units
* • •, un selected at random from U. In principle the
sampling process goes as follows: a random number device inde�
pendently selects integers ji, ■ * *, jn, each of which equals any
value between 1 and N with probability 1/N. These integers deter�
mine which members of U are selected to be in the random sample,
m = Uj1, u2 = Uj2, • • •, un = Ujn. In practice the selection process
is seldom this neat, and the population U may be poorly defined,
but the conceptual framework of random sampling is still useful for
understanding statistical inference. (The methodology of good ex�
perimental design, for example the random assignment of selected
units to Treatment or Control groups as was done in the mouse
experiment, helps make random sampling theory more applicable
to real situations like that of Table 2.1.)
Our definition of random sampling allows a single unit �7* to ap�
pear more than once in the sample. We could avoid this by insisting
that* the integers j\,j2, • • * ,jn be distinct, called “sampling with�
out replacement.” It is a little simpler to allow repetitions, that is
to “sample with replacement”, as in the previous paragraph. If the
size n of the random sample is much smaller than the population
size N, as is usually the case, the probability of sample repetitions
will be small anyway. See Problem 3.1. Random sampling always
means sampling with replacement in what follows, unless otherwise
stated.
Having selected a random sample ui, U2 , • • •, un, we obtain one
or more measurements of interest for each unit. Let Xi indicate
the measurements for unit u*. The observed data are the collec�
tion of measurements Xi,X2, • • •, �n. Sometimes we will denote the
observed data (#i, #2, ’ • *, �n) by the single symbol x.
We can imagine making the measurements of interest on ev�
ery member I7i, f/2 , • • •, Un of W, obtaining values Xi, X 2, • • •, X^.
This would be called a census of U.
The symbol X will denote the census of measurements
(Xi, X 2 ,• • •, X n ). We will also refer to X as the population of mea�
surements, or simply the population, and call x a random sample of
size n from X. In fact, we usually can’t afford to conduct a census,
which is why we have taken a random sample. The goal of statisti�
cal inference is to say what we have learned about the population X
from the observed data x. In particular, we will use the bootstrap
to say how accurately a statistic calculated from �1 , ^2 ? • • •, xn (for
instance the sample median) estimates the corresponding quantity
for the whole population.