Gregersen E. (editor) The Britannica Guide to Statistics and Probability
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proportion of defectives in the sample, an np-chart or a
p-chart can be used.
All control charts are constructed in a similar fashion.
For example, the centre line of an x¯-chart corresponds
to the mean of the process when the process is in control
and producing output of acceptable quality. The vertical
axis of the control chart identifies the scale of measure-
ment for the variable of interest. The upper horizontal
line of the control chart, referred to as the upper control
limit, and the lower horizontal line, referred to as the
lower control limit, are chosen so that when the process
is in control there will be a high probability that the value
of a sample mean will fall between the two control lim-
its. Standard practice is to set the control limits at three
standard deviations above and below the process mean.
The process can be sampled periodically. As each sample
is selected, the value of the sample mean is plotted on
the control chart. If the value of a sample mean is within the
control limits, the process can be continued under
the assumption that the quality standards are being main-
tained. If the value of the sample mean is outside the
control limits, an out-of-control conclusion points to
the need for corrective action in order to return the pro-
cess to acceptable quality levels.
saMple suRvey MeThods
Statistical inference is the process of using data from a
sample to make estimates or test hypotheses about a popula-
tion. The field of sample survey methods is concerned with
effective ways of obtaining sample data. The three most
common types of sample surveys are mail surveys, telephone
surveys, and personal interview surveys. All involve the use
of a questionnaire, for which a large body of knowledge
exists concerning the phrasing, sequencing, and grouping of
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questions. There are other types of sample surveys that do
not involve a questionnaire. For example, the sampling of
accounting records for audits and the use of a computer to
sample a large database are sample surveys that use direct
observation of the sampled units to collect the data.
A goal in the design of sample surveys is to obtain a
sample that is representative of the population so that
precise inferences can be made. Sampling error is the
difference between a population parameter and a sample
statistic used to estimate it. For example, the difference
between a population mean and a sample mean is sam-
pling error. Sampling error occurs because a portion,
and not the entire population, is surveyed. Probability
sampling methods, where the probability of each unit
appearing in the sample is known, enable statisticians to
make probability statements about the size of the sam-
pling error. Nonprobability sampling methods, which are
based on convenience or judgment rather than on prob-
ability, are frequently used for cost and time advantages.
However, one should be extremely careful in making
inferences from a nonprobability sample. Whether or
not the sample is representative is dependent on the
judgment of the individuals designing and conduct-
ing the survey and not on sound statistical principles.
In addition, there is no objective basis for establishing
bounds on the sampling error when a nonprobability
sample has been used.
Most governmental and professional polling surveys
employ probability sampling. It can generally be assumed
that any survey that reports a plus or minus margin of
error has been conducted using probability sampling.
Statisticians prefer probability sampling methods and rec-
ommend that they be used whenever possible. A variety of
probability sampling methods are available. A few of the
more common ones are reviewed here.
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Simple random sampling provides the basis for many
probability sampling methods. With simple random sam-
pling, every possible sample of size n has the same
probability of being selected.
Stratified simple random sampling is a variation of
simple random sampling in which the population is parti-
tioned into relatively homogeneous groups called strata
and a simple random sample is selected from each stra-
tum. The results from the strata are then aggregated to
make inferences about the population. A side benefit of
this method is that inferences about the subpopulation
represented by each stratum can also be made.
Cluster sampling involves partitioning the population
into separate groups called clusters. Unlike in the case
of stratified simple random sampling, it is desirable for
the clusters to be composed of heterogeneous units. In
single-stage cluster sampling, a simple random sample of
clusters is selected, and data are collected from every unit
in the sampled clusters. In two-stage cluster sampling, a
simple random sample of clusters is selected and then
a simple random sample is selected from the units in each
sampled cluster. One of the primary applications of clus-
ter sampling is called area sampling, where the clusters
are counties, townships, city blocks, or other well-defined
geographic sections of the population.
decision analysis
Decision analysis, also called statistical decision theory,
involves procedures for choosing optimal decisions in the
face of uncertainty. In the simplest situation, a decision
maker must choose the best decision from a finite set of
alternatives when there are two or more possible future
events, called states of nature, that might occur. The list of
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possible states of nature includes everything that can hap-
pen, and the states of nature are defined so that only one
of the states will occur. The outcome resulting from the
combination of a decision alternative and a particular
state of nature is referred to as the payoff.
When probabilities for the states of nature are avail-
able, probabilistic criteria may be used to choose the best
decision alternative. The most common approach is to use
the probabilities to compute the expected value of each
decision alternative. The expected value of a decision
alternative is the sum of weighted payoffs for the decision.
The weight for a payoff is the probability of the associated
state of nature and therefore the probability that the pay-
off occurs. For a maximization problem, the decision
alternative with the largest expected value will be chosen.
For a minimization problem, the decision alternative with
the smallest expected value will be chosen.
Decision analysis can be extremely helpful in sequen-
tial decision-making situations—that is, situations in
which a decision is made, an event occurs, another deci-
sion is made, another event occurs, and so on. For instance,
a company trying to decide whether or not to market a
new product might first decide to test the acceptance of
the product using a consumer panel. Based on the results
of the consumer panel, the company then decides whether
or not to proceed with further test marketing. After ana-
lyzing the results of the test marketing, company
executives decide whether or not to produce the new
product. A decision tree is a graphical device that helps in
structuring and analyzing such problems. With the aid of
decision trees, an optimal decision strategy can be devel-
oped. A decision strategy is a contingency plan that
recommends the best decision alternative depending on
what has happened earlier in the sequential process.
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GAMe tHeoRY
G
ame theory is the branch of applied mathematics
that provides tools for analyzing situations in which
parties, called players, make decisions that are interde-
pendent. This interdependence causes each player to
consider the other player’s possible decisions, or strate-
gies, in formulating his own strategy. A solution to a game
describes the optimal decisions of the players, who may
have similar, opposed, or mixed interests, and the out-
comes that may result from these decisions.
Although game theory can be and has been used to
analyze parlour games, its applications are much broader.
In fact, game theory was originally developed by the
Hungarian-born American mathematician John von
Neumann and his Princeton University colleague Oskar
Morgenstern, a German-born American economist, to
solve problems in economics. In their book The Theory of
Games and Economic Behavior (1944), von Neumann and
Morgenstern asserted that the mathematics developed
for the physical sciences, which describes the workings of
a disinterested nature, was a poor model for economics.
They observed that economics is much like a game,
wherein players anticipate each other’s moves, and there-
fore requires a new kind of mathematics, which they called
game theory. (The name may be somewhat of a misnomer,
because game theory generally does not share the fun or
frivolity associated with games.)
Game theory has been applied to a wide variety of sit-
uations in which the choices of players interact to affect
the outcome. In stressing the strategic aspects of decision
making, or aspects controlled by the players rather than
CHAPTER 4
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Oskar Morgenstern coauthored Theory of Games and Economic Behavior,
applying Neumann’s theory of games of strategy to competitive business. Ralph
Morse/Time & Life Pictures/Getty Images
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by pure chance, the theory both supplements and goes
beyond the classical theory of probability. It has been
used, for example, to determine what political coalitions
or business conglomerates are likely to form, the optimal
price at which to sell products or services in the face of
competition, the power of a voter or a bloc of voters,
whom to select for a jury, the best site for a manufacturing
plant, and the behaviour of certain animals and plants in
their struggle for survival. It has even been used to chal-
lenge the legality of certain voting systems.
It would be surprising if any one theory could address
such an enormous range of “games,” and in fact there is no
single game theory. Many theories have been proposed,
each applicable to different situations and each with its
own concepts of what constitutes a solution. This chapter
describes some simple games, discusses different theories,
and outlines principles underlying game theory.
classificaTion of gaMes
Games can be classified according to certain significant
features, the most obvious of which is the number of play-
ers. Thus, a game can be designated as being a one-person,
two-person, or n-person (with n greater than two) game,
with games in each category having their own distinctive
features. In addition, a player need not be an individual. It
may be a nation, corporation, or team comprising many
people with shared interests.
In games of perfect information, such as chess, each
player knows everything about the game at all times. Poker
is an example of a game of imperfect information, however,
because players do not know all of their opponents’ cards.
The extent to which the goals of the players coin-
cide or conflict is another basis for classifying games.
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Constant-sum games are games of total conflict, which are
also called games of pure competition. For example, poker
is a constant-sum game because the combined wealth of
the players remains constant, but its distribution shifts in
the course of play.
Players in constant-sum games have completely
opposed interests, whereas in variable-sum games they
may all be winners or losers. In a labour-management dis-
pute, for example, the two parties certainly have some
conflicting interests, but both benefit if a strike is averted.
Variable-sum games can be further distinguished as
being either cooperative or noncooperative. In coopera-
tive games players can communicate and, most important,
make binding agreements. In noncooperative games play-
ers may communicate, but they cannot make binding
agreements, such as an enforceable contract. An automo-
bile salesperson and a potential customer will be engaged
in a cooperative game if they agree on a price and sign a
contract. However, the dickering that they do to reach
this point will be noncooperative. Similarly, when people
bid independently at an auction they are playing a non-
cooperative game, even though the high bidder agrees to
complete the purchase.
Finally, a game is said to be finite when each player
has a finite number of options, the number of players is
finite, and the game cannot go on indefinitely. Chess,
checkers, poker, and most parlour games are finite. Infinite
games are more subtle and will only be touched upon in
this section.
A game can be described as in extensive, normal, or
characteristic-function form. (Sometimes these forms are
combined.) Most parlour games, which progress step by
step, one move at a time, can be modeled as games in
extensive form. Extensive-form games can be described by
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a “game tree,” in which each turn is a vertex of the tree,
with each branch indicating the players’ successive choices.
The normal (strategic) form is primarily used to
describe two-person games. In this form a game is repre-
sented by a payoff matrix, wherein each row describes the
strategy of one player and each column describes the
strategy of the other player. The matrix entry at the inter-
section of each row and column gives the outcome of each
player choosing the corresponding strategy. The payoffs
to each player associated with this outcome are the basis
for determining whether the strategies are “in equilib-
rium,” or stable.
The characteristic-function form is generally used to
analyze games with more than two players. It indicates the
minimum value that each coalition of players—including
single-player coalitions—can guarantee for itself when
playing against a coalition made up of all the other
players.
one-peRson gaMes
One-person games are also known as games against nature.
With no opponents, the player only needs to list available
options and then choose the optimal outcome. When
chance is involved the game might seem to be more com-
plicated, but in principle the decision is still relatively
simple. For example, a person deciding whether to carry
an umbrella weighs the costs and benefits of carrying or
not carrying it. Although this person may make the wrong
decision, there does not exist a conscious opponent. That
is, nature is presumed to be completely indifferent to the
player’s decision, and the person can base his decision on
simple probabilities. One-person games hold little inter-
est for game theorists.
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Two-peRson
consTanT-suM gaMes
Games of Perfect Information
The simplest game of any real theoretical interest is a
two-person constant-sum game of perfect information.
Examples of such games include chess, checkers, and the
Japanese game of go. In 1912 the German mathemati-
cian Ernst Zermelo proved that such games are strictly
determined. By making use of all available information,
the players can deduce optimal strategies, which makes the
outcome preordained (strictly determined). In chess, for
example, exactly one of three outcomes must occur if the
players make optimal choices: (1) White wins (has a strat-
egy that wins against any strategy of Black), (2) Black wins,
or (3) White and Black draw. In principle, a sufficiently
By using all available information, players of the Japanese game of go can
deduce optimal strategies, making the outcome preordained (strictly deter-
mined). Shutterstock.com