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C USTOMER & C OMPETITIVE I NTELLIGENCE. S. S. IX. IGMA. FOR S YSTEMS I NNOVATION & D ESIGN. D EPARTMENT OF S TATISTICS. REDGEMAN@UIDAHO.EDU OFFICE: +1-208-885-4410. D R. R ICK E DGEMAN, P ROFESSOR & C HAIR – S IX S IGMA B LACK B ELT. S. S. IX.

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slide1

CUSTOMER&COMPETITIVEINTELLIGENCE

S

S

IX

IGMA

FOR SYSTEMSINNOVATION&DESIGN

DEPARTMENT OFSTATISTICS

REDGEMAN@UIDAHO.EDU OFFICE: +1-208-885-4410

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

DR. RICK EDGEMAN, PROFESSOR& CHAIR– SIX SIGMA BLACK BELT

slide2

S

S

IX

IGMA

HypothesisTesting & Confidence Intervals

DEPARTMENT OFSTATISTICS

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide3

S

S

IX

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 OFSTATISTICS

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide4

Conjectures 

(Hypotheses)

… or …

B

A

Consequences

Meaning

& Action(s)

Information & Risk

Requirements

Evaluation

(Test Method)

ZoneofBelief

Decision

Criteria

Informed

Decision

Gather & Evaluate

Facts

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

The Hypothesis Testing Approach

slide5

The Scientific Method

Noninformative Event

Informative Event

Little or

Nothing Learned

No Observer

or Uninformed

Observer

Nothing Learned

Scientific Method

of Investigation

Informed

Observer

Little or

Nothing Learned

Discovery!

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide6

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

slide7

The American Trial SystemIn Truth, the Defendant is: H0: Innocent HA: Guilty

CorrectDecisionIncorrectDecision

Innocent Individual Guilty Individual

Goes Free Goes Free

IncorrectDecisionCorrectDecision

Innocent Individual Guilty Individual

Is Disciplined Is Disciplined

Innocent

Guilty

Verdict

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide8

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

slide9

True, But Unknown State of the WorldH0 is True HA is True

Correct Decision Incorrect Decision

Type II Error Probability =

Incorrect Decision Correct Decision

Type I Error Probability = 

Ho is True

Decision

HA is True

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide10

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

slide11

H0:  = <>0 vs. HA: ≠ > < 0

  • n = _______  = _______
    • 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, FTR H0.

Z-test & C.I. for µ

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide12

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

slide13

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

slide14

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

slide15

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

slide16

Z-test for a proportion

^

  • Zcalc = (p - p0)/(  p0(1-p0)/n )
  • _____ Reject H0 in Favor of HA. _______ FTR H0.
  • The Confidence Interval for p is Given by:

p + Z/2(  p(1-p)/n )

  • Interpretation

^

^

^

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide17

Advance, Inc.

Integrated Circuit Manufacturing

Methods & Materials

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide18

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

slide19

Training Effect Example

  • H0: < 100 HA:  > 100
  • n = 50  = .05
  • (i) testing a mean (ii) normal distribution (iii)  = 10 is known so that this is a Z-test
  • DR: Reject H0in 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?”

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide20

Loan Application Processing

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide21

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

slide22

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 H0in 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

slide23

Small Business

Loan Defaults

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide24

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 a = .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

slide25

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 H0in favor of HA iff Zcalc< 1.282. Otherwise, FTR H0
  • Zcalc = (p - p0)/( p0(1-p0)/n ) = (.06 - .12)/ (.12(.88)/150 ) = -.06/.026533 = -2.261
  • X Reject H0 in favor of HA. _______ FTR H0
  • The 95% Confidence Interval is Given by: p + Z/2 ( p(1-p)/ n ) which is .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

slide26

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 H0in favor of HAiff2calc>2large,Otherwise, FTR H0.
  • DR: (< in HA) Reject H0in favor of HAiff2calc<2small,Otherwise, FTR H0.

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide27

2Test & 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

slide28

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

slide29

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.
  • 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.

FFFExample

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide30

Two Sample Tests

and

Confidence Intervals

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide31

H0: μ1 – μ2 = ≥ ≤ μd

HA: μ1 – μ2 < > μd

n1 = _____ n2 = _____ α = 0

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.

Tests and Intervals for Two Means

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide32

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

slide33

Z = [(X1 – X2) – μd]

  • σ√(1/n1 + 1/n2)
  • Z = [(X1 – X2) – μd]
  • √(σ21/n1 + σ22/n2)
  • (b) t = [(X1 – X2) – μd] (assume equal variances)
  • Sp√(1/n1 + 1/n2)where df = n1+n2 – 2
  • and Sp2 = (n1-1)S12 + (n2-1)S22
  • (c ) t = [(X1 – X2) – μd] (do not assume equal variances)
  • √(S12/n1 + S22/n2)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

slide34

Equality of Variances: The F-Test

H0: s1= ≥ ≤s2 vs. HA: s1 < > s2

n1 = _____ n2 = _____ a = _____

Test of equality of variances  F-test

___ > in HA: reject H0 in favor of HA iff Fcalc > Fa,big.Otherwise, FTR H0.

___ < in HA: reject H0 in favor of HA iff Fcalc < Fa,small. Otherwise, FTRH0.

___ in HA: reject H0 in favor of HA iff Fcalc < Fa/2,small or if Fcalc > Fa/2,big. Otherwise, FTR H0.

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide35

Fcalc = S12/S22

Make a decision.

Fcalc/ Fn1-1,n2-1,a/2 large≤ s12/s22 ≤ Fcalc/Fn1-1,n2-1,a/2 small

Conclusions / Implications

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

slide36

Tests & Intervals for

Two Proportions

H0: p1 – p2 = ≥ ≤ pd

HA: p1 – p2 < > pd

n1 = _____ n2 = _____ 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

slide37

Z = [(p1 – p2)] IF pd = 0

√ p(1-p)(1/n1 + 1/n2) where p = (X1+X2)/(n1 + n2)

^ ^

Z = [(p1 – p2) – pd] 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

slide38

S

S

IX

IGMA

Endof Session

DEPARTMENT OFSTATISTICS

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho