Chapter 5 Chapter 5 Process Analysis PART B 1 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 2 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 3 Important Probability Distributions Discrete Binomial Poisson Continuous Normal Exponential 44 Binomial Distribution The binomial distribution describes the probability of obtaining exactly x successes in a sequence of n identical experiments called trials 5 Computing the Binomial Distribution using Excel BINOM DIST number s trials probability s cumulative 6 Poisson Distribution l expected value or average number of occurrences x 0 1 2 3 e 2 71828 7 Computing the Poisson Distribution Using Excel POISSON DIST x mean cumulative 8 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 9 Normal Distribution Familiar bell shaped curve 10 Standard Normal Distribution If a normal random variable has a mean 0 and a standard deviation 1 it is called a standard normal distribution represented by z 11 Calculating Normal Probabilities If x is any value from a normal distribution with mean and standard deviation we may easily convert it to an equivalent value from a standard normal distribution using 12 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 13 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 14 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 15 Calculating the Exponential Distribution Using Excel The Excel function EXPON DIST x lambda true can be used to compute cumulative exponential probabilities 16 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 17 Central Limit Theorem 18 Illustrating the Central Limit Theorem 1919 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 20 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 21 Excel Procedures 22 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 23 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 24 Analysis of Variance Analysis of Variance or ANOVA is a hypothesis testing methodology for drawing conclusions about equality of means of multiple populations 25 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 26 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 27 Five 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 28 Cause 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 29 Cause and Effect Example 30