statistical processes

ENMA 420/520Statistical ProcessesSpring 2007

Michael F. Cochrane, Ph.D.Dept. of Engineering ManagementOld Dominion University

statistical processes

Class NineReadings & Problems

Reading assignmentM & S

Chapter 8 Sections 8.1 – 8.12

B&C Chapter 6

Recommended problemsM & S Chapter 8

88, 90, 91, 94, 98, 100

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Hypothesis Testing:Format Problem as Two Hypothesis

Starting point of processClaim you wish to test

Is a new design lowering warranty costs?

Pose as two alternative hypothesisNull hypothesis (H0)

The status quo (warranty costs stay the same)

Alternative hypothesis (HA) The research question (warranty costs are reduced)

Question is answered by contradictionDoes data show that H0 so improbable that you reject

What do we mean?

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Hypothesis Testing:Verifying Claims Through Observations

Claim: a new design reduces warranty costs.

HypothesisThere is no difference between average warranty costs for new & old designs

Collect data

Is data consistent withhypothesis?Yes

Do not reject hypothesis

No

Reject hypothesisWhy not, “accept?”

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Hypothesis Testing:H0 Assumed True Unless Shown Otherwise

Data cannot prove H0

Could show H0 so unlikely cannot believe true

Burden of proof is to support HA

Basis: test statistic

Implies probabilistic conclusions: % probability of error

Is this similarto our system of law?

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Simple Illustration of Concept

Management claim: Average wages $16 Call H0

Union claim: Ave wage $16 Call HA

Use hypothesistesting to see ifclaim is true!

Worker wages y

Sample40 workers

Assuming H0

true, what ispdf of y_bar?

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Constructing the PDF of y_bar:Assume H0 is True

PDF of y_bar

0.0

0.3

0.6

12.4 13.9 15.3 16.7 18.1 19.6

Why is the distribution normal?

What is its mean?

What is itsvariance?

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Continuing With Example:Assume Some Results

Sample results:n = 40

y_bar = 14.50s = 4.50

Can we use this pdf to test assumption

H0 is true?

PDF of y_bar

0.0

0.3

0.6

12.4 13.9 15.3 16.7 18.1 19.6

Recap, assuming H0 is true:- What is mean of distribution for y_bar?- What is its variance?- What are we missing to test validity of assumption?

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Testing Validity of Assumption:H0 is True

PDF of y_bar

0.0

0.3

0.6

12.4 13.9 15.3 16.7 18.1 19.6

Assume prob of error = 0.05 Create region for rejecting H0

Rejection region Rejection region

$64,000 question: what are these critical values??

If H0 true: = 162 = 2/40 s2/40 = 0.506

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PDF of y_bar

PDF of y_bar

0.0

0.3

0.6

12.4 13.9 15.3 16.7 18.1 19.6

Rejection region Rejection region

14.60 17.40

Given sample mean = 14.50, what do you conclude?

Are you 100% sure of conclusions?

Let’s restate these

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Rejection Region

40.1716)96.1)(712.0(712.0

16

96.1

0

0

0

02

y

yns

y

yz

y

y

y

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Rejection Region In Terms of Z

y_bar Converted to Z

0.0

0.2

0.4

0 1 2 3 4 5-1-2-3-4-5

Rejection region Rejection region

-z/2= -1.96 z/2=1.96

Next, convert sample statistic to equivalent z value& see if falls in rejection region.

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Converting Test Statistic to Z0

11.2712.0

165.14

0

00

ns

y

yz

y

y

y

What are possibleerrors in our conclusion?

Again,is the assumption(H0 is true)likely valid?

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Quite a Tale:Two & one Tail Tests

Previous hypothesis test called two tailedRejection region both tails of y_bar pdf

Assuming H0 is true

Two tailed test take formH0: = 0

HA: 0

One tailed test take formH0: = 0

HA: > 0 (or, < 0)Modify previous example:

H0: = 16HA: < 16

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PDF of y_bar

0.0

0.3

0.6

12.4 13.9 15.3 16.7 18.1 19.6

The assumption that H0 is true still holds, so pdf of y_bar is the same!

Given that = 0.05, why is critical value for rejection = 14.83?

Rejection region

Now HA: < 16, why is rejection region only left tail?

What isthis area

Modifying the Rejection Region:One Tail Test

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Back to Decision:What Are Potential Errors?

H0 True HA TrueNot Reject

H0

OKProb= 1-

Type II errorProb=

Reject H0 Type I errorProb=

OKProb= 1-

True States of Nature

Decision made

This is defined as power of the test

Who defines & ? What are ideal values of & ?

What do you think happens to as decreases?

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Type I error = p(reject H0 H0 true)

Complement1- = p(not reject H0 H0 true)

Type II error = p(not reject H0 HA true)

Complement1- = p(reject H0 HA true)

Types of Errors and Their Complements

These are errors,want as small as possible. Note mutually exclusiveconditions.

These are OK,want as large aspossible. Notemutually exclusiveconditions

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Hypothesis Testing:Designing the Experiment

As experimenter can modifyn - the number of observations - the probability of Type I error - the probability of Type II error

What are impacts of increasing n?What is impact of decreasing ? What is impact of decreasing ?

Previously set n = 40, = 0.05 can we determine ?

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= p(not reject H0 HA true)Since conditional on HA what value should we use?

Why did we not have same problem with ? = p(reject H0 H0 true)

Calculating :A Function Not a Single Value

Calculate over a range of possible values of HA

If H0 true then know mean of pdf for y_bar!

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Calculating :Making Assumption for HA

Estim Mean 0 'True' Mean -2.010027067

Hypoth Mean 0 Std Err Mean 0.7115124735 Beta 0.1291114742

Power 0.8708885258

1614

Reject Region Not Reject Region

PDF if H0 is assumed

true

PDF if HA is assumed

true

14.83

This is P(not reject H0| =14.0)

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Calculating :Assuming HA: = 14.0

= p(not reject H0 HA true)Now assume that = 14.0

1 = p(not reject H0 = 14.0 )

= p( y_bar0 > 14.83 = 14.0)

= p( z > [(14.0 - 14.83) / 0.712] = 14.0) = p( z > 1.16573 = 14.0) = 0.122

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Discussion PointsHypothesis Testing

How do you choose H0 and HA?

Why is hypothesis testing a conservative approach?Why can’t you prove H0?

What is relationship between hypothesis testing and confidence intervals?What are Type I and Type II errors?In 1-tail test is actually maximum probability of Type I error, why?Why is calculated as a function of actual y?

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Principal Hypothesis Tests

Testing population meansLarge sampleSmall sample

Testing differences between 2 population meansIndependent samples

Large samples Small samples

Note use of Student’s tdistribution & normalityassumption for y

Note use of tdistributions,what are theassumptions made?

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Principal Hypothesis TestsTesting differences between 2 population means

Matched pairs Large sample Small sample

Testing population proportionLarge sample

Note use of tdistributions,why no assumption about population variances?

Why is there no “small sample size” test for population proportion?

Note sample size assumption!