statistics hypothesis testing & confidence intervals

7/29/2019 Statistics Hypothesis Testing & Confidence Intervals

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HYPOTHESIS TESTING &

DEPARTMENT OFSTATISTICS

DR. RICKEDGEMAN, PROFESSOR& CHAIR SIX SIGMABLACKBELT

[email protected] OFFICE: +1-208-885-4410

CONFIDENCE INTERVALS


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The Hypothesis Testing Approach

Conjectures(Hypotheses)

Evalu

ation

(TestMethod)

Gather&EvaluateFacts

Zone

of

Belief

ConsequencesA B or


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The Scientific MethodNo Observer

or UninformedObserver

InformedObserver

Noninformative Event Informative Event

Scientific Method

of Investigation

Nothing Learned

Little orNothing Learned

Little or

Nothing Learned

Discovery!


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Motivation for Hypothesis Testing

The intent of hypothesis testing is formally examine twoopposing 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 thetwo hypotheses is true and which of the two hypotheses isfalse.

Beyond the issue of truth, addressed statistically, is the issueof justice. Justice is beyond the scope of statisticalinvestigation.


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The American Trial SystemIn Truth, the Defendant is:

H0: Innocent HA: Guilty

CorrectDecision IncorrectDecision

Innocent Individual Guilty Individual

Goes Free Goes Free

IncorrectDecision CorrectDecision

Innocent Individual Guilty IndividualIs Disciplined Is Disciplined

Innocent

Guilty

Verd

ict


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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 WorldH0 is True HAis True

Ho is True

Decision

HAis True

Correct Decision Incorrect Decision

Type II Error Probability =

Incorrect Decision Correct Decision

Type I Error Probability =


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Hypothesis Testing Algorithm

1) Specify H0 and HA2) Specify n and 3) What Type of Test and Why?

4) Critical Value(s) and Decision Rule (DR)5) Collect Pertinent Data and Determine the Calculated Value of the

Test Statistic (e.g. Zcalc, tcalc, 2calc, etc)6) Make a Decision to Either Reject H0 in Favor of HA or to Fail to

Reject (FTR) H0.7) Construct & Interpret the Appropriate Confidence Interval

8) Conclusions? Implications & Actions


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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


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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: = < > 0 vs. HA: > < 0

n = _______ = _______ Testing a Hypothesis About a Mean;

Process Performance Measure is Approximately Normally Distributed or WeHave a Large Sample;

We Do Not KnowWhich 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 < -t/2 or if tcalc > +t/2.Otherwise, FTR H

0.

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.


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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 = < > p0 vs. HA: p > < p0

n = _______ = _______ Testing a Hypothesis About a Proportion;

We have a large samplethat 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, FTRH0.

DR: (< in HA) Reject H0 in favor of HA iff Zcalc < -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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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 historicalproductivity 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.

Z-Test & Confidence Interval:

Training Effect Example


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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 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 productivityfrom 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 asample of 25 applications submitted after the measures were

implemented gave an average processing time of 15.2 days and astandard deviation of 2.0 days.


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First Peoples 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 is15.2 + 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.


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Small Business

Loan Defaults


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First Peoples Bank of Central CitySmall Business Loan Defaults

Historically, 12% of Small Business Loans granted result indefault. Three years ago, FPB of Central City purchasedsoftware which they hope will assist in reducing the defaultrate by more effectively discriminating between small

business loan applicants who are likely to default and thosewho are not likely to do so.

After adequately training their loan officers in use ofsoftware, FPB sampled 150 small business loan applications

processed using the software and found 9 to be in default atthe end of two years.

Using = .10, does it appear that the software is of value?


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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

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%.

^

^ ^ ^

Small BusinessLoan Default Rate


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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 if2calc < 2small,/2 or if2calc >2large,/2. Otherwise, FTR H0.

DR: (> in HA) Reject H0in favor of HAiff

2

calc > 2

large,Otherwise, FTR H0. DR: (< in HA) Reject H0in favor of HAiff2calc < 2small,Otherwise, FTR H0.


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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


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Fast Facts Financial, Inc.Fast Facts Financial (FFF), Inc. provides credit reports to lendinginstitutions 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, butare 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 willreduce 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 = .05level of significance. Assume that the time factor is approximately

normally distributed.


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FFFExample H0: = < > 0 vs. HA: > < 0 where0 = 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 iff2calc < 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 that1.405 days < < 2.50 days

Evidence is inconclusive. Work should continue on this.


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Two Sample Testsand

Confidence Intervals


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H0: 12= dHA: 12< > d

n1 = _____ n2= _____ = 0Comparison 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


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C.I. for1

2

X1 X2 ZX1-X2

or

X1 X2 tSX1-X2Decisions Same as any other Z or T test.

Implications Context Specific


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(a) Z = [(X1 X2)d]

(1/n1 + 1/n2)

Z = [(X1 X2)d]

(21/n1+ 22/n2)

(b) t = [(X1 X2)d] (assume equal variances)

Sp(1/n1 + 1/n2) where df = n1+n2 2and Sp

2 = (n1-1)S12 + (n2-1)S2

2

(c ) t = [(X1 X2)d] (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


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Equality of Variances: The F-Test

H0: 1= 2 vs. HA: 1 < > 2

n1 = _____ 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, FTRH0.

___in HA: reject H0 in favor of HA iff Fcalc < F/2,small or if Fcalc > F/,big.Otherwise, FTR H

0.


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Fcalc = S12/S2

2

Make a decision.

Fcalc/ Fn1-1,n2-1,/2 large 12/22 Fcalc/Fn1-1,n2-1,/2 small

C.I. for1/2 is obtained by taking squareroots of the endpoints of the above C.I. for

1

2/2

2

Conclusions / Implications Context Specific.


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Tests & Intervals for Two Proportions

H0

: p1

p2

= pd

HA: p1 p2 < > pd

n1 = _____ n2= _____ = 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 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

^ ^


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HYPOTHESIS TESTING &

DEPARTMENT OFSTATISTICS

DR RICK EDGEMAN PROFESSOR & CHAIR SIX SIGMA BLACK BELT

[email protected] OFFICE: +1-208-885-4410

CONFIDENCE

INTERVALS

Endof Session

statistics hypothesis testing & confidence intervals

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