An extension on a series about hypothesis testing, this lesson reviews the 1 Proportion Test as a measurement of proportions.

1. Section & Lesson #:
Pre-Requisite Lessons:
Complex Tools + Clear Teaching = Powerful Results
Hypothesis Testing: Proportions
(Compare 1:Standard)
Six Sigma-Analyze – Lesson 13
An extension on a series about hypothesis testing, this lesson reviews the 1
Proportion Test as a measurement of proportions.
Six Sigma-Analyze #12 – Hypothesis Testing – Finding the Right
Statistical Test
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.

2. 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)

3. 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:
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.
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).

4. 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:
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.
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Lesson
Current

5. Proportions Redefined
o What are proportions?
• They are a measurement of discrete values.
• They are generally measured as percentages.
 Though percentages are numeric, they are not considered continuous values.
– Although they can still be used in non-proportional statistical tests that require continuous values, it’s important to
understand that doing so may put your analysis at risk. Be sure any potential findings from it are fully validated.
o Defects and Defectives are proportional measurements.
• Below are some examples of how proportions can be measured:
o Why are statistical tests needed for proportions?
• Simple proportions using percentages may be insufficient.
• Statistical tests on proportions help assess the level of confidence we can have in them.
 For example, a 50% proportion of 100,000 records can have very different implications from a sample that
only has 100 records.
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.
5
5 Units:
Defect?
Defectives:
Yes
2
Yes
3
Yes
1
No
0
Yes
3
DPMO = 800K (80%)
DPU = 1.8 or
20% Defective

6. 1 Proportion Test: Introduction
o When should I use it?
• To compare one proportion value with a standard or target value.
o How do I find it in Minitab?
• Stat > Basic Statistics > 1 Proportion…
o What are the inputs for the test?
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.
6
Now
A portion that is being measured (e.g., a defect).
The total # of samples from which the events were measured.
The standard or target the events are being compared to.
Defines how the test will compare the events to the target.

7. 1 Proportion Test: Call Center Example
o Example: Repeat Calls in a Call Center Meeting a Target
• Background:
 A call center in Phoenix wants to know if they’re meeting their goal where 60%
of their calls do not have a repeat call within 2 days. They sampled 130 calls and
found 82 did not have repeat calls. Though that’s 63%, does it meet their goal?
• Practical Problem:
 Do the 82 of 130 sampled calls prove the Phoenix call center is meeting their goal?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: the proportion <= 60% (or 0.60) and Ha: the proportion > 60% (or 0.60)
– What does this mean? If the goal is met, then we expect to reject the H0 and accept the Ha proving they met the goal.
 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 > Basic Statistics > 1 Proportion…
– Select “Summarized data”, and type 82 for “Number of events” and 130 for “Number of trials”.
– Check the box to “Perform Hypothesis Test” and type in the goal of 0.60
– Click “Options…” and ensure the “Confidence Level” is 95.0 and “Alternative” is “greater than”.
• Statistical Solution:
 Refer to the session window results. What does this mean?
– Since P-value is > 0.05 (α), then we fail to reject H0.
– “95% Lower Bound” is a confidence internal; what does it mean?
• Practical Solution:
 The sample is insufficient to prove that the Phoenix call
center is meeting its goal of 60%.
 Why did this happen? How can you change the results?
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.
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Now

8. 1 Proportion Test: Billing Example
o Example: Billing Quality Meeting a Target
• Background:
 The Billing department has a goal of 98% accuracy for customer bills which each
have multiple opportunities for error. A sample of 1000 bills found only 17 errors.
• Practical Problem:
 Do the 17 of 1000 sampled bills prove the Billing dept. is meeting their goal?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: the proportion <= 98% and Ha: the proportion > 98%
– What does this mean? If the goal is met, then we expect to reject the H0 and accept the Ha proving they met the goal.
 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 > Basic Statistics > 1 Proportion…
– Select “Summarized data”, and type 983 for “Number of events” and 1000 for “Number of trials”. Why 983 and not 17?
– Check the box to “Perform Hypothesis Test” and type in the goal of 0.98
– Click “Options…” and ensure the “Confidence Level” is 95.0 and “Alternative” is “greater than”.
• Statistical Solution:
 Refer to the session window results. What does this mean?
– Since P-value is > 0.05 (α), then we fail to reject H0.
– “95% Lower Bound” is a confidence internal; what does it mean?
– What happens if you use 2% goal and 17 errors in the test?
• Practical Solution:
 The sample is insufficient to prove that the Billing dept
is meeting its goal of 98%.
 Why did this happen? How can you change the results?
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.
8
Now

9. 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 percentage value, try applying the 1 Proportion Test.
 To do this, you’ll need to compare that factor with a goal for that factor typically set by the organization.
 Other factors in your organization can be used for this exercise.
• Before running the 1 Proportion Test, does the factor meet or exceed the goal?
• After running the 1 Proportion Test, does the factor statistically meet or exceed the goal?
• 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 that factor meeting or not meeting the goal 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 1 Proportion Test be used to influence your organization?
– Should they change how the goals are set?
– Should they change how the factor is measured?
– Should they change how they react when they compare the metric to the goal?
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.
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