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Understanding Process Analysis and Probability Concepts

This chapter explores process analysis techniques and basic probability concepts such as experimental outcomes, sample space, probability properties, calculating probabilities, conditional probability, random variables, probability distributions, and sampling distributions.

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Understanding Process Analysis and Probability Concepts

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  1. Chapter 5 Process Analysis

  2. Analysis • Analysis is the examination of processes, facts, and data to gain an understanding of why problems occur and where opportunities for improvement exist. • Statistical inference is the process of drawing conclusions about unknown characteristics of a population from which data were taken. • Predictive statisticsfocus on cause-and-effect relationships and predictions of future performance from historical data.

  3. Basic Probability Concepts • An experiment is a process that results in some outcome. • The outcome of an experiment is a result that we observe • The collection of all possible outcomes of an experiment is called the sample space. • Probability is the likelihood that an outcome occurs.

  4. Probability Properties • Label the n outcomes in a sample space as O1, O2, … On, where Oi represents the ith outcome in the sample space. • The probability associated with any outcome must be between 0 and 1 • 0 ≤ P(Oi) ≤ 1 for each outcome Oi • The sum of the probabilities over all possible outcomes must be 1.0 • P(O1) + P(O2) + … + P(On) = 1

  5. Events • An event is a collection of one or more outcomes from a sample space • If A is any event, the complement of A, denoted as Ac, consists of all outcomes in the sample space not in A. • Two events are mutually exclusive if they have no outcomes in common.

  6. Calculating Probabilities • Rule 1: The probability of any event is the sum of the probabilities of the outcomes that compose that event. • Rule 2: The probability of the complement of any event A is P(Ac) =1 – P(A).

  7. Calculating Probabilities • Rule 3: If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B) • Rule 4: If two events A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) – P(A and B)

  8. Conditional Probability • Conditional probability is the probability of occurrence of one event A, given that another event B is known to be true or have already occurred. • Multiplication rule of probability:

  9. Tree Diagram

  10. Calculation of Joint Probabilities

  11. Independent Events • Two events A and B are independent if P(A | B) = P(A).

  12. Random Variables • A random variable, X,is a numerical description of the outcome of an experiment. Formally, a random variable is a function that assigns a numerical value to every possible outcome in a sample space.

  13. Probability Distributions • A probability distribution, f(x), is a characterization of the possible values that a random variable may assume along with the probability of assuming these values. • The cumulative distribution function, F(x),specifies the probability that the random variable X will assume a value less than or equal to a specified value, x, denoted as P(X ≤ x).

  14. Important Probability Distributions • Discrete • Binomial • Poisson • Continuous • Normal • Exponential 14

  15. Binomial Distribution • The binomial distribution describes the probability of obtaining exactly x “successes” in a sequence of n identical experiments, called trials.

  16. Computing the Binomial Distribution using Excel BINOM.DIST(number_s, trials, probability_s, cumulative)

  17. Poisson Distribution • = expected value or average number of occurrences • x = 0, 1, 2, 3, … • e = 2.71828…

  18. Computing the Poisson Distribution Using Excel POISSON.DIST(x, mean, cumulative)

  19. Probability Density Function • A curve that characterizes outcomes of a continuous random variable is called a probability density function, and is described by a mathematical function f(x). • Probabilities are only defined over intervals. • The cumulative distribution function, F(x), represents the probability P(X ≤ x).

  20. Normal Distribution • Familiar bell-shaped curve.

  21. Standard Normal Distribution • If a normal random variable has a mean μ = 0 and a standard deviation σ= 1, it is called a standard normal distribution, represented by z.

  22. Calculating Normal Probabilities • If x is any value from a normal distribution with mean μ and standard deviation σ, we may easily convert it to an equivalent value from a standard normal distribution using:

  23. Calculating Normal Probabilities Using Excel • Excel function NORM.DIST(x, mean, standard deviation, true) calculates the cumulative probability F(x)for a specified mean and standard deviation. • The Excel function NORM.S.DIST(z) calculates the cumulative probability for the standard normal distribution.

  24. NORM.INV Function • The Excel function NORM.INV(probability, mean, standard_dev) can be used when we know the cumulative probability (probability) but don’t know the value of x.

  25. Exponential Distribution • The exponential distribution models the time between randomly occurring events, such as the time to or between failures of mechanical or electrical components.

  26. Calculating the Exponential Distribution Using Excel • The Excel function EXPON.DIST(x, lambda, true) can be used to compute cumulative exponential probabilities.

  27. Sampling Distributions • A sampling distribution is the distribution of a statistic for all possible samples of a fixed size. • Sampling distribution of the mean • Expected value of the sample mean is the population mean • Standard deviation of the sample mean (called the standard error of the mean) is the population standard deviation divided by the square root of the sample size

  28. Central Limit Theorem

  29. Illustrating the Central Limit Theorem 29

  30. Confidence Intervals • A confidence interval (CI) is an interval estimate of a population parameter that also specifies the likelihood that the interval contains the true population parameter. This probability is called the level of confidence, denoted by 1 − α, and is usually expressed as a percentage. 30

  31. Common Confidence Intervals

  32. Confidence Interval Template • The Student Companion Site provides an Excel workbook, Confidence Intervals.xlsx, with worksheet templates for formulas (5.17) through (5.19).

  33. Hypothesis Testing • Hypothesis testing involves drawing inferences about two contrasting propositions (hypotheses) relating to the value of a population parameter, one of which is assumed to be true in the absence of contradictory data (called the null hypothesis), and the other which must be true if the null hypothesis is rejected (called the alternative hypothesis).

  34. Hypothesis Testing Process • Steps 1. Formulate the hypotheses to test. 2. Select a level of significance. 3. Determine a decision rule on which to base a conclusion. 4. Collect data and calculate a test statistic. 5. Apply the decision rule to the test statistic and draw a conclusion.

  35. Excel Procedures

  36. Regression Analysis • Regression analysis is a tool for building statistical models that characterize relationships between a dependent variable and one or more independent variables, all of which are numerical. • A regression model that involves a single independent variable is called simple regression. A regression model that involves several independent variables is called multiple regression.

  37. Correlation • Correlation is a measure of a linear relationship between two variables, X and Y, and is measured by the (population) correlation coefficient. • Correlation coefficients will range from −1 to +1.

  38. Analysis of Variance • Analysis of Variance, or ANOVA, is a hypothesis-testing methodology for drawing conclusions about equality of means of multiple populations.

  39. One Way Analysis of Variance • In its simplest form—one-way ANOVA—we are interested in comparing means of observed responses of several different levels of a single factor. • ANOVA tests the hypothesis that the means of all populations are equal against the alternative hypothesis that at least one mean differs from the others.

  40. Multi-Vari Studies • A multi-vari study investigates three types of process variation: positional (variation within the same item or sample), cyclical (variation between parts or samples), and temporal (over time, such as between different production shifts).

  41. Design of Experiments • A designed experiment is a test or series of tests that enables the experimenter to compare two or more methods to determine which is better, or determine levels of controllable factors to optimize the yield of a process or minimize the variability of a response variable.

  42. Factorial Experiments • Factorial experiment – one that considers all combinations of levels of each factor • The simplest type of factorial experiment is one with two factors at two levels • Each combination of different levels of the factor is called a treatment.

  43. Main Effects • A main effect measures the difference that a factor has on the response. • What is the effect of increasing the temperature regardless of the value of the reaction time? • What is the effect of increasing the reaction time regardless of the temperature? Main effect = (Average response at high level) - (Average response at low level) (5.22)

  44. Interactions • An interaction is the effect of changing one factor has on the level of other factors. • For example, increasing factor 1 when factor 2 is at the low level might result in an increase in the response variable; however, increasing factor 1 when factor 2 is at the high level might result in a decrease in the response variable. Interaction effect = (Average response with both factors at the same level) – (Average response with both factors at opposite levels) (5.23)

  45. Excel Templates for Factorial Experiments • The Student Companion Site has Excel templates for factorial experiments.

  46. Note to Instructors • The following slides provide an experiential exercise found on the Web for applying DOE. • Students usually have a lot of fun with this. 46

  47. Paper Helicopter Design • Cut along all the solid lines on the diagram to the right. • Fold flap A forward and flap B to the back. • Fold flaps C and D both forward along the dotted lines. • Fold along the line E upward to give a weight at the bottom. 47

  48. Making the Helicopter 48

  49. Helicopter Project Description • Response goal: Maximize the length of time before hitting the ground from a fixed height. • Identify 3 potential sources that might influence the response, for example, AB length, CD length, and width • Develop an experimental design; collect data, and analyze results. Each group should submit a report of their findings next class. Show all results; calculate the main effects and interactions, and draw conclusions about what is the best design.

  50. Root Cause Analysis • Root cause – “that condition (or interrelated set of conditions) having allowed or caused a defect to occur, which once corrected properly, permanently prevents recurrence of the defect in the same, or subsequent, product or service generated by the process.”

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