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## ENGM 720 - Lecture 06

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**ENGM 720 - Lecture 06**Multiple Comparisons, 7 Tools of Ishikawa ENGM 720: Statistical Process Control**Assignment:**• Reading: • Chapter 4.5, Chapter 5 – 5.2, 5.4 • Finish reading • Review for Exam I • Covers material through hypothesis tests and seven tools • Assignments: • Obtain the Hypothesis Test (Chart &) Tables • Access Previous Assignment Solutions & Prepare Notebook: • Download Assignment 4 & Assignment 4 Solutions ENGM 720: Statistical Process Control**Multiple Comparisons**• Analysis-of-variance (ANOVA) is a statistical method used to test hypotheses regarding more than two sample means. • For a one-factor experiment the hypothesis tested is: ENGM 720: Statistical Process Control**Multiple Comparisons**• The strategy in an analysis of variance is to compare the variability between sample means to the variability within sample means. If they are the same, the null hypothesis is accepted. If the variability between is bigger than within, the null hypothesis is rejected. Null Hypothesis Alternative Hypothesis ENGM 720: Statistical Process Control**Definitions**• An experimental unitis the item measured during an experiment. The errors in these measurements are described by random variables. • It is important that the error in measurement be the same for all treatments (random variables be independent and have the same distribution). • The easiest way to assure the error is the same for all treatments is to randomly assign experimental units to treatment conditions. ENGM 720: Statistical Process Control**Definitions**• The variable measured in an experiment is called the dependent variable. • The variable manipulated or changed in an experiment is called the independent variable. • Independent variables are also called factors, and the sample means within a factor are called levelsortreatments. ENGM 720: Statistical Process Control**Definitions**• Random samples of size n are selected from each of k different populations. The k different populations are classified on the basis of a single criterion or factor. (one-factor and k treatments) • It is assumed that the k populations are independent and normally distributed with means µ1, µ2, ... , µk, and a common variance σ2. • Hypothesis to be tested is: ENGM 720: Statistical Process Control**Definitions**• A fixed effects modelassumes that the treatments have been specifically chosen by the experimenter, and our conclusions apply only to the levels chosen • Fixed Effect Statistical Model:where eij are independent and identically distributed N(0,σ2). • Because the fixed effects model assumes that the experiment is performed in a random manner, a one-way ANOVA with fixed effects is often called a completely randomized design. Overall Mean ith Treatment Effect Error in Measurement Observed Value ENGM 720: Statistical Process Control**Definitions**• For a fixed effects model, if we restrict: • Thenis equivalent to: ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**. . . . . . . . . . . . TM 720: Statistical Process Control**Analysis of the Fixed Effects Model**• Sum of Squares Treatments:The sum of squares treatments is a measure of the variability between the factor levels. • Sum of Squares Errors:The error sum of squares is a measure of the variability within the factor levels. Sum of Squares Errors (SSE) Factor level 1 Factor level 2 Factor level 3 X3· X1· X2· Sum of Squares Treatments (SSTr) ENGM 720: Statistical Process Control**BASE**Larger SSTr Smaller SSTr Larger SSE Smaller SSE Analysis of the Fixed Effects Model • P-values:The plausibility of the null hypothesis (that the factor level means are all equal)depends upon the relative size of the sum of squares for treatments (SSTr) tothe sum of squares for errors (SSE). ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**• Sum of Squares Partition for One Factor Layout:In a one factor layout, the total variability in the data observations is measured by the total sum of squares (SST) which is defined to be Total Sum of Squares SST Treatment Sum of Squares SSTr Error Sum of Squares SSE ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**• Sum of Squares Partition for One Factor Layout:This can be partitioned into two components: SST = SSTr + SSE,where the sum of squares for treatments (SSTr) measures the variability between the factor levels, • and the sum of squares for error (SSE) measures the variability within the factor levels. ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**• Sum of Squares Partition for One Factor Layout: On an intuitive level, the plausibility of the null hypothesis that the factor level means (µi) are all equal depends upon the relative size of the sum of squares for treatments (SSTr) to the sum of squares for error (SSE) Possibly Likely the Same Definitely NOT Likely the Same Definitely VERY Likely the Same ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**• F-Test for One Factor Layout:In a one factor layout with k levels and n replications gives a total sample size kn = N, the treatments are said to have k - 1 degrees of freedom and the error is said to have N - k degrees of freedom. Mean squares are obtained by dividing a sum of squares by its respective degrees of freedom so thatand ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**• F-Test for One Factor Layout:A p-value for the null hypothesis that the factor level means µi, are all equal is calculated as p-value = P(X ³ F) where the F-statistic is:and the random variable X has a Fk-1, N - k distribution. ENGM 720: Statistical Process Control**Analysis of the Fixed Effects Model**ENGM 720: Statistical Process Control**ANOVA Example**• The tensile strength of a synthetic fiber used to make cloth for shirts is of interest to a manufacturer. It is suspected that strength is affected by the percentage of cotton in the fiber. • Five levels of cotton percentage are of interest: 15%, 20%, 25%, 30%, and 35%. • Five observations are to be taken at each level of cotton percentage and the 25 total observations are to be run in random order. ENGM 720: Statistical Process Control**ANOVA Example**RANDOMIZATIONPROCEDURE . . . . . . . . . ENGM 720: Statistical Process Control**ANOVA Example**Tensile Strength of Synthetic Fiber (lb/in2) ENGM 720: Statistical Process Control**118.94 =14.8**8.06 161.20 =8.06 20 475.76 =118.94 4 ANOVA Example 5-1= 4 25-5= 20 25-1= 24 ENGM 720: Statistical Process Control**Critical Points for the F-Distribution**Alpha = 0.05 ENGM 720: Statistical Process Control**ANOVA Example**ENGM 720: Statistical Process Control**ANOVA Example**ENGM 720: Statistical Process Control**Ishikawa’s “Magnificent Seven” Tools**• The Seven Tools are: • Histogram / Stem & Leaf Diagram • Cause & Effect (Fishbone) Diagram • Defect Concentration Diagram • Check Sheet • Scatter (Plot) Diagram • Pareto Chart • Control Chart - not covered on exam! • The tools were not invented by Ishikawa, but were very successfully put into methodical use by him • The first six are used before starting to use the seventh • They are also reused when needed to find an assignable cause ENGM 720: Statistical Process Control**Ishikawa’s Tools: Histogram**• A histogram is a bar chart that takes the shape of the distribution of the data. The process for creating a histogram depends on the purpose for making the histogram. • One purpose of a histogram is to see the shape of a distribution. To do this, we would like to have as much data as possible, and use a fine resolution. • A second purpose of a histogram is to observe the frequency with which a class of problems occurs. The resolution is controlled by the number of problem classes. ENGM 720: Statistical Process Control**Histogram of Lab 01 Results**ENGM 720: Statistical Process Control**Ishikawa’s Tools: Fishbone Diagram**• Cause & Effect diagram constructed by brainstorming • Identified problem at the “head” • Connects potential causes along the spine • Sub-causes are listed along the major “bones” • Man • Material • Method • Machine • Environment ENGM 720: Statistical Process Control**Man**Method Skill Level Low RPM Attention Level Travel Limits Dusty Environment Poor Conductor Temperature Humidity Poor Mixing Orifice Clogs Poor Vendor Worn Parts Machine Material Bad Paint Cause & Effect Diagram, Cont. • The purpose of the cause and effect diagram is to obtain as many potential influencers of a process, so that the problem solving can take a more directed approach. ENGM 720: Statistical Process Control**Ishikawa’s Tools: Defect Diagram**• A defect concentration diagram graphically records the frequency of a defect with respect to product location. • Obtain a digital photo or multi-view part print showing all product faces. • Operator tallies the number and location of defects as they occur on the diagram. ENGM 720: Statistical Process Control**Title**Header Info: Date, Time, Location, Operator, etc. Times of Problem Occurrence (periodic) List of Prob Types Raw Data recorded here Statistics For Prob Types Time of Occurrence Statistics Overall Statistics Instructions, settings, comments, etc. Ishikawa’s Tools: Check Sheet • Check sheets are used to collect data (values or pieces of information) in a consistent manner. • List each of the known / possible problems • Record each occurrence including time-orientation. ENGM 720: Statistical Process Control**Y**Y Y X X X Ishikawa’s Tools: Scatter Plot • A scatter plot shows the relationship between any two variables of interest: • Plot one variable along the X-axis and the other along the Y-axis • The presence of a relationship can be inferred or ruled out, but it cannot determine if a cause and effect relationship exists ENGM 720: Statistical Process Control**Ishikawa’s Tools: Pareto Chart**• 80% of any problem is the result of 20% of the potential causes • Histogram categories are sorted by the magnitude of the bar • A line graph is overlaid, and depicts the cumulative proportion of defects • Quickly identifies where to focus efforts ENGM 720: Statistical Process Control**Statistical Quality Control and Improvement**Improving Process Capability and Performance Continually Improve the System Characterize Stable Process Capability Head Off Shifts in Location, Spread Time Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability Center the Process LSL 0 USL Use of Ishikawa’s Tools • Removing special causes of variation • Preparation for: • hypothesis tests • control charts • process improvement ENGM 720: Statistical Process Control**Questions & Issues**ENGM 720: Statistical Process Control