STATISTICS AND
OPTIMIZATION
Dr. Asawer A. Alwasiti
Contenet
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Chapter One: introduction
fundamental elements of statistics, types of data, Methods of describing data, measures of central tendency,
measures of variation measures of relative standing
Chapter Two: Probability
discreet of random variable: the probability distribution, the binomial probability distribution, The hypogeometric
probability distribution, Poison distribution.
Continuous random distribution: the continuous random variable, uniform probability distribution, normal probability
distribution
Chapter Three: Testing of Hypothesis
Sampling distributions – Testing of hypothesis for mean, variance, proportions and differences using Normal, t, Chisquare and F distributions - Tests for independence of attributes and Goodness of fit.
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Chapter Four: Simple linear regression
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Regression model, methods of least squares, the coefficient of correlation, The coeffiecnt of determination
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Chapter Five DESIGN OF EXPERIMENTS
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Experimental design terminology, factorial design Completely randomized design – Randomized block design
Chapter One
Introduction
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Statistics: is concerned with scientific methods for collecting,
organizing, summarizing, presenting and analyzing data as well as
drawing valid conclusions and making reasonable decisions.
Types of data:
Quantitative data : are those that represent the quantity or amount of
something, measured on a numerical scales. For example; the power
frequency
Qualitative data: it’s the data that can only classified i.e. posses no
numerical representation
Population: refers to all the persons, objects, source or measurements
under consideration, or it is a data set that is our target of interest.
Sample: refers to any portion of the population
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Descriptive Statistics: used to organize, summarize and describe
measures of sample. It uses numbers to summarize information which
is known about some situation.
Inductive (inference) statistics: are used to predict population
parameters from sample measures.
Variables: is a symbol such as X, Y ,H…. which can assume any of
the prescribed set of values. It contains qualitative and quantitative
variables
Continuous variable: can theoretically assume any value between
two given values depending on accuracy of measurements
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Discrete variable: all data can be obtained from counting
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Parameter: the measures which describe population characteristics.
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Example:
The reliability of computer system is measured in terms of
life length of a specific hardware component (e.g hard disk
life). To estimate the reliability of a particular system , 100
computer component are tested until they fail, under their
life length are recorded.
What is the population of interest?
What is the sample?
Are the data are qualitative or quantitative?
How could the sample information be used to estimate the
reliability of the computer system?
Graphical method of representation
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Qualitative Data
They are usually achieved using Bar graph or Pie chart
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Bar graph: the category (class) of the qualitative variable is represented by Bar
graph in which the height of each bar is either the class frequency, class relative
frequency or class percentage.
Pie chart: the category (class) of the quantitative variable is represented by Pie
chart. The size of each slice is proportional to the class relative frequency.
Pareto diagram: a bar graph with the category (class) of the qualitative variable
arranged by height in descending order from left to right.
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Example:
Group of researchers investigating the safety of nuclear power reactors
and the hazard of using energy, they discovered 45 energy related
accident worldwide since1977 that resulted in multi factories as:
category
frequency
Coal mine collapse
7
Dam frailer
4
Gas explosion
28
lightning
1
Nuclear reactor
1
Oil fire
4
total
45
Chart of Causes
30
28
26
24
22
Frequency
20
18
16
14
12
10
8
6
4
2
0
Coal mine collapse
Gas explosion
Nuclear reactor
Dam frailer
lightning
Causes
Oil fire
Pie Chart of Causes
Oil fire
Nuclear reactor
lightning
Coal mine collapse
Dam frailer
Gas explosion
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Quantitative Data
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It can be represented in graphical or numerical way
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Graphical representation
Quantitative Data can be represented graphically by Histogram
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Frequency distribution
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Raw data: are collected data which have been collected numerically
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Array: arranged of raw data in ascending or descending order.
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Range: the difference between the largest and smallest value
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Frequency distribution: a table arrangement of data by classes together with the corresponding class frequencies.
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Class interval: A symbol defining the class.
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Class mark: is the mid point of the class interval
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Formation of frequency distribution:
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Determine the largest and smallest observation
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Take total width = range + 1 unit in the last significant digit
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Dived total width in 5-20 class of equal width
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Calculate class width, interval and class mark
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Calculate frequencies
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Histogram
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Graphical representation of frequency distribution consist of a set of rectangular having:
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Basis with centers at class marks and lengths equal to the class width
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Area proportional to class frequencies
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Frequency polygon
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Formed by connecting the mid points of the tops of the rectangular in the histogram
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Relative frequency
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Is the frequency of the class divided by the total frequency and expressed as a percentage
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Example
The pH level of drilling mud of well that determined within 24 hr is shown in table below, make the frequency
distribution table and graph the data
7.25
7.37
7.33
7.38
7.3
7.35
7.34
7.38
7.41
7.39
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7.26
7.3
7.39
7.31
7.39
7.34
7.34
7.32
7.42
7.43
7.36
7.35
7.34
7.32
7.24
7.35
7.37
7.35
7.45
7.46
7.36
7.26
7.39
7.35
7.33
7.3
7.34
7.39
7.4
7.4
7.34
7.34
7.39
7.3
7.37
7.25
7.33
7.33
7.41
7.4
7.3
7.29
7.28
7.29
7.32
7.36
7.32
7.38
7.43
7.45
Example
The viscosity of 40 sample of drilling mud measured in cp is shown below.
50.2them in frequency
49.3 table and
49.9with histogram.
50.1
50.5
Represent
51.1
49.7
50.3
49.9
51.4
49.8
49.6
49.5
49.8
50.7
50.2
50.4
50
50.7
48.6
48.9
50
50
50.3
49.4
49.9
48.6
50
49.4
50.6
50
49.9
50.6
50.8
49
49.5
51.3
50.8
50.2
50.3