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Determine Critical X’s Statistical Tests for a Continuous
Single VariableDeliverable 10A
Define Module Roadmap
Define1D – Define VOC, VOB, and CTQ’s2D – Define Project Boundaries3D – Quantify Project Value4D – Develop Project Mgmt. Plan
Measure5M – Document Process6M – Prioritize List of X’s7M – Create Data Collection Plan8M – Validate Measurement System9M – Establish Baseline Process Cap.
Analyze 10A – Determine Critical X’s
Improve12I – Prioritized List of Solutions13I – Pilot Best Solution
Control14C – Create Control System15C – Finalize Project Documentation
Green11G – Identify Root Cause Relationships
Queue 1
Queue 2
Deliverables – Analyze
# Deliverable Deliverable Concept & TasksPrimary 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 (Chi2, ANOVA, Regression, etc.)
• C&E Diagram
• Supplemental statistical tests
10A – Determine Critical X’s
# Deliverable Deliverable Concept & TasksPrimary 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 (Chi2, ANOVA, Regression, etc.)
• C&E Diagram
• Supplemental statistical tests
Steps to Complete Deliverable:1. 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 TestTest for Equal Variance
One-Way ANOVA (Tukeys)Moods MedianPaired t Test
Two Way ANOVAGLM
CHI Square TOATwo Proportion
Continuous XCorrelation
Simple Linear RegressionMultiple Linear Regression
These tools are not taught as part of Black Belt training
Vs. TargetNormality 1 Sample T
One Sample Sign
CHI Square GOFOne 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
NYDone
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?
NY
Testing for Normality
One Variable, Continuous DataHo: The data is normally distributed
Ha: 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.o 95.0, 98.1, 102.2, 88.6, 94.3, 100.6, 86.0, 96.2
• This can be accomplished in two locations:o Stat > Basic Statistics > Graphical Summaryo Stat > Basic Statistics > Normality test
Descriptive StatisticsGraphical Output
Is the data normal?
Normality Test Graphical Output
Alternate Normality Tests
• Anderson-Darlingo The Anderson-Darling test uses an ECDF (Empirical Cumulative
Density Function) approach to testing normality.o The AD test should be used to test for normality unless directed
otherwise by the MBB.• Ryan-Joiner
o The Ryan-Joiner normality test uses a regression approach of fitting the data points to a line representing a normal distribution.
• Kolmogorov-Smirnovo The Kolmogorov-Smirnov test is also an ECDF approach.
1 Sample t Test
Normally Distributed Continuous Data vs. TargetHo: a = TargetHa: 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 distributedo 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
One-Sample T: PPM Chlorine
Test of mu = 0.3 vs > 0.395%LowerVariable N Mean StDev SE Mean Bound T PPPM Chlorine 55 0.335889 0.045808 0.006177 0.325552 5.81 0.000
1 Sample t Test
P value < 0.05, reject the null and conclude the mean is
greater than 0.3
Target Value (Ho) Confidence interval for the lower bound
Ho: =0.3Ha: >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 TargetHo: Median = TargetHa: 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.o 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 testo Choose Stat > Nonparametrics > 1-Sample Sign.o In Variables, enter the column(s) containing the data.o 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
Ho: median=15Ha: 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 MedianVacation 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%LowerVariable N Mean StDev SE Mean Bound T PVacation 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
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