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## STATISTICS & NUMERICAL METHODS FOR PLANT ENGINEERS AGE-214

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**STATISTICS & NUMERICAL METHODSFOR PLANT ENGINEERSAGE-214**By S. O. Duffuaa Systems Engineering Department**Salih Duffuaa**• Dr. Duffuaa is a Professor of Industrial and Systems Engineering at the Department of Systems Engineering at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. He received his PhD in Operations Research from the University of Texas at Austin, USA. His research interests are in the areas of Operations research, Optimization, quality control, process improvement and maintenance engineering and management. He teaches course in the areas of Statistics, Quality control, Production and inventory control, Maintenance and reliability engineering and Operations Management. He consulted to industry on maintenance , quality control and facility planning. He authored a book on maintenance planning and control published by John Wiley and Sons and edited a book on maintenance optimization and control. He is the Editor of the Journal of Quality in Maintenance Engineering, published by Emerald in the United Kingdom.**King Fahd University of Petroleum & MineralsDepartment of**Systems Engineering A closedshort course for Saudi Aramco Employees On Statistics for Plant Engineers and Lab Scientists Oct, 18-22, 2008**Dr. Salih Duffuaa,Introduction to probability ,frequency &**Probability distributions, mean and variance. Saturday Dr. Mohammad Haboubi, The normal distribution, the central limit theorem and sampling distributions.. Sunday Dr. Hesham Al-Fares: ,Point and interval estimation, statistical significance tests. Monday Dr. Mohammad Al Salamah,Simple regression, residual analysis Tuesday Dr. Shokri Selim:Multiple regression, adequacy of a regression model, Applications Wednesday**Course Outcomes**• Apply probability concepts and laws to solve basic lab problems. • Summarize and present data in meaningful ways. • Compute probabilities from probability distributions. • Construct confidence intervals for sample data. • Test statistical hypothesis. • Construct regression models and use them in various applications such as equipment calibration and prediction.**Day 1 Module Objectives**• Concept and definition of probability. • Axioms of probability • Laws of probability**Day 1 Module Objectives**• Data Summary • Measures of central tendency. • Mean X-bar and Median M • Measures of variability • Range R, Variance S2, Standard deviation S and coefficient of variation (CoV). • Frequency Distribution. • Distributions • Expected value**Day 1 Module Objectives**• Random variables. • Mass and distribution functions • Expected value**Examples of a Random Experiment**• Measuring a current in a wire. • Number of samples analyzed per day . • Time to do a task. Time to analyze a sample. • Yearly rain fall in Dhahran**Examples of a Random Experiment**• Throwing a coin • Number of accidents on campus per month. • Students must generate at least 5 examples.**Random Experiments**• Every time the experiment is repeated a different out come results. • The set of all possible outcomes is call Sample Space denoted by S. • In the experiment of throwing the coin the sample space S = { H, T}.**Random Experiments**• In the experiment on the number of defective parts in three parts the sample space S = { 0, 1, 2, 3}. • Number of weekly traffic accidents on KFUPM campus.**Event**• An event E is a subset of the sample space. • Example of Events in the experiment of the number of defective in a sample of 3 parts are: • E1 = { 0}, E2 = { 0,1}, E3 = { 1, 2}**Example of Events**• A sample of polycarbonate plastic is analyzed for scratch resistance and shock resistance. The results from 49 samples are: • Shock resistance H L H 40 4 Scratch Resistance L 2 3 Let A denote the event a sample has high shock resistance and B denote the event a sample has high scratch resistance. Determine the the number of samples in AB, AB and A`**Solution of Example**• IAI = 42, IBI = 44 • IABI = 40 • IABI = 46 • A = 7 , B = 5**Exercise**• Refer to the event example and answer the following: • Find the number in AB • Find the number of elements in AB • Find the number of elements in AB**Listing of Sample Spaces**• Tree Diagrams • Experience**Listing of Sample Spaces**• The experiment of throwing a coin twice H H T H 1 T T S = { HH, HT, TH, TT}**Example on Listing Sample Spaces**• Draw the tree diagram for finding the sample space for the number of defect item in a sample of size three taken from a production line producing chips.**Types of Sample Spaces**• A sample space is discrete if it consists of a finite ( or countable infinite ) set of outcomes. Examples are: • S = { H, T}, S = { 1, 2, 3, …} • Sample space is continuous if contains an interval (finite or infinite): { T: 0 ≤ T ≤ 60}. • Students should give more examples**Notation**• P - denotes a probability • A, B, ...- denote a specific event • P (A)- denotes the probability of an event occurring**Concepts and Definition of Probability**Four definitions of probability: • Classical or a priori probability • Statistical or a posteriori probability • Subjective probability (used in Bayesian methods). • Mathematical probability**CLASSICAL OR A PRIORI PROBABILITY**P(A) = # of ways A can occur (# favorable cases) Total number of possible cases (# of total possible cases)**STATISTICAL OR A POSTERIORIPROBABILITY**# of successes Pr (A) = Number of trials In the limit as # oftrials Infinity**SUBJECTIVE PROBABILITY**• A measure of the degree of belief. • There is a 10% chance it will rain today. • There is a 95% chance you can see the new moon tomorrow morning. • Subjective probability is the basis for Bayesian methods.**Probability Limits**• The probability of an impossible event is 0. • The probability of an event that is certain to occur is 1. 0 ≤ P(A) ≤ 1 Impossible to occur**Probability Limits**• The probability of an impossible event is 0. • The probability of an event that is certain to occur is 1. 0 ≤ P(A) ≤ 1 Impossible to occur Certain to occur**MATHEMATICAL PROBABILITY**• A measure of uncertainty ( or possibility) that satisfy the following conditions: • 0 ≤ P(A) ≤ 1 • P(S) = 1 • Pr (A U B) = Pr (A) + Pr (B) If A Π B = Ǿ**Possible Values for Probabilities**Certain 1 Likely 0.5 50-50 Chance Unlikely Impossible 0**Probability of an Event**• For discrete a sample space, the probability of an event denoted as P(E) equals the sum of the probabilities of the outcomes in E. • Example: S = { 1, 2, 3, 4, 5} each outcome is equally likely. E is even numbers within S. E = { 2, 4}, P(E) = 2/5.**Axioms of Probability**• If S is the sample space and E is any event then the axioms of probability are: 1. P(S) = 1 2. 0 P(E) 1 3. If E1 and E2 are event such thatE1 E2 = , then, P(E1 E2) = P(E1 ) + P(E2)**mutually exclusive**• Events A and B are mutually exclusiveif they cannot occur simultaneously.**Definition**Total Area = 1 P(A) P(B) P(A and B) Overlapping Events**Definition**Total Area = 1 Total Area = 1 P(A) P(B) P(A) P(B) P(A and B) Overlapping Events Nonoverlapping Events**Complementary Events**P(A) The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. P(A) (read “not A”)**Rules for Complementary Events**P(A) + P(A) = 1**Rules for Complementary Events**P(A) + P(A) = 1 = 1 – P(A) P(A)**Rules for Complementary Events**P(A) + P(A) = 1 = 1 – P(A) P(A) = 1 – P(A) P(A)**Venn Diagram for the Complement of Event A**Total Area = 1 P(A) P(A) = 1 – P(A)**Probability of ‘At Least One’**• ‘At least one’ is equivalent to one or more. • The complement of getting atleast oneitem of a particular type is that you get no itemsof that type.**Probability of ‘At Least One’**• If P(A) = P (getting at least one), then • P(A) = 1 – P(A) • where P(A) is P (getting none)**Definitions**• Any event combining 2 or more events • Compound Even Notation • P(A or B) = P (event A occurs or event B occurs or they both occur) • P(A and B) =P (event A occurs and event B occurs)**Addition Rule**• P(A or B) = P (event A occurs or event B occurs or they both occur). • P(AB) = P(A) + P(B) – P( AB) • If AB) = , then, • P(AB) = P(A) + P(B)**Addition Rule**A B P(AB) = P(A) + P(B) – P( AB)**Addition Rule: Example**• Let S = { 1, 2, 3, 4, 5, 6, 7, 8,9,10) • A = { 2,3,4,5,6}, B = {4, 5,6,7,9,10} • AB = { 2,3,4,5,6,7,9,10} • P(A) = 5/10 =0.5 P(B) = 6/10 = 0.6 • P(AB) = 0.8 • P(AB) = 0.5 + 0.6 – 0.3 = 0.8**Problem**• Let assume A, B and C are events from the sample space S. P(A) = 0.4. P(B) = 0.5, P(C) = 0.3, P(BC) = 0.1, P(AC) = 0.2, A and B are mutually exclusive. Compute the following: (i) P(A'), (ii) P(AυC), (iii) P[(AυB)C) (iv) P(A υB υC), (v) P(B'UC‘) Note A' means A compliment.**Conditional Probability**• Conditional Probability Concept • P(A B) = P(A B)/ P(B) for P(B) > 0 • Give Examples • Solve problems**Example on Conditional Probability**• Let S = { 1, 2, 3, 4, 5, 6, 7, 8,9,10) • A = { 2,3,4,5,6}, B = {4, 5,6,7,9,10} • P(A|B) = P(AB)/P(B) = 0.3/0.6 = 0.5 • This is as if we consider B our sample space and see how many elements from A in B. This will make P(B) = 1**Conditional Probability**Dependent Events P(A and B) = P(A) • P(B|A)**Conditional Probability**Dependent Events P(A and B) = P(A) • P(B|A) • Formal P(A and B) • P(B|A) = • Intuitive • The conditional probability of B given A can be • found by assuming the event A has occurred and, • operating under that assumption, calculating the • probability that event B will occur. P(A)