A05 Continuous One Variable Stat Tests

Determine Critical X’s Statistical Tests for a Continuous Single Variable Deliverable 10A

Define Module Roadmap Define 1D – Define VOC, VOB, and CTQ’s 2D – Define Project Boundaries 3D – Quantify Project Value 4D – Develop Project Mgmt. Plan Measure 5M – Document Process 6M – Prioritize List of X’s 7M – Create Data Collection Plan 8M – Validate Measurement System 9M – Establish Baseline Process Cap. Analyze 10A – Determine Critical X’s Improve 12I – Prioritized List of Solutions 13I – Pilot Best Solution Control 14C – Create Control System 15C – Finalize Project Documentation Green 11G – Identify Root Cause Relationships Queue 1 Queue 2

Deliverables – Analyze # Deliverable Deliverable Concept & Tasks Primary Tool(s) Secondary Tool(s) 10A Determine Critical X’s Using the data gathered after deliverable 8M, we need to assess which X’s cause changes in the Y. When viewing raw data or even charts, people can come to incorrect conclusions. 10A uses statistics to be the arbiter in deciding which X’s are important. Common statistical tests (Chi 2 , ANOVA, Regression, etc.) C&E Diagram Supplemental statistical tests

10A – Determine Critical X’s # Deliverable Deliverable Concept & Tasks Primary Tool(s) Secondary Tool(s) 10A Determine Critical X’s Using the data gathered after deliverable 8M, we need to assess which X’s cause changes in the Y. When viewing raw data or even charts, people can come to incorrect conclusions. 10A uses statistics to be the arbiter in deciding which X’s are important. Common statistical tests (Chi 2 , ANOVA, Regression, etc.) C&E Diagram Supplemental statistical tests Steps to Complete Deliverable: Gather the list of probable X’s listed in the Root Cause Investigation Matrix (deliverable 7M). For the non-data based X’s, complete the Cause and Effect Diagram to identify the root cause using the “5 why’s”. For X’s that will be assessed using data, apply the appropriate Hypothesis test to verify if it is a critical X. List Xs you determine to be critical.

Objectives – Determine Critical Xs Upon completion of this module, the student should be able to: Use and interpret a test for Normality Use and interpret a 1 Sample t Test Use and interpret a One Sign Test

Statistical Tests Continuous Y Discreet Y Discreet X 2 Sample t Test Test for Equal Variance One-Way ANOVA (Tukeys) Moods Median Paired t Test Two Way ANOVA GLM CHI Square TOA Two Proportion Continuous X Correlation Simple Linear Regression Multiple Linear Regression These tools are not taught as part of Black Belt training Vs. Target Normality 1 Sample T One Sample Sign CHI Square GOF One Proportion

Hypothesis Test Categories Continuous Y, Continuous X(s) Tests Continuous Y, Discrete X(s) Tests Discrete Y, Continuous X(s) Tests Discrete Y, Discrete X(s) Tests Continuous Y? Y N Continuous X(s)? Continuous X(s)? Y N N Y Start

Continuous Y, Discrete X(s) Test for Normality ( Shape = normal) Residuals Normal? ResidualsEqual Variance? Residuals Stable? Y Y See MBB N N N Y Done Testing vs. a Target Value(s)? Y N 1 Sample t  1 Sample Sign (m = #) Data Symmetric? 1 Sample Wilcoxon (m = #) Y N Not Normal Normal Done No of X’s? 1 > 2 2 Sample t (Assume equal variance) (  No of levels? 2 Data Paired? N Y > 3 Paired t (  See MBB 1 Way ANOVA (  General Linear Model (  Go to “B” Perform Box-Cox Transform and Reanalyze Data already Transformed? N Y

Testing for Normality One Variable, Continuous Data H o : The data is normally distributed H a : The data is not normally distributed

Review – Test for Normality Some statistical tests require that the data is normally distributed. As a review, enter the following sample data into Minitab and determine if the population the sample came from is normally distributed. 95.0, 98.1, 102.2, 88.6, 94.3, 100.6, 86.0, 96.2 This can be accomplished in two locations: Stat > Basic Statistics > Graphical Summary Stat > Basic Statistics > Normality test

Descriptive Statistics Graphical Output Is the data normal?

Normality Test Graphical Output

Alternate Normality Tests Anderson-Darling The Anderson-Darling test uses an ECDF (Empirical Cumulative Density Function) approach to testing normality. The AD test should be used to test for normality unless directed otherwise by the MBB. Ryan-Joiner The Ryan-Joiner normality test uses a regression approach of fitting the data points to a line representing a normal distribution. Kolmogorov-Smirnov The Kolmogorov-Smirnov test is also an ECDF approach.

1 Sample t Test Normally Distributed Continuous Data vs. Target H o :  a = Target H a :  a ≠ Target

1 Sample t Test A t Test is used to compare a mean against a target value. The t Test can be one tailed or two tailed. The Minitab worksheet Chlorine Residuals.mtw contains samples of chlorine residuals. Assume we have a requirement for the average chlorine residual in water to average more than 0.3 ppm. Is the system meeting the requirement? First verify that the data is normally distributed Stat > Basic Stats > Graphical Summary

Chlorine Residuals Graphical Summary Data is normal!

1 Sample t Test Choose Stat > Basic Statistics > 1 Sample t

1 Sample t Test Check ‘histogram of the data’ Make the test one-tailed by choosing ‘greater than’ The test mean is 0.03 Summarized data would go here

1 Sample t Test One-Sample T: PPM Chlorine Test of mu = 0.3 vs > 0.3 95% Lower Variable N Mean StDev SE Mean Bound T P PPM Chlorine 55 0.335889 0.045808 0.006177 0.325552 5.81 0.000 P value < 0.05, reject the null and conclude the mean is greater than 0.3 Target Value (H o ) Confidence interval for the lower bound H o :  =0.3 H a :  >0.3

Class Exercise You are in charge of three street light repair crews and are required to have a productivity of at least 10 street lights replaced per day for all crews. The Minitab file Street Light Repairs.mtw has data on the last 30 working days. Are you meeting your goal? 10 Min

One Sample Sign Test Non-Normally Distributed Continuous Data vs. a Target H o : Median = Target H a : Median ≠ Target

One Sample Sign Test A one sample sign test is used to compare the median of non-normal data to a target The test can be either one-tailed or two-tailed. The one-sample sign test is a nonparametric alternative to the 1 sample t test. This is the method Minitab uses to calculate the confidence interval in it’s “Graphical Summary” chart. Since you can determine if your test value falls in the confidence interval by looking at the graphical summary, you only need to run the one sample signed test if you wish to obtain a specific P-Value.

1-Sample Sign Test Be sure to test the data for normality before using the 1-sample signed test. Use the 1-sample sign test only if the data is not normally distributed. Running the 1-sample sign test Choose Stat > Nonparametrics > 1-Sample Sign. In Variables, enter the column(s) containing the data. Choose one of the following: To calculate a sign confidence interval for the median, choose ‘Confidence Interval’ To perform a sign test, choose ‘Test Median’

1-Sample Sign Test Management would like to ensure the emotional health of their employees by setting policies that encourage employees to take their vacation days. They have determined that employees should take at least 15 days of vacation per year. The sampled payroll records of 78 employees is located in the Minitab worksheet Vacation.mtw . Is the typical employee taking at least 15 days?

Vacation Days Example First, determine if the data is normal… Did this catch you? Don’t forget to check for normality. Technically, we should use a 1 sample t test of the mean for this data. However, let’s proceed with a 1- sample sign test for the median for illustration.

Vacation Days Example Stat > Nonparametrics> 1-Sample Sign H o : median=15 H a : median>15

Vacation Days Example Sign Test for Median: Vacation Days Sign test of median = 15.00 versus > 15.00 N Below Equal Above P Median Vacation Days 78 30 5 43 0.0801 18.00 Note the lack of power in a nonparametric statistical test. 43 of 78 data points are above 15 yet there is still insufficient evidence to prove the population is above 15. What if we had used the 1-sample t after all? The number of observations below the test median. The number of observations above the test median. The number of observations equal to the test median.

Vacation Days Revisited One-Sample T: Vacation Days Test of mu = 15 vs > 15 95% Lower Variable N Mean StDev SE Mean Bound T P Vacation Days 78 16.6154 7.6145 0.8622 15.1800 1.87 0.032 There is enough data to confirm the mean is >15!

1-Sample Sign Homework Our old friend Pat is back. This time, Pat is worried about the cycle time required to process invoices. Pat has collected data for multiple employees to process a batch of invoices over the last few weeks and stored at in “ Invoices.mtw ”. Can Pat tell management that the processing time is less than 25 minutes?

Learning Check – Determine Critical Xs Upon completion of this module, the student should be able to: Use and interpret a test for Normality Use and interpret a 1 Sample t Test Use and interpret a One Sign Test

A05 Continuous One Variable Stat Tests

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A05 Continuous One Variable Stat Tests

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