In general, if the true population mean μis known exactly , so that μ=^1
N
∑N
j=1
Xj,
where N is the population size, then the population standard deviation is given by
σ=
√∑
N
j=1(Xj−μ)^2
N (11 .16)
Use Nif the exact population mean is known and use n− 1 if samples of size n << N
are selected from a population where the true mean μis unknown.
Probability
An experiment or observation produces samples from a population where the
outcome recorded either belongs or does not belong to a prescribed collection of
events being studied. A sample space Sis the set of all possible outcomes from an
experiment. A sample space can be either finite or infinite. An example of a sample
space with a finite collection of events is the roll of a single die. Here the sample
space is S={ 1 , 2 , 3 , 4 , 5 , 6 }corresponding to the numbers on the six faces of the die.
An example of an infinite sample space is that of an experiment where the outcome
from an single event can be a real number within a specified range.
Figure 11-4. Venn diagrams
A Venn diagram consists of representing the sample space by a rectangle, then
any event within Scan be represented by the interior of a circle Awithin the rect-
angle. The set of all events not in Ais called the complement of Aand is denoted
using the notation Ac. The null set, empty set or impossible event is denoted by
the symbol ∅. The union of two events A and B is denoted A∪B and represents
all events or experiments of Scontained in Aor Bor both. The intersection of two
events Aand Bis denoted A∩B and represents all events in Scontained in both A
and B. The concepts of a complement, union and intersection of sets is illustrated
in the figure 11-4. If two sets Aand B have no events in common, then this is
