Slide 1Random VariablesProbability DistributionsImportant Probability DistributionsBinomial DistributionComputing the Binomial Distribution using ExcelPoisson DistributionComputing the Poisson Distribution Using ExcelProbability Density FunctionNormal DistributionStandard Normal DistributionCalculating Normal ProbabilitiesCalculating Normal Probabilities Using ExcelNORM.INV FunctionExponential DistributionCalculating the Exponential Distribution Using ExcelSampling DistributionsCentral Limit TheoremIllustrating the Central Limit TheoremHypothesis TestingHypothesis Testing ProcessExcel ProceduresRegression AnalysisCorrelationAnalysis of VarianceOne Way Analysis of VarianceRoot Cause AnalysisFive Why TechniqueCause-and-Effect DiagramsSlide 301Chapter 5Chapter 5Process Analysis - PART BRandom 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.2Probability 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).3Important Probability DistributionsDiscrete–Binomial–PoissonContinuous–Normal–Exponential44Binomial DistributionThe binomial distribution describes the probability of obtaining exactly x “successes” in a sequence of n identical experiments, called trials.5Computing the Binomial Distribution using ExcelBINOM.DIST(number_s, trials, probability_s, cumulative)6Poisson Distributionl= expected value or average number of occurrences x = 0, 1, 2, 3, … e = 2.71828…7Computing the Poisson Distribution Using ExcelPOISSON.DIST(x, mean, cumulative)8Probability 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).9Normal DistributionFamiliar bell-shaped curve.10Standard 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.11Calculating 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:12Calculating 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.13NORM.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.14Exponential DistributionThe exponential distribution models the time between randomly occurring events, such as the time to or between failures of mechanical or electrical components.15Calculating the Exponential Distribution Using ExcelThe Excel function EXPON.DIST(x, lambda, true) can be used to compute cumulative exponential probabilities.16Sampling 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 size17Central Limit Theorem18Illustrating the Central Limit Theorem1919Hypothesis 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).20Hypothesis Testing ProcessSteps1. 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.21Excel Procedures22Regression 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. 23Correlation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 +1.24Analysis of VarianceAnalysis of Variance, or ANOVA, is a hypothesis-testing methodology for drawing conclusions about equality of means of multiple populations. 25One 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.26Root 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.”27Five Why TechniqueRedefine a problem statement as a chain of causes and effects to identify the source of the symptoms by asking why, ideally five times.28Cause-and-Effect DiagramsCause-and-effect diagram – a simple graphical method for presenting a chain of causes and effects and for sorting out causes and organizing relationships between variables.2930Cause and Effect -