Handout Probability and Statistics
4
Probability and Statistics Introduction Profound changes in science and engineering education recognized the needs for introducing the ideas of probabilit y theory in a wide variety of scientific fields. One of the most significant is the greater emphasis that has been placed upon complexity and precision. A scientist recognizes the importance of studying scientific phenomena having complex interrelations among their components; these components are often n ot only mechanical or electrical parts but also ‘soft-science’ in nature, such as those stemming from behavioral and social sciences. Moreover, precision is stressed – precision in describing interrelationships among factors involved in a scientific phenomenon and precisi on in predicting its behavior. This, coupled with increasing complexity in the problems we face, leads to the recogn ition that a great deal of uncertainty and variability are inevitably present in problem formulation, and one of the mathematical tools that is effective in dealing with them is probability and statistics. The use of statistical methods in manufacturing, development of food products, computer software, and many other areas involves the gathering of information or scientific data. These statistical methods are designed to contribute to the process of making scientific judgments in the face of uncertainty and variation. The product density of a particular material from a manufactur ing process will not always be the same. Indeed, if the process involved is a batch process rather than continuous, there will be variation in material density between not only the batches (batch-to-batch variation) that come off the line, but also within-batch var iation. Statistical methods are used to analyze data from a process to improve the quality of the process. The development of basic principles in constructing probability models and the subsequent analysis of these models is schematically presented in Figure 1. A basic understanding of probability theory and random variables is central to the whole modeling process as they provide the required mathematical machinery with which the modeling process is carri ed out and consequences deduced. The step from B to C is the induction step by which the structure of the model is formed from factual observations of the scientific phenomenon under study. Model verification and parameter es timation (E) on the basis of observed data ( D) fall within t he framework of stati stical inference. A model may be rejected at this stage as a result of inadequate inductive reasoning or insufficient or deficient data. A reexamination of factual obser vations or additional data may be required here. Finally, model analysis and deduction are made to yield desired answers upon model substation.
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Probability and Statistics
Introduction
Profound changes in science and engineering education recognized the needs for
introducing the ideas of probability theory in a wide variety of scientific fields. One of the most significant is the greater emphasis that has been placed upon complexity and
precision.
A scientist recognizes the importance of studying scientific phenomena having complex
interrelations among their components; these components are often not only mechanical
or electrical parts but also ‘soft-science’ in nature, such as those stemming from
behavioral and social sciences.
Moreover, precision is stressed – precision in describing interrelationships among factors
involved in a scientific phenomenon and precision in predicting its behavior. This,
coupled with increasing complexity in the problems we face, leads to the recognition thata great deal of uncertainty and variability are inevitably present in problem formulation,
and one of the mathematical tools that is effective in dealing with them is probability andstatistics.
The use of statistical methods in manufacturing, development of food products, computer software, and many other areas involves the gathering of information or scientific data.
These statistical methods are designed to contribute to the process of making scientific
judgments in the face of uncertainty and variation. The product density of a particular
material from a manufacturing process will not always be the same. Indeed, if the process involved is a batch process rather than continuous, there will be variation in
material density between not only the batches (batch-to-batch variation) that come off theline, but also within-batch variation. Statistical methods are used to analyze data from a process to improve the quality of the process.
The development of basic principles in constructing probability models and thesubsequent analysis of these models is schematically presented in Figure 1. A basic
understanding of probability theory and random variables is central to the whole
modeling process as they provide the required mathematical machinery with which the
modeling process is carried out and consequences deduced. The step from B to C is theinduction step by which the structure of the model is formed from factual observations of
the scientific phenomenon under study. Model verification and parameter estimation (E)
on the basis of observed data (D) fall within the framework of statistical inference. Amodel may be rejected at this stage as a result of inadequate inductive reasoning or
insufficient or deficient data. A reexamination of factual observations or additional data
may be required here. Finally, model analysis and deduction are made to yield desiredanswers upon model substation.
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Figure 1. Basic Cycle of Probabilistic Modeling and Analysis
Inferential statistics has produced an enormous number of analytical methods that allow
for analysis of data from the system. This reflects the true nature of science that allowsus to go beyond merely reporting data but, rather, allow the drawing of conclusions about
the scientific system. Information is gathered in the form of samples, or collections of
observations. Samples are collected from populations that are collections of allindividuals or individual items of a particular type. Often, it is very important to collect
scientific data in a systematic way.
The use of application of concepts in probability allows real-life interpretation of the
results of statistical inference. As a result, it can be said that statistical inference makes
use of concepts in probability. Figure 2 shows the fundamental relationship between probability and inferential statistics. For a statistical problem the sample along with the
inferential statistics allows us to draw conclusions about the population, with inferential
statistics making clear use of elements of probability. This reasoning is inductive innature. In the case where population clearly described with all features of the population
known, then questions of importance will focus on the nature of data that might
hypothetically be drawn from the population. Thus one can say that problems in
probability allow us to draw conclusions about characteristics of hypothetical data takenfrom the population based on known features of the population. This type of reasoning is
deductive in nature. In the grand scheme of things both the field of probability and
statistics are very important and clearly are complimentary.
Figure 2. Fundamental Relationship between Probability and Inferential Statistics
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