Review
p. 13
10. Sampling.
How do we select samples from a population? This, in its own right, is the subject of a
course. One of the most common procedures is
simple random sampling.
For an
infinite
population,
this means each element in the sample is selected independently. Selection bias must be avoided.
Populations associated with ongoing processes, such as the set of all calls to a technical support
center, can be regarded as infinite. For finite populations, it means that each sample of size n
has the same chance of being selected. This leads
to a question: Given a population of size N, how
many different samples of size n are possible?
Answer:
N
n
N!
C
n!(N
n)!
=

Exercises:
1. How many samples of size 5 are possible for a population of size 40?
2. Of the 10 most active issues on the NYSE with market caps greater than $500 million,
4 are to be selected for an indepth study of business practices. How many possible such
choices are there?
11. Population parameters, sample statistics and sampling distributions. The Central Limit
Theorem.
What may we infer about a population, based on limited information about it?
As we have seen, if we have a set of numbers and know its mean and standard deviation, but nothing
else at all, we may conclude that the proportion of the data set falling within k standard deviations of
the mean ( provided that k > 1) is no less than

2
1
1
k
.
This is Chebyshev's Theorem. It cannot fail, but is of limited utility. Since probability is proportion, it
gives us a kind of weak predictive power. For example, if a number were selected at random from the
data set, one could say that the probability is at least .75 that the number falls within 2 s.d. of the
mean. More knowledge about the population gives better predictive power. If the population ( data
set, random variable) is also known to have a normal probability distribution, by which we mean that
this mathematical model fits the population well, then we can make more precise predictions.
The better the model fits, the better the precision. A rough and ready (and limited) version of this is
the socalled Empirical Rule, for which the various
texts require that the data set be " mound
shaped", " symmetric moundshaped", " bellshaped", etc. You know the specified proportions for
intervals within 1, 2, or 3 s.d.
of the mean ( 68%, 95%, and 99+%, respectively).
In general, we study populations by sampling. What do we study? Since we would like to be able to
make predictions, we need estimates of certain numerical properties, especially the mean
μ
, the s.d.
σ
, and the proportion p (of the population having a certain characteristic). Inferences about
these
population parameters are based on their corresponding sample statistics,
ˆ
x, s, and p.
Much of our
focus is on
μ
.
The Sampling Distribution of x
Suppose that we wish to use
x
, the sample mean of n randomly selected observations of the random
variable x ( values from the population) as an estimate of the population mean
μ
X
.
Many different
samples are possible. This means that
x
is itself a random variable, and as such, has its own mean