 1  Outline  input analysis  goodness of fit  randomness  independence of factors  homogeneity of data  Model 05-01

 2  Chi-Square Test  arbitrary data grouping  possibly good fit in one but bad in other groupings

 3  Kolmogorov-Smirnov Test  advantages  no arbitrary data grouping as in the Chi-square test  goodness of fit test for continuous distributions  universal, same criterion for all continuous distributions  disadvantages  not designed for discrete distributions, being distribution dependent in that case  not designed for unknown parameters, biased goodness of fit decision for estimated parameters

 4  KS Test  F(x): underlying (continuous) distribution  F n (x): empirical distribution of n data points  F(x) & F n (x) being close in some sense  define D n = sup x |F(x) - F n (x)|  if D n being too large: data not from F(x)

 5  Idea of KS Test  continuous distribution F  F n empirical distribution of F for n data points  F (x) = p  |F n (x) - F(x)| ~ |Y p - p| for Y p ~ Bin(n, p)  sup x |F n (x) - F(x)| ~ sup p |Y p - p|

 6  Test for Randomness Do the data points behave like random variates from i.i.d. random variables?

 7  Test for Randomness  graphical techniques  run test  run up and run down test

 8  Background  random variables X 1, X 2, …. (assumption X i  constant)  if X 1, X 2, … being i.i.d.  j-lag covariance Cov(X i, X i+j )  c j = 0  V(X i )  c 0  j-lag correlation  j  c j /c 0 = 0

 9  Graphical Techniques  estimate j-lag correlation from sample  check the appearance of the j-lag correlation

 10  Run Test  Does the following pattern of A and B appear to be random?  AAAAAAAAAAAAAAABBBBBAAAAA  Any statistical test for the randomness of the pattern?  # of permutations with 20A’s & 5B’s =  # of permutations with 5B’s together = 21  an event of probability

 11  Run Test for Two Types of Items  for two types of items  R: number of runs  AABBBABB: 4 runs by this 8 items  for n a of item A and n b of item B  E(R) = 2n a n b /(n a +n b ) + 1  V(R) =  if min(n a, n b ) > 10, R ~ normal

 12  Run Test for Continuous Data  (43.2, 7.4, 5.4, 25.3, 27.3, 13.9, 67.5, 35.4)  sign changes:   + +  +   3 runs down & 2 runs up, a total 5 runs  R: number of runs, for n sample values  E(R) = (2n-1)/3  V(R) = (16n-29)/90  Dist(R)  normal

 13  Test for Independence It is easier to simulate a system if the classifications are independent. Are the classifications of the random quantities independent?

 14  Tax reform Income level LowMediumHighTotal For Against Total Test for Independence  for two classifications  e.g., Is voting behavior independent of income levels  easier to simulate for independent voting opinion and income levels 2 ╳ 3 Contingency Table

 15  Test for Independence  independent income level and opinion  generate income level: 3 types (i.e., m types)  generate opinion: 2 types (i.e., n types)  generate an entity: 5 types (m+n types)  dependent income level and opinion  generate income level: 3 types (i.e., m types)  generate opinion: 2 types (i.e., n types)  generate an entity: 6 types (mn types)  for k factors (classifications)  independent: m 1 + m 2 + … + m k  dependent: m 1 m 2 … m k

 16  Test for Independence  H 0 : voting opinion and income levels are independent  H 1 : voting opinion and income levels are dependent

 17  Test for Independence  marginal distribution:  If H 0 is true,

 18  Test for Independence  expected frequency: cell probability multiplies the total number of observations  in general, the expected frequency of any cell is:

 19  Test for Independence Observed and Expected Frequencies  d.o.f. associated with the chi-squared test is Tax reform Income level LowMediumHighTotal For182(200.9)213(209.9)203(187.2)598 Against154(135.1)138(141.1)110(125.8)402 Total r number of rows c number of columns

 20   dependent voting opinion and income levels Test for Independence  Calculate for the r ╳ c Contingency Table  reject H 0 if ; otherwise accept

 21  Test for Homogeneity Are the entities of the same type?

 22  Test for Homogeneity 3 ╳ 3 Contingency Table Abortion law Political affiliation DemocratRepublicIndependentTotal For Against Undecided Total predetermined

 23  Test for Homogeneity  row (or column) totals are predetermined  H 0 : same proportion in each row (or column)  H 1 : different proportions across rows (or columns)  analysis: same as the test of independence

 24   H 0 : Democrats, Republicans, and Independents give the same opinion (proportion of options)  H 1 : Democrats, Republicans, and Independents give different opinion (proportion of options)   = 0.05  critical region: χ 2 > with v = 4 d.o.f.  computations: find the expected cell frequency Test for Homogeneity

 25  Observed and Expected Frequencies Decision: Do not reject H 0. Abortion law Political affiliation DemocratRepublicIndependentTotal For82(85.6)70(64.2)62(64.2)214 Against93(88.8)62(66.6)67(66.6)222 Undecided25(25.6)18(19.2)21(19.2)64 Total Test for Homogeneity

 26  Model 5-1: An Automotive Maintenance and Repair Shop  additional maintenance and repair facility in the suburban area  customer orders (calls)  by appointments, from one to three days in advance  calls arrivals ~ Poisson process, mean 25 calls/day  distribution of calls: 55% for the next day; 30% for the days after tomorrow; 15% for two days after tomorrow  response missing a desirable day: 90% choose the following day; 10% leave

 27  An Automotive Repair and Maintenance Shop  service  Book Time, (i.e., estimated service time) ~ *BETA(2, 3) min  Book Time also for costing  promised wait time to customers  wait time = Book Time + one hour allowance  actual service time ~ GAMM(book time/1.05, 1.05) min  first priority to wait customers  customer behavior  20% wait, 80% pick up cars later  about 60% to 70% of customers arrive on time  30% to 40% arrive within 3 hours of appointment time

 28  Costs and Revenues  schedule rules  at most five wait customers per day  no more than 24 book hours scheduled per day (three bays, eight hours each)  normal cost: $45/hour/bay, 40-hour per week  overtime costs $120/hour/bay, at most 3 hours  revenue from customers: $78/ book hour  penalty cost  each incomplete on-going car at the end of a day: $35  no penalty for a car whose service not yet started

 29  System Performance  simulate the system 20 days to get  average daily profit  average daily book time  average daily actual service time  average daily overtime  average daily number of wait appointments not completed on time

 30  Relationship Between Models  Model 5-1: An Automotive Maintenance and Repair Shop  a fairly complicated model  non-queueing type  Model 5-2: Enhancing the Automotive Shop Model  two types of repair bays for different types of cars  customer not on time