Hypothesis Testing: Central Tendency – Normal (Compare 2+ Factors)

Section & Lesson #:
Pre-Requisite Lessons:
Complex Tools + Clear Teaching = Powerful Results
Hypothesis Testing: Central Tendency –
Normal (Compare 2+ Factors)
Six Sigma-Analyze – Lesson 18
An extension on a series about hypothesis testing, this lesson reviews the
ANOVA test as a central tendency measurement for normal dist.
Six Sigma-Analyze #17 – Hypothesis Testing: Central Tendency –
Normal (Compare 1:1)
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted by any means
(electronic, mechanical, photographic, photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.

Why do we need hypothesis testing?
o Remember, our project goal is to resolve a problem by first building a transfer function.
• We don’t want to just alleviate symptoms, we want to resolve the root cause.
 Remember Hannah? We don’t want to alleviate the arthritis pain in her leg, but heal the strep throat.
• If we don’t know what the root cause is, then we need to build a transfer function.
 By building a transfer function, we can know what changes (improvements) should fix the root cause.
o Remember, the Transfer Function is defined as Y = f(X).
• This is described as “output response Y
is a function of one or more input X’s”.
• It’s part of the IPO flow model where we
described the IPO flow model as one or
more inputs feeding into a process that
transforms it to create a new output.
o How does a transfer function fit with hypothesis testing?
• Hypothesis testing tells us which X’s (inputs) are independently influencing the Y (output).
 When we reject a null hypothesis, we’re building evidence proving which X’s are “guilty” of driving the Y.
 We’ll compile all the evidence in the Improve phase of DMAIC and begin to fix those root causes.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
Y = f(X)
Input (X) > Process > Output (Y)

Review Hypothesis Testing: 4 Step Process
o Remember, the 4 high-level steps for hypothesis testing begin/end with being practical:
o As the heart of hypothesis testing, steps 2 & 3 can be drilled to the following 6 steps:
3
Practical Statistical
Problem Problem
Solution Solution
 

Practical Problem
State the problem as a
practical Yes/No question.
 Statistical Problem
Convert the problem to an
analytical question identifying
the statistical tool/method.

Practical Solution
Interpret the analytical
answer in a practical way.
Statistical Solution
Interpret the results of the
hypothesis test with an
analytical answer.

1. Define the objective.
2. State the Null Hypothesis (H0) and Alternative Hypothesis (Ha).
3. Define the confidence (1-α) and power (beta or 1-β).
4. Collect the sample data.
5. Calculate the P-value.
6. Interpret the results: accept or reject the null hypothesis (H0).

Is the data type for both values discrete?
What are you measuring?
Is the data normal?
SpreadCentral Tendency
Compare 1:Standard
1 Proportion Test
Compare 1:1
2 Proportion Test
Compare 2+ Factors
Chi2 Test
Compare 1:Standard
1 Sample T Test
Compare 1:1
2 Sample T Test
or Paired T Test
Compare 2+ Factors
One-way ANOVA Test
Compare 1:Standard
1 Sample Wilcoxon
or 1 Sample Sign
Compare 1:1
Mann-Whitney Test
Compare 2+ Factors
Mood’s Median Test
or Kruskal-Wallis Test
Compare 1:Standard
1 Variance Test
Compare 1:1
2 Variance Test
Compare 2+ Factors
Test for Equal Variances
Yes No
Yes No
Proportions
Compare 1:1
Pearson Correlation
or Fitted Line Plot
Compare 2+ Factors
Multiple Regression or
General Linear Model
Is the data type for both values continuous?
Relationships
No Yes
Review Finding the Right Statistical Test
o What statistical test do I use for my hypothesis testing?
• The type of statistical test depends on the data to be tested, as described in the chart below:
4
Lesson
Current

Confidence Intervals (CI) Redefined
o Remember, confidence intervals (CI) are an estimated upper/lower range of the mean.
• The CI narrows when you add more samples:
• Remember, statistics are intended to help you make inferences about a population.
 A mean of 57 with a 95% confidence interval of 55 and 59 implies that although the sample mean is only
57, you can be 95% confident that the population mean will fall somewhere between 55 and 59.
• The first step in hypothesis testing is to define the objective by asking “is there a difference…”
 Statistical tests generally compare the different CIs between factors to see if a difference exists.
5
With fewer samples, the population
mean falls within a wide interval.
Add more samples and the population
mean falls in a more narrow interval.
With more samples, the more confident
(narrow) the mean interval becomes.
Factor A
Factor B
2.5 2.75 3.0 3.25 3.5
Large overlap indicates the population
mean may be the same for both
groups; therefore there is no difference
between them (high P-value).
N = 20
Factor A
Factor B
2.5 2.75 3.0 3.25 3.5
Small overlap indicates a smaller chance
the population mean may be the same for
both groups; therefore there is may be no
difference between them (small P-value).
N = 50
Factor A
Factor B
2.5 2.75 3.0 3.25 3.5
No overlap indicates the population
mean is different for both groups;
therefore there is a difference between
them (low or no P-value).
N = 100

One-Way ANOVA: Introduction
o When should I use it?
• To compare mean values for multiple factors.
o How do I find it in Minitab?
• Stat > ANOVA > One-Way…
o What are the inputs for the test?
6
Now
The column containing the continuous
data to test; it should represent the Y or
CTQ output being measured.
The factor having 2 or more discrete values that each defines
the set of values for which each mean will be analyzed.
Creates a visual display of 4 different charts used to analyze
the residuals created by the ANOVA.

One-Way ANOVA: Interpreting Results
o How do I interpret the ANOVA results?
• Below is an example of the output from an ANOVA test.
• We’ll separately explore each summary and detail portion.
7
Now
Summary:
This portion summarizes
the results of the test.
Details:
This portion lists the
detailed results of the test.

One-Way ANOVA: Summary Portion
o What does the summary portion of the ANOVA results mean?
o What values do I use to interpret these results?
• If P-value < α (0.05), statistical difference exists.
• High R-sq may mean “practical” difference exists.
8
NowSource DF SS MS F P
CategoryA 4 426446 106611 2.07 0.091
Error 95 4900934 51589
Total 99 5327380
S = 227.1 R-Sq = 8.00% R-Sq(adj) = 4.13%
Between
analysis
results
Within
analysis
results
Sources of
variation
Degrees of
Freedom per
Source
(# of unique
values – 1)
Sum of Squares
(variation) per
Source
Estimate of
variation from
each source
(Calc’d as SS/DF)
P valueF ratio
(MSBetween
)MSWithin
Unexplained
Variation
(equals 1σ)
% of variation in MetricA (Response)
explained by CategoryA (Factor)
(Cald’d as SSFactor / SSTotal)

One-Way ANOVA: Detail Portion
o What does the detailed portion of the ANOVA results mean?
o How do I interpret these results?
• Examine the difference in each unique value’s mean
to see it’s significant (probably confirmed by R-sq).
9
Now
Individual 95% CIs For Mean Based on Pooled StDev
Level N Mean StDev +---------+---------+---------+---------
Blue 14 524.9 182.5 (---------*---------)
Green 14 362.5 141.3 (---------*---------)
Red 19 559.0 226.4 (--------*-------)
White 30 557.3 240.9 (-----*------)
Yellow 23 539.2 270.0 (-------*-------)
+---------+---------+---------+---------
240 360 480 600
Factor’s
unique
values
Count for each
unique value
Means per
unique value
Std Deviation
per unique value
Illustration of
Confidence Intervals
per unique value
The amount of overlap in these
CIs are what affect the p-value.

One-Way ANOVA: Residuals Introduction
o What are residuals?
• They represent all the deviations for each data point.
• Remember, a deviation is the distance a data point is from the mean.
o Why are residuals used in an ANOVA test?
• They help identify potential serious problems in the analysis.
• It helps validate the ANOVA test to ensure the results are reliable.
o What do we look for in the residuals?
• Residuals should have these characteristics:
 They should be normally distributed.
 They should be independent.
 They should have equal variances.
o How do I measure the residuals?
• In the ANOVA dialog box, select “Graphs…”.
• In the Graphs box, select “Four in One” from
the “Residual Plots” section (example at right).
10
Now
Deviation = 9 – 5 = 4
8004000-400-800
99.9
99
90
50
10
1
0.1
Residual
Percent
550500450400350
500
250
0
-250
-500
Fitted Value
Residual
6004002000-200-400
20
15
10
5
0
Residual
Frequency
1009080706050403020101
500
250
0
-250
-500
Observation Order
Residual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for MetricA

One-Way ANOVA: Residuals Interpretation
o Interpreting the residual plots for normality:
• These plotted residuals should be normally distributed.
11
Now
8004000-400-800
99.9
99
90
50
10
1
0.1
Residual
Percent
550500450400350
500
250
0
-250
-500
Fitted Value
Residual
6004002000-200-400
20
15
10
5
0
Residual
Frequency
1009080706050403020101
500
250
0
-250
-500
Observation Order
Residual
8004000-400-800
99.9
99
90
50
10
1
0.1
Residual
Percent
550500450400350
500
250
0
-250
-500
Fitted Value
Residual
6004002000-200-400
20
15
10
5
0
Residual
Frequency
1009080706050403020101
500
250
0
-250
-500
Observation Order
Residual
Though there’s no Anderson-
Darling test, these distributions
should still appear to be normal.

One-Way ANOVA: Residuals Interpretation
o Interpreting the residual plots for equal variance and independence:
• These plotted residuals should appear random yet evenly spread.
12
Now
8004000-400-800
99.9
99
90
50
10
1
0.1
Residual
Percent
550500450400350
500
250
0
-250
-500
Fitted Value
Residual
6004002000-200-400
20
15
10
5
0
Residual
Frequency
1009080706050403020101
500
250
0
-250
-500
Observation Order
Residual
800 550500450400350
500
250
0
-250
-500
Fitted Value
Residual
600 1009080706050403020101
500
250
0
-250
-500
Observation Order
Residual
Versus Fits
Versus Order
al Plots for MetricA
The spread of plots on
either side of the line
should look similar.
The spread of plots
should look random with
no patterns or trends.

One-Way ANOVA: Boxplots
o What are boxplots (a.k.a., “box and whiskers”)?
• A graphical summary of a distribution’s shape, central tendency & spread.
• They help compare multiple distributions and statistical characteristics.
• It’s like a birds-eye view (looking down from the top) of a distribution.
13
Now
YellowWhilteRedGreenBlue
1200
1000
800
600
400
200
0
CategoryA
MetricA
Boxplot of MetricA
****
Minitab draws each boxplot this way
and lays out multiple boxplots on
the same scale for easy comparison
The mean is included on each boxplot
with a line drawn between them.
Median

ANOVA Test: MetricA & CategoryA Example
o Example: MetricA and CategoryA sample values
• Background:
 Use the arbitrary values in the “MetricA” and “CategoryA” columns of the
Minitab Sample Data file.
• Practical Problem:
 Is the mean for MetricA different between the various CategoryA values?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: μCategoryA1 = μCategoryA2 = μCategoryA3 etc., and Ha: = μCategoryA1 ≠ μCategoryA2 ≠ μCategoryA3 etc.
 Define the confidence (1-α) and power (1-β):
– For confidence, we’ll accept the default of 95% (which means α = 5%) and power of 90% (which means β = 10%).
 Type the statistical problem into Minitab:
– In Minitab, go to Stat > ANOVA > One Way…
– Select MetricA for “Response” and select CategoryA for “Factor”.
– Click “Graphs…” and for “Residual Plots” select Four in one.
• Statistical Solution:
 Refer to the session window results.
– Since P-value is > 0.05 (α), then we fail to reject H0.
– R-sq(adj) suggests only 4% of variation can be explained.
• Practical Solution:
 The sample is insufficient to prove that the
means for MetricA between each type of CategoryA
value are different.
 If we assumed power=90% (β=10%), then how do
we apply that to this test? What does it mean?
14
Now

ANOVA Test: MetricA & CategoryB Example
o Example: MetricA and CategoryB sample values
• Background:
 Use the arbitrary values in the “MetricA” and “CategoryB” columns of the
Minitab Sample Data file.
• Practical Problem:
 Is the mean for MetricA different between the various CategoryB values?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: μCategoryB1 = μCategoryB2 = μCategoryB3 etc., and Ha: = μCategoryB1 ≠ μCategoryB2 ≠ μCategoryB3 etc.
 Define the confidence (1-α) and power (1-β):
– For confidence, we’ll accept the default of 95% (which means α = 5%) and power of 90% (which means β = 10%).
 Type the statistical problem into Minitab:
– In Minitab, go to Stat > ANOVA > One Way…
– Select MetricA for “Response” and select CategoryB for “Factor”.
– Click “Graphs…” and for “Residual Plots” select Four in one.
• Statistical Solution:
 Refer to the session window results.
– Since P-value is > 0.05 (α), then we fail to reject H0.
– R-sq(adj) suggests 0% of variation can be explained.
• Practical Solution:
 The sample is insufficient to prove that the
means for MetricA between each type of CategoryB
value are different.
 If we assumed power=90% (β=10%), then how do
we apply that to this test? What does it mean?
15
Now

Practical Application
o Refer to the critical metric (output Y) and at least 5 factors (input X’s) you identified in
a previous lesson for applying to this hypothesis testing.
• For any factor that is a continuous value, try applying the ANOVA Test.
 To do this, you’ll need a discrete factor that has 2 or more sets of values (e.g., across multiple periods of
time, or different locations, or different groups, etc.).
 Other factors in your organization can be used for this exercise.
• Before running the ANOVA Test, do the means for each factor group appear to be different?
• After running the ANOVA Test, are the means for each factor group statistically different?
• If the answers to the above 2 questions are different, then how does that affect how you’d
typically measure and communicate that factor in the organization?
 For example, does the difference between the means affect financial decisions (e.g., how people are
compensated), or process changes (e.g., how the process may be modified), or other critical actions?
 If so, then how should the results from this statistical test be used to influence your organization?
– Should they change how each factor group is compared (e.g., between different times, locations, groups, etc.)?
– Should they change how each factor group is measured?
– Should they change how they react when they compare the metric this way?
16

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