【正文】 ’ll learn how to analyze stratified data. Here is one example: 13 Population vs. Process A Population ?Situation: You can operationally define the boundaries of an existing (whole) group so that each unit in the group can be identified and, theoretically, numbered. ?Sampling Purpose: To describe characteristics of that group. ?Example: University alumni (as of Aug 31st) are sampled to determine what percentage will send at least one child to college within the next two years. Use the sample to draw inferences about the whole group: ., Average = X, Proportion = p Sample 14 Representative Samples For conclusions to be valid, samples must be representative. ?Data should fairly represent the population or process ?No systematic differences should exist between the data you collect and the data you don’t collect 15 Sampling Approaches Random Sampling Stratified Random Sampling S a m p leP o p u la t i o nEach unit has the same chance of being selected Randomly sample a proportionate number from each group AABBBBCDDD Population Sample C A B D A A A C D D D D D B B B B B B B 16 Sampling Approaches Sample Population or Process Preserve time order Sample Process 9:00 9:30 10:30 10:00 Preserve time order Systematic Sampling Subgroup Sampling Sample every nth one (., every 3rd) Sample n units every tth time (., 3 units every hour)。 calculate the mean (proportion ) for each subgroup 17 Types of Variation ?Special Cause: something different happening at a certain time or place ?Common Cause: always present to some degree in the process 18 Process Capability Statistics Mean Standard Deviation Specification Cp amp。 Cpk Pp amp。 Ppk Estimated PPM Actual Observed PPM 19 ANALYZE IMPROVE 20 The Focus of Analyze Y = f (X1, X2, X3, ..., Xn) X1, X3, X5 Identification Verification Quantification What vital few process and input variables (Xs) affect critical to quality process performance or output measures (Ys)? 21 Process or Data Door? Process Door Data Door ?To understand the drivers of variation in the process ?To tackle quality problems and waste ?To understand the root cause of differences between outputs ?To improve the understanding of process flow ?To tackle cycle time problems ?To identify opportunities to reduce process costs Stratification Scatter Diagrams MultiVari Analysis Detailed Process Map Value Added Analysis Cycle Time Analysis It is remended to go through both doors to make sure that potential causes are not overlooked. 22 Two Types of Errors in Hypothesis Testing A ctual (Truth )Groups areSameGroups areD if f eren tAc c ept H 0 :Gro u p s a r e S a m eN o Error T y pe I I Err orC o n cl u si o norD eci si o n R ejec t H0 :Gro u p s a r e Diffe r e n tT y pe I Error N o ErrorThere are four possible outes to any decision we make based on a hypothesis test: We can decide the groups are the same or different, and we can be right or wrong. ? Both types of error are important. ? Guarding too heavily against one error increases the risk of the other error. ? Increasing the sample size: ? Reduces the risk of Type II errors. ?Allows you to detect smaller differences. Actual (Truth) Innocent Guilty Innocent Innocent Guilty amp。 you walk Conclusion or Decision Guilty Innocent amp。 in jail Guilty Example: Court Cases .. 23 Types of Hypothesis Tests Hypothesis Test Purpose ttest Compare two group averages. paired ttest Compare two group averages when data is matched. ANOVA (Analysis Of Variance) Compare two or more group averages. Test for equal variances (Ftest, Bartlett’s test, Levene’s test) Compare two or more group variances. Chisquare test Compare two or more group proportions. 24 Regression: Quantifies the Relationship Between X and Y Regression analysis generates a line that quantifies the relationship between X and Y. 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 X (input) Y (output) Appropriate Data for X or Y In Regression Data Type Minitab Format DiscreteOrdinal ranks 1, 2, ..., 5 Numerical DiscreteCount or Percents of defects, % defective Numerical Continuous Amounts Cycle time Numerical The line, or regression equation, is represented by mathematical equation of a straight line as: Y= bo+ b1X bo = intercept (where the line crosses X= 0) b1 = slope (rise over run, or change in Y per unit increase in X) 25 Confidence and Prediction Intervals 200 250 300165175185195205215225T e m p (F )Seal (g/cm2)Y = 1 0 1 . 6 1 1 + 0 . 3 5 4 2 3 7 XR S q = 8 3 . 3 %R e g r e s s i o n9 5 % C I9 5 % P IR e g r e ssi o n P l o tConfidence Interval ? . = An interval likely to contain the “best fit” line. ? Gives a range of the predicted values for the fitted Y if the regression is repeated again (average). ? Based on a given Xvalue ? For a given confidence Prediction Interval ? . = An interval likely to contain the actual Y values for a given X. ? Gives a range of likely actual values for Y (individual). ? Based on a given Xvalue. ? For a given confidence. 26 Three Factors: Cube Layout ? A cube helps us visualize the experimental space covered by 3 factors ? Each corner represents one of the eight sets of experimental conditions Lower left front corner = Recycled paper, no paper clip, ” (This corner represents all the Low levels: –, –, –) Upper right back corner