18 CHAPTER 2 Sampling from PopulationsBut we shall draw inference about the population based on a sample
of 1000 subjects with type II diabetes that we were able to enroll in a
clinical trial. If the trial is properly conducted statistically, we may
estimate a treatment effect in the population based on an estimate from
the sample of 1000 subjects in the trial. This estimate, if favorable, may
lead the FDA to approve the drug for treatment of type II diabetes to
any American with type II diabetes.
Without a proper statistical design and analysis, the inference to
the population would not be valid and would not lead to an approval
even if the results are positive for the sample. The sample estimate
could be biased, and the probability that a decision favors the conclu-
sion of effectiveness when the drug is really not effective (called the
type I error or signifi cance level) would not be appropriately
controlled.
So to summarize, a population is a collection of things or people
that have similarities and possibly subgroup differences that you are
interested in learning about. A sample is simply a subset of the popula-
tion that you take measurements on to draw inferences about those
measurements for the population the sample was taken from.
2.2 SIMPLE RANDOM SAMPLING
One of the easiest and most convenient ways to take a sample that
allows statistical inference is by taking a simple random sample. As
mentioned earlier, many methods of sampling can create biases. Simple
random sampling assures us that sample estimates like the arithmetic
mean are unbiased.
Simple random sampling involves selecting a sample of size n from
a population of size N. The number of possible ways to draw a sample
of size n out of a population of size N is the binomial coeffi cient CnN
(read as “ combinations of N choose n ” ), the number of combinations
of N things taken n at a time. This is known in combinatorial mathemat-
ics to be N !/[ n !( N – n )!]. By “ n! ” , we mean the product n ( n − 1 )
( n − 2)... 3 2 1. In simple random sampling, we make the selection
probability the same for each possible choice for the sample n. So
the probability that any particular set occurs is 1/CnN. In Section 2.3 ,
we will show a method for taking simple random samples based on