chosen the first hospital in our simple random sample, we cannot choose it again to be in our simple random sample. Thus the events "Choose the first hospital in the sample; choose the second hospital in the sample; ... ," are not independent events: The choice of first hospital restricts the choice of the second and subsequent hospitals in the sample.

If we now consider the random variable Y = the number of heart bypass surgeries performed in 2008, then a consequence is that the random variables Y

The Bottom Line:

In many cases, the Moore-McCabe definition does not coincide with the more precise definition.

More
specifically, the
Moore-McCabe definition allows sampling without replacement, whereas the
more precise definition requires sampling with replacement.

The Bad News: The precise definition is the one that is used in the mathematical theorems that justify the procedures of statistical inference.

The Good News: If the population is large enough, the Moore -McCabe definition is close enough for all practical purposes.

Unfortunately, the
question,"How large is large
enough?" does not have a simple answer.