CUSTOMER &S
Cix
OMPETITIVE
SigmaINTELLIGENCE
S S
IX
FOR
IGMA
SYSTEMS INNOVATION & DESIGN
DEPARTMENT
OF STATISTICS
REDGEMAN@UIDAHO.EDU
OFFICE: +1-208-885-4410
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
DR. RICK EDGEMAN,
PROFESSOR & CHAIR – SIX SIGMA BLACK BELT
Dr. Rick L. Edgeman, University of Idaho
S S
IX
Six Sigma
IGMA
Hypothesis Testing & Confidence Intervals
DEPARTMENT
OF STATISTICS
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
S S
IX
Six Sigma
IGMA
a highly structured strategy for acquiring, assessing, and
applying customer, competitor, and enterprise
intelligence for the purposes of product, system or
enterprise innovation and design.
DEPARTMENT
OF STATISTICS
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
… or …
B
Conjectures 
(Hypotheses)
Evaluation
(Test Method)
Zone of Belief
Consequences
A
Gather & Evaluate
Facts
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
The Hypothesis Testing Approach
Six Sigma
The Scientific Method
Noninformative Event
No Observer
or Uninformed
Observer
Informed
Observer
Nothing Learned
Little or
Nothing Learned
Informative Event
Little or
Nothing Learned
Scientific Method
Discovery!
of Investigation
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Motivation for Hypothesis Testing
• The intent of hypothesis testing is formally examine two
opposing conjectures (hypotheses), H0 and HA.
• These two hypotheses are mutually exclusive and exhaustive so
that one is true to the exclusion of the other.
• We accumulate evidence - collect and analyze sample
information - for the purpose of determining which of the two
hypotheses is true and which of the two hypotheses is false.
• Beyond the issue of truth, addressed statistically, is the issue of
justice. Justice is beyond the scope of statistical investigation.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
The American Trial System
In Truth, the Defendant is:
H0: Innocent
HA: Guilty
Verdict
Innocent
Guilty
Correct Decision
Incorrect Decision
Innocent Individual
Goes Free
Guilty Individual
Goes Free
Incorrect Decision
Correct Decision
Innocent Individual
Is Disciplined
Guilty Individual
Is Disciplined
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Hypothesis Testing & The American Justice System
• State the Opposing Conjectures, H0 and HA.
• Determine the amount of evidence required, n, and the risk
of committing a “type I error”, 
• What sort of evaluation of the evidence is required and
what is the justification for this? (type of test)
• What are the conditions which proclaim guilt and those
which proclaim innocence? (Decision Rule)
• Gather & evaluate the evidence.
• What is the verdict? (H0 or HA?)
• Determine a “Zone of Belief” - Confidence Interval.
• What is appropriate justice? --- Conclusions
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
True, But Unknown State of the World
H0 is True
Ho is True
Correct Decision
HA is True
Incorrect Decision
Type II Error Probability =
Decision
HA is True
Incorrect Decision
Correct Decision
Type I Error Probability = 
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Hypothesis Testing Algorithm
•
•
•
•
•
Specify H0 and HA
Specify n and 
What Type of Test and Why?
Critical Value(s) and Decision Rule (DR)
Collect Pertinent Data and Determine the Calculated Value of the
Test Statistic (e.g. Zcalc, tcalc, 2calc, etc)
• Make a Decision to Either Reject H0 in Favor of HA or to Fail to
Reject (FTR) H0.
• Construct & Interpret the Appropriate Confidence Interval
• Conclusions? Implications & Actions
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
• H0:  = < > 0 vs. HA:  ≠ > < 0
• n = _______
 = _______
–
–
–
–
Z-test & C.I. for µ
Testing a Hypothesis About a Mean;
Process Performance Measure is Approximately Normally Distributed;
We “Know” 
Therefore this is a “Z-test” - Use the Normal Distribution.
• DR: (≠ in HA) Reject H0 in favor of HA if Zcalc < -Z/2 or if Zcalc >
+Z/2. Otherwise, FTR H0.
• DR: (> in HA) Reject H0 in favor of HA iff Zcalc > +Z . Otherwise,
FTR H0.
• DR: (< in HA) Reject H0 in favor of HA iff Zcalc < -Z. Otherwise,
Client,
& Competitive Intelligence for Product, Process & Systems Innovation
FTR
HEnterprise
0.
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Z-test Algorithm (Continued)
• Zcalc = (X - 0)/(/ /n)
• _____ Reject H0 in Favor of HA. _______ FTR H0.
• The Confidence Interval for  is Given by:
X + Z/2(/ n )
• Interpretation
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
t-test and Confidence Interval for 


H0:  = < > 0 vs. HA:   > < 0
n = _______
 = _______
• Testing a Hypothesis About a Mean;
• Process Performance Measure is Approximately Normally Distributed or We
Have a “Large” Sample;
• We Do Not Know Which Must be Estimated by S.
• Therefore this is a “t-test” - Use Student’s T Distribution.



DR: ( in HA) Reject H0 in favor of HA if tcalc < -t/2 or if tcalc > +t/2.
Otherwise, FTR H0.
DR: (> in HA) Reject H0 in favor of HA iff tcalc > +t . Otherwise, FTR H0.
DR: (< in HA) Reject H0 in favor of HA iff tcalc < -t Otherwise, FTR H0.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
t-test Algorithm (Continued)
• tcalc = (X - 0)/(s/ /n )
• _____ Reject H0 in Favor of HA. _______ FTR H0.
• The Confidence Interval for  is Given by:
• X + t/2(s/ n )
• Interpretation
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Z-test & C.I. for p


H0: p = < > p0 vs. HA: p  > < p0
n = _______
 = _______
•
•
•



Testing a Hypothesis About a Proportion;
We have a “large” samplethat is, both np0 and n(1-p0) > 5
Therefore this is a “Z-test” - Use the Normal Distribution.
DR: ( in HA) Reject H0 in favor of HA if Zcalc < -Z/2 or if Zcalc > +Z/2.
Otherwise, FTR H0.
DR: (> in HA) Reject H0 in favor of HA iff Zcalc > +Z . Otherwise, FTR H0.
DR: (< in HA) Reject H0 in favor of HA iff Zcalc < -Z. Otherwise, FTR H0.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Z-test for a proportion




^ - p )/(  p (1-p )/n )
Zcalc = (p
0
0
0
_____ Reject H0 in Favor of HA. _______ FTR H0.
The Confidence Interval for p is Given by:
^ + Z/2(  ^p(1-p)/n
^
p
)
Interpretation
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Advance, Inc.
Integrated Circuit
Manufacturing
Methods & Materials
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Z-Test & Confidence Interval: “Training Effect Example”
• Interested in increasing productivity rating in the integrated circuit
division, Advance Inc. determined that a methods review course would be
of value to employees in the IC division.
• To determine the impact of this measure they reviewed historical
productivity records for the division and determined that the average level
was 100 with a standard deviation of 10.
• Fifty IC division employees participated in the course and the post-course
productivity of these employees was measured, on average, to be 105.
• Assume that productivity ratings are approximately distributed. Did the
course have a beneficial effect. Test the appropriate hypothesis at the  =
.05 level of significance.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
• H0:  < 100
HA:  > 100
• n = 50
 = .05
• (i) testing a mean (ii) normal distribution (iii)  = 10 is known so that this is a Ztest
• DR: Reject H0 in favor of HA iff Zcalc > 1.645. Otherwise, FTR H0
• Zcalc = (X - 0)/( / n) = (105 - 100)/ (10/ 50 ) = 5/1.414 = 3.536
•
X Reject H0 in favor of HA. _______ FTR H0
• The 95% Confidence Interval is Given by: X + Z/2 (/  n) which is 105 +
1.96(1.414) = 105 + 2.77 or 102.23 <  < 107.77
• Thus the course appears to have helped improve IC division employee productivity
from an average level of 100 to a level that is at least 102.23 and at most 107.77.
• A follow-up question: “is this increase worth the investment?”
Training Effect Example
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Loan Application Processing
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
First People’s Bank of Central City
• First People’s Bank of Central City would like to improve their loan
application process. In particular currently the amount of time
required to process loan applications is approximately normally
distributed with a mean of 18 days.
• Measures intended to simplify and speed the process have been
identified and implemented. Were they effective? Test the
appropriate hypothesis at the  = .05 level of significance if a sample
of 25 applications submitted after the measures were implemented
gave an average processing time of 15.2 days and a standard deviation
of 2.0 days.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma








First People’s Bank
of Central City
H0:  > 18
HA:  < 18
n = 25
 = .05
(i) testing a mean (ii) normal distribution (iii)  is unknown and must
be estimated so that this is a t-test
DR: Reject H0 in favor of HA iff tcalc < -1.711. Otherwise, FTR H0
tcalc = (X - 0)/(s / √n) = (15.2 - 18)/ (2/ √ 25 ) = -2.8/.4 = -7.00
X Reject H0 in favor of HA. _______ FTR H0
The 95% Confidence Interval is Given by: X + t/2 (s/√n) which is
15.2 + 2.064(.4) = 15.2 + .83 or 14.37 <  < 16.03
Thus the course appears to have helped decrease the average time
required to process a loan application from 18 days to a level that is at
least 14.37 days and at most 16.03 days.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Small Business
Loan Defaults
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
First People’s Bank of Central City
Small Business Loan Defaults
• Historically, 12% of Small Business Loans granted result in default. Three
years ago, FPB of Central City purchased software which they hope will
assist in reducing the default rate by more effectively discriminating between
small business loan applicants who are likely to default and those who are
not likely to do so.
• After adequately training their loan officers in use of software, FPB sampled
150 small business loan applications processed using the software and
found 9 to be in default at the end of two years.
• Using  = .10, does it appear that the software is of value?
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma








Small Business
Loan Default Rate
H0: p > .12
HA: p < .12
n = 150
 = .10
(i) testing a proportion (ii) np0 = 150(.12) = 18 and n(1-p0 ) = 132
DR: Reject H0 in favor of HA iff Zcalc < 1.282. Otherwise, FTR H0
^ - p0)/( p0(1-p0)/n ) = (.06 - .12)/ (.12(.88)/150 ) =
Zcalc = (p
-.06/.026533 = -2.261
X Reject H0 in favor of HA. _______ FTR H0
^ n ) which is
The 95% Confidence Interval is Given by: ^
p + Z/2 ( ^
p(1-p)/
.06 + 1.645( .06(.94)/150 ) = .06 + 1.645(.0194) or .06 + .032 or
.028 < p < .092
Thus the course appears to have helped decrease the small business loan
default rate from a level of 12% to a level that is between 2.8% and 9.2%
with a best estimate of 6%.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
2-test & C.I. for 
 H0:  = < > 0 vs. HA:  = > < 0
 n = _______
 = _______
 Testing a Hypothesis About a Standard Deviation (or Variance);
 The Measured Trait (e.g. the PPM) is Approximately Normal;
 Therefore this is a “2-test” - Use the Chi-Square Distribution.
 DR: (in HA) Reject H0 in favor of HA if 2calc < 2small,/2 or if 2calc >
2large,/2. Otherwise, FTR H0.
 DR: (> in HA) Reject H0 in favor of HA iff 2calc > 2large, Otherwise, FTR H0.
 DR: (< in HA) Reject H0 in favor of HA iff 2calc < 2small, Otherwise, FTR H0.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
2

Test & C.I. (continued)
 2calc = (n-1)s2/(20 )
 _____ Reject H0 in Favor of HA. _______ FTR H0.
 The Confidence Intervals for and are Given by:
 (n-1)s2/2large,/2 < 2 < (n-1)s2/2small,/2
and
 (n-1)s2/2large,/2 <  <
(n-1)s2/2small,/2
 Interpretation
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Fast Facts Financial, Inc.
Fast Facts Financial (FFF), Inc. provides credit reports to lending institutions that
evaluate applicants for home mortgages, vehicle, home equity, and other loans.
A pressure faced by FFF Inc. is that several competing credit reporting companies
provide reports in about the same average amount of time, but are able to promise a
lower time than FFF Inc - the reason being that the variation in time required to
compile and summarize credit data is smaller than the time required by FFF.
FFF has identified & implemented procedures which they believe will reduce this
variation. If the historic standard deviation is 2.3 days, and the standard deviation for
a sample of 25 credit reports under the new procedures is 1.8 days, then test the
appropriate hypothesis at the  = .05 level of significance. Assume that the time
factor is approximately normally distributed.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
• H0:  = < > 0 vs. HA:  > < 0 where 0 = 2.3
• n = 25  = .05 .
•
•
•
Testing a Hypothesis About a Standard Deviation (or Variance);
The Measured Trait (e.g. the PPM) is Approximately Normal;
Therefore this is a “2-test” - Use the Chi-Square Distribution.
FFF
Example
• DR: (< in HA) Reject H0 in favor of HA iff 2calc < 2small, = 13.8484.
Otherwise, FTR H0.
 2calc = (n-1)s2/20 = (24)( 1.82 )/ (2.32) = 77.76/5.29 = 14.70
•
Reject H0 in favor of HA. X FTR H0.
• 77.76/39.3641 < 2 < 77.76/12.4011 or 1.975 < 2 < 6.27 so that
1.405 days <  < 2.50 days
• Evidence is inconclusive. Work should continue on this.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Two Sample Tests
and
Confidence Intervals
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
H0: μ1 – μ2 = ≥ ≤ μd
HA: μ1 – μ2  < > μd
n1 = _____
n2 = _____
α=0
Tests and Intervals
for Two Means
Comparison of Means from Two Processes
Normality Can Be Reasonably Assumed
Are the two variances known or unknown?
(a) Known  Z-test
(b) Unknown but Similar in Value  t-test with n1+n2 – 2 df
(c) Unknown and Unequal  t-test with “complicated df”
Critical Values and Decision Rules are the same as for any Z-test or t-test.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
C.I. for μ1 – μ2
X1 – X2  ZσX1-X2
or
X1 – X2  tSX1-X2
Decisions – Same as any other Z or T test.
Implications – Context Specific
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
(a) Z = [(X1 – X2) – μd]
σ√(1/n1 + 1/n2)
Z = [(X1 – X2) – μd]
√(σ21/n1 + σ22/n2)
(b) t = [(X1 – X2) – μd]
Sp√(1/n1 + 1/n2)
(c ) t = [(X1 – X2) – μd]
√(S12/n1 + S22/n2)
(assume equal variances)
where df = n1+n2 – 2
and Sp2 = (n1-1)S12 + (n2-1)S22
(do not assume equal variances)
where df = [(s12 /n1) + (s22/n2)]
2
(s12 /n1)2 + (s22/n2)2
n1 – 1
n2 – 1
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Equality of Variances: The F-Test
H 0:  1 = ≥ ≤  2
n1 = _____
vs.
HA: 1  < > 2
n2 = _____
 = _____
Test of equality of variances  F-test
___ > in HA: reject H0 in favor of HA iff Fcalc > F,big. Otherwise, FTR H0.
___ < in HA: reject H0 in favor of HA iff Fcalc < F,small. Otherwise, FTR H0.
___  in HA: reject H0 in favor of HA iff Fcalc < F/2,small or if Fcalc > F/,big.
Otherwise, FTR H0.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Fcalc = S12/S22
Make a decision.
Fcalc/ Fn1-1,n2-1,/2 large ≤ 12/22 ≤ Fcalc/Fn1-1,n2-1,/2 small
Conclusions / Implications
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
H0: p1 – p2 = ≥ ≤ pd
HA: p1 – p2  < > pd
n1 = _____
n2 = _____
Tests & Intervals for
Two Proportions
α=0
Comparison of Proportions from Two Processes
n1p1, n2p2, n1(1-p1) and n2(1-p2) all ≥ 5
 Z-test
Critical Values and Decision Rules are the same as for
any Z-test.
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Z=
[(p1 – p2)]
IF pd = 0
√ p(1-p)(1/n1 + 1/n2)
Z=
^
^
[(p1 – p2) – pd]
where p = (X1+X2)/(n1 + n2)
IF pd  0
^
^
^
^
√ (p1(1--p1)/n1 + p2(1-p2)/n2
^
^
C.I. for p1-p2 is (p1 – p2)  Z/2
^
^
^
^
√ (p1(1--p1)/n1 + p2(1-p2)/n2
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
S S
IX
Six Sigma
IGMA
End of Session
DEPARTMENT
OF STATISTICS
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho