1. statistical processes ENMA 420/520 Statistical Processes Spring 2007 Michael F. Cochrane, Ph.D. Dept. of Engineering Management Old Dominion University

2. statistical processes Class Nine Readings & Problems Reading assignment M & S Chapter 8 Sections 8.1 – 8.12 B&C Chapter 6 Recommended problems M & S Chapter 8 88, 90, 91, 94, 98, 100

3. 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?

4. statistical processes Hypothesis Testing: Verifying Claims Through Observations Claim: a new design reduces warranty costs. Hypothesis There is no difference between average warranty costs for new & old designs Hypothesis There is no difference between average warranty costs for new & old designs Collect data Is data consistent with hypothesis? Yes Do not reject hypothesis No Reject hypothesis Why not, “accept?”

5. statistical processes Hypothesis Testing: H 0 Assumed True Unless Shown Otherwise Data cannot prove H 0 Could show H 0 so unlikely cannot believe true Burden of proof is to support H A Basis: test statistic Implies probabilistic conclusions:   % probability of error Is this similar to our system of law?

6. statistical processes Simple Illustration of Concept Management claim: Average wages $16 Call H 0 Union claim: Ave wage  $16 Call H A Use hypothesis testing to see if claim is true! Use hypothesis testing to see if claim is true! Worker wages y Sample 40 workers Sample 40 workers Assuming H 0 true, what is pdf of y_bar? Assuming H 0 true, what is pdf of y_bar?

7. statistical processes Constructing the PDF of y_bar: Assume H 0 is True Why is the distribution normal? What is its mean? What is its variance?

8. statistical processes Continuing With Example: Assume Some Results Sample results: n = 40 y_bar = 14.50 s = 4.50 Can we use this pdf to test assumption H 0 is true? Recap, assuming H 0 is true: - What is mean of distribution for y_bar? - What is its variance? - What are we missing to test validity of assumption?

9. statistical processes Testing Validity of Assumption: H 0 is True Assume   prob of error = 0.05 Create region for rejecting H 0 Rejection region $64,000 question: what are these critical values?? If H 0 true:  = 16  2 =  2 /40  s 2 /40 = 0.506 If H 0 true:  = 16  2 =  2 /40  s 2 /40 = 0.506

10. statistical processes PDF of y_bar Rejection region 14.6017.40 Given sample mean = 14.50, what do you conclude? Are you 100% sure of conclusions? Let’s restate these

11. statistical processes Rejection Region

12. statistical processes Rejection Region In Terms of Z Rejection region -z  /2 = -1.96 z  /2 =1.96 Next, convert sample statistic to equivalent z value & see if falls in rejection region.

13. statistical processes Converting Test Statistic to Z 0 What are possible errors in our conclusion? Again, is the assumption (H 0 is true) likely valid?

14. statistical processes Quite a Tale: Two & one Tail Tests Previous hypothesis test called two tailed Rejection region both tails of y_bar pdf Assuming H 0 is true Two tailed test take form H 0 :  =  0 H A :    0 One tailed test take form H 0 :  =  0 H A :  >  0 (or,  <  0 ) Modify previous example: H 0 :  = 16 H A :  < 16

15. statistical processes The assumption that H 0 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 H A :  < 16, why is rejection region only left tail? What is this area Modifying the Rejection Region: One Tail Test

16. statistical processes Back to Decision: What Are Potential Errors? 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?

17. statistical processes Type I error  = p(reject H 0  H 0 true) Complement 1-  = p(not reject H 0  H 0 true) Type II error  = p(not reject H 0  H A true) Complement 1-  = p(reject H 0  H A true) Types of Errors and Their Complements These are errors, want as small as possible. Note mutually exclusive conditions. These are OK, want as large as possible. Note mutually exclusive conditions

18. statistical processes Hypothesis Testing: Designing the Experiment As experimenter can modify n - 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  ?

19. statistical processes  = p(not reject H 0  H A true) Since conditional on H A what value should we use? Why did we not have same problem with  ?  = p(reject H 0  H 0 true) Calculating  : A Function Not a Single Value Calculate  over a range of possible values of H A If H 0 true then know mean of pdf for y_bar!

20. statistical processes Calculating  : Making Assumption for H A 1614 Reject RegionNot Reject Region PDF if H 0 is assumed true PDF if H A is assumed true 14.83 This is P(not reject H 0 |  =14.0)

21. statistical processes Calculating  : Assuming H A :  = 14.0  = p(not reject H 0  H A true) Now assume that  = 14.0  1 = p(not reject H 0   = 14.0 ) = p( y_bar 0 > 14.83   = 14.0) = p( z > [(14.0 - 14.83) / 0.712]   = 14.0) = p( z > 1.16573   = 14.0) = 0.122

22. statistical processes Discussion Points Hypothesis Testing How do you choose H 0 and H A ? Why is hypothesis testing a conservative approach? Why can’t you prove H 0 ? 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 ?

23. statistical processes Principal Hypothesis Tests Testing population means Large sample Small sample Testing differences between 2 population means Independent samples Large samples Small samples Note use of Student’s t distribution & normality assumption for y Note use of t distributions, what are the assumptions made?

24. statistical processes Principal Hypothesis Tests Testing differences between 2 population means Matched pairs Large sample Small sample Testing population proportion Large sample Note use of t distributions, why no assumption about population variances? Why is there no “small sample size” test for population proportion? Note sample size assumption!