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statistical processes Hypothesis Testing: Format Problem as Two Hypothesis Starting point of process Claim you wish to test Is a new design lowering warranty costs? Pose as two alternative hypothesis Null hypothesis (H 0 ) The status quo (warranty costs stay the same) Alternative hypothesis (H A ) The research question (warranty costs are reduced) Question is answered by contradiction Does data show that H 0 so improbable that you reject What do we mean?
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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
statistical processes
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?
statistical processes
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?”
statistical processes
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?
statistical processes
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?
statistical processes
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?
statistical processes
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?
statistical processes
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
statistical processes
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
statistical processes
Rejection Region
40.1716)96.1)(712.0(712.0
16
96.1
0
0
0
02
y
yns
y
yz
y
y
y
statistical processes
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.
statistical processes
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?
statistical processes
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
statistical processes
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
statistical processes
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?
statistical processes
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
statistical processes
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 ?
statistical processes
= 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!
statistical processes
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)
statistical processes
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
statistical processes
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?
statistical processes
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?
statistical processes
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!