HYPOTHESIS TESTING

1
HYPOTHESIS TESTING
CONFIDENCE INTERVALS
DEPARTMENT OF STATISTICS
REDGEMAN_at_UIDAHO.EDU
OFFICE 1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR CHAIR SIX SIGMA
BLACK BELT
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Conjectures ? (Hypotheses)
B
or
A
Consequences
Meaning Action(s)
Information Risk Requirements
Evaluation (Test Method)
Zone of Belief
Decision Criteria
Informed Decision
Gather Evaluate Facts
The Hypothesis Testing Approach
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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!
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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.

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The American Trial System
In Truth, the Defendant is
H0 Innocent HA
Guilty
Correct Decision Incorrect
Decision Innocent Individual
Guilty Individual Goes Free
Goes Free
Incorrect Decision Correct Decision
Innocent Individual Guilty
Individual Is Disciplined
Is Disciplined
Innocent Guilty
Verdict
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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 Zone of Belief Confidence Interval.
• What is appropriate justice? --- Conclusions

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True, But Unknown State of the World
H0 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
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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

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Z-test C.I. for µ

• H0 ? lt gt ?0 vs. HA ? ? gt lt ?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
  lt -Z?/2 or if Zcalc gt Z?/2. Otherwise, FTR H0.
• DR (gt in HA) Reject H0 in favor of HA iff Zcalc
  gt Z? . Otherwise, FTR H0.
• DR (lt in HA) Reject H0 in favor of HA iff Zcalc
  lt -Z?. Otherwise, FTR H0.

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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

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t-test and Confidence Interval for ?

• H0 ? lt gt ?0 vs. HA ? ? gt lt ?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 Students T
  Distribution.
• DR (? in HA) Reject H0 in favor of HA if tcalc
  lt -t?/2 or if tcalc gt t?/2. Otherwise, FTR H0.
• DR (gt in HA) Reject H0 in favor of HA iff tcalc
  gt t? . Otherwise, FTR H0.
• DR (lt in HA) Reject H0 in favor of HA iff tcalc
  lt -t? Otherwise, FTR H0.

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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

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Z-test C.I. for p

• H0 p lt gt p0 vs. HA p ? gt lt p0
• n _______ ? _______
• Testing a Hypothesis About a Proportion
• We have a large sample???that is, both np0 and
  n(1-p0) gt 5
• Therefore this is a Z-test - Use the Normal
  Distribution.
• DR (? in HA) Reject H0 in favor of HA if Zcalc
  lt -Z?/2 or if Zcalc gt Z?/2. Otherwise, FTR H0.
• DR (gt in HA) Reject H0 in favor of HA iff Zcalc
  gt Z? . Otherwise, FTR H0.
• DR (lt in HA) Reject H0 in favor of HA iff Zcalc
  lt -Z?. Otherwise, FTR H0.

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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

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Advance, Inc.
Integrated Circuit Manufacturing Methods
Materials
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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.

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Training Effect Example

• H0 ? lt 100 HA ? gt 100
• n 50 ? .05
• (i) testing a mean (ii) normal distribution
  (iii) ? 10 is known so that this is a Z-test
• DR Reject H0 in favor of HA iff Zcalc gt
  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 lt ? lt 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?

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Loan Application Processing
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First Peoples Bank of Central City

• First Peoples 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.

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First Peoples Bank of Central City

• H0 ? gt 18 HA ? lt 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 lt
  -1.711. Otherwise, FTR H0
• tcalc (X - ?0)/(s / vn) (15.2 - 18)/ (2/
  v 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/vn) which is 15.2 2.064(.4)
  15.2 .83 or 14.37 lt ? lt 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.

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Small Business Loan Defaults
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First Peoples 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?

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Small Business Loan Default Rate

• H0 p gt .12 HA p lt .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 lt
  -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 lt p lt .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.

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?2-test C.I. for ?

• H0 ? lt gt ?0 vs. HA ? ? gt lt ?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 lt
  ?2small,?/2 or if ?2calc gt ?2large,?/2.
  Otherwise, FTR H0.
• DR (gt in HA) Reject H0 in favor of HA iff ?2calc
  gt ?2large,? Otherwise, FTR H0.
• DR (lt in HA) Reject H0 in favor of HA iff ?2calc
  lt ?2small,? Otherwise, FTR H0.

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?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 lt ?2 lt (n-1)s2/?2small,?/2
  
• and
  
  
• (n-1)s2/?2large,?/2 lt ? lt
  (n-1)s2/?2small,?/2
• Interpretation

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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.
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FFF Example

• H0 ? lt gt ?0 vs. HA? ? gt lt ?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 (lt in HA) Reject H0 in favor of HA iff ?2calc
  lt ?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 lt ?2 lt 77.76/12.4011 or 1.975 lt
  ?2 lt 6.27 so that 1.405 days lt ? lt
  2.50 days
• Evidence is inconclusive. Work should continue
  on this.

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Two Sample Tests and Confidence Intervals
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Tests and Intervals for Two Means
H0 µ1 µ2 µd HA µ1 µ2 ? lt gt µd n1
_____ n2 _____ a 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 n1n2 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.
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C.I. for µ1 µ2 X1 X2 ? ZsX1-X2
or X1 X2 ? tSX1-X2 Decisions
Same as any other Z or T test. Implications
Context Specific
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• Z (X1 X2) µd
• sv(1/n1 1/n2)
• Z (X1 X2) µd
• v(s21/n1 s22/n2)
• (b) t (X1 X2) µd (assume equal
  variances)
• Spv(1/n1 1/n2) where df
  n1n2 2
• 
  and Sp2 (n1-1)S12 (n2-1)S22
• (c ) t (X1 X2) µd (do not assume
  equal variances)
• v(S12/n1 S22/n2) where df
  (s12 /n1) (s22/n2)2
• (s12 /n1)2 (s22/n2)2

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• Equality of Variances The F-Test
• H0 s1 s2 vs. HA s1 ? lt gt s2
• n1 _____ n2 _____ a _____
• Test of equality of variances ? F-test
• ___ gt in HA reject H0 in favor of HA iff Fcalc gt
  Fa,big. Otherwise, FTR H0.
• ___ lt in HA reject H0 in favor of HA iff Fcalc lt
  Fa,small. Otherwise, FTR H0.
• ___ ? in HA reject H0 in favor of HA iff Fcalc lt
  Fa/2,small or if Fcalc gt Fa/2,big. Otherwise, FTR
  H0.

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Fcalc S12/S22 Make a decision. Fcalc/
Fn1-1,n2-1,a/2 large s12/s22
Fcalc/Fn1-1,n2-1,a/2 small C.I. for s1/s2 is
obtained by taking square roots of the endpoints
of the above C.I. for s12/s22 Conclusions /
Implications Context Specific.
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Tests Intervals for Two Proportions
H0 p1 p2 pd HA p1 p2 ? lt gt pd n1
_____ n2 _____ a 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.
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Z (p1 p2)
IF pd 0 v p(1-p)(1/n1 1/n2)
where p (X1X2)/(n1 n2)
Z
(p1 p2) pd IF pd ? 0

v (p1(1--p1)/n1 p2(1-p2)/n2

C.I. for p1-p2 is (p1
p2) ? Z?/2 v (p1(1--p1)/n1 p2(1-p2)/n2
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HYPOTHESIS TESTING
CONFIDENCE INTERVALS
End of Session
DEPARTMENT OF STATISTICS
REDGEMAN_at_UIDAHO.EDU
OFFICE 1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR CHAIR SIX SIGMA
BLACK BELT