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