ECON 413
Fall 2004
Toward Statistical Inference
Statistics is ultimately concerned with drawing inferences about some population of
interest. We can learn about a population by taking a sample from it and by using the
information in the sample to learn about the population. There are many different
methods of sampling from a population.
In simple random sampling, all possible samples from the population have an equal
chance of being the actual sample selected. Simple random sampling also gives each
member of the population the same chance of being included in the sample. Random
sampling can be done either with replacement or without replacement.
Suppose we are interested in learning about a variable
X
, say, e.g. the rate of return on
the stock of companies that trade on the New York Stock Exchange (NYSE). The
distribution of the values of
X
for all companies that trade on the NYSE is called the
population distribution. If we pick a stock at random from the population of all NYSE
stocks, the value of
X
for that stock is a random variable. The distribution of this
random variable is also the population distribution.
A random sample of size n is a sequence of independent observations 12
(, ,, )
n
X
XX…,
where i
X
represents the value of
X
for the ith individual and each i
X
has the
distribution of the population being sampled. We can write the joint p.d.f. (or p.f.) of the
observations in a random sample as
12 1 2
( , ,..., ) ( ) ( )... ( )
nX X Xn
f
xx x f xf x f x=
where X
f
denotes the population p.d.f. (or p.f.).
We are usually interested in learning about certain characteristics of populations that are
called parameters. For example, the mean
µ
and the standard deviation
σ
of a
population distribution are examples of parameters. A parameter is a fixed number.
Because it is based on all the observations in the population, its value is almost always
unknown.
On the other hand, a statistic is a numerical descriptive measure of a sample. Sample
mean
X
, sample median m, and sample standard deviation s are examples of statistics.
Statistics are calculated from the observations in the sample. A statistic can be used to
estimate an unknown parameter.
Example: A recent survey asked a nationwide random sample of 2500 adults if they
agreed or disagreed with the following statement: “I like buying new clothes, but
shopping is often frustrating and time-consuming.” 66% of he respondents (1650 people
out of 2500) agreed.
