Homework 5

Introduction

In this assignment, you will perform both input modeling and output analysis. When answering the following questions, please show all work (equations, explanations, etc.) not just the final answer. List any software you use and explain what you used it for.

Input Modeling

1. The following data are generated randomly from a Normal distribution. Compute the maximum-likelihood estimators for μ and σ².

   -0.3182	0.1913	1.3174	0.7215	0.0155	1.7381
   -0.0895	-1.8758	2.8889	-0.5848	-1.231	0.1327
   0.3708	1.1832	0.0492	0.2236	1.0263	1.4639
   2.0838	-0.0495	-2.0548	0.2184	-3.3413	-0.5454
   0.3755	0.6597	0.7597	-0.868	0.3406	-1.4124

2. The highway between Atlanta, Georgia and Athens, Georgia has a high incidence of accidents along its 100 kilometers. Public safety officers say that the occurrence of accidents along the highway is randomly (uniformly) distributed, but the news media says otherwise. The Georgia Department of Public Safety published records for the month of September. These records indicated the point at which 30 accidents involving an injury or death occurred, as shown below (the data points representing the distance from the city limits of Atlanta). Use the Kolmogorov-Smirnov test to discover whether the distribution of location of accidents is uniformly distributed. Use the level of significance α = 0.05.

   88.3	40.7	36.3	27.3	36.8
   91.7	67.3	7.0	45.2	23.3
   98.8	90.1	17.2	23.7	97.4
   32.4	87.8	69.8	62.6	99.7
   20.6	73.1	21.6	6.0	45.3
   76.6	73.2	27.3	87.6	87.2

3. The time required for 50 different employees to compute and record the number of hours worked during the week was measured, with the following results in minutes. Use the chi-square test to test the hypothesis that these service times are exponentially distributed. Use six intervals. Use the level of significance α = 0.05.

   Employee	Time (min)
   1	1.88
   2	0.54
   3	1.9
   4	0.15
   5	0.02
   6	2.81
   7	1.5
   8	0.53
   9	2.62
   10	2.67
   11	3.53
   12	0.53
   13	1.80
   14	0.79
   15	0.21
   16	0.8
   17	0.26
   18	0.63
   19	0.36
   20	2.03
   21	1.42
   22	1.28
   23	0.82
   24	2.16
   25	0.05
   26	0.04
   27	1.49
   28	0.66
   29	2.03
   30	1.00
   31	0.39
   32	0.34
   33	0.01
   34	0.10
   35	1.10
   36	0.24
   37	0.26
   38	0.45
   39	0.17
   40	4.29
   41	0.80
   42	5.5
   43	4.91
   44	0.35
   45	0.36
   46	0.90
   47	1.03
   48	1.73
   49	0.38
   50	0.48

4. At a small store, you record the service time (in minutes) for 30 transactions (shown below). How are these service times distributed? Develop and test a suitable model. Use one of the goodness-of-fit tests to decide. Use the level of significance α = 0.05.

   4.6093	3.8583	5.2921	11.8326	3.3745	14.3956
   2.4541	1.1305	1.1191	2.9973	4.141	2.0066
   2.7272	8.7227	2.1489	3.0065	2.338	2.6579
   2.6083	4.8375	0.2878	2.6055	1.6949	0.3578
   9.5841	1.7347	2.9482	6.7692	12.1024	4.1612

Output Analysis

1. The small store from #4 above desires their service time to be faster, closer to 2.5 minutes. You implement a simulation of their store. Over 10 runs, you record the service time for 30 customers:

   6.4678	5.6306	5.3717	6.9439	3.7322
   5.2842	5.6135	3.9969	6.1131	4.2907

   The store owner has thoughts on how to improve service times. You implement these thoughts in the simulated system and rerun the simulation for 10 runs of 30 customers each. You do your best to keep the same random numbers in run i this time as run i had for the first run. The average service times recorded were:

   3.1879	2.9015	2.7428	3.997	3.3992
   3.4287	3.5636	4.7058	3.1817	3.1958

   Is there a difference in service times? Construct the appropriate 95% confidence interval to decide.
2. The store owner is curious if her improvements bring the service time close enough to 2.5 minutes. Using the data above for the improvements, construct a 95% confidence interval to decide.
3. The store owner wants to know the 95% confidence interval on the service times from your original simulation run (before the improvements). After seeing the range, she is disappointed that it is too large. She wants the confidence interval to be within 15 seconds (0.25 minutes). How many simulation runs are needed to have to have a confidence interval that is 30 seconds wide?

Submission

The assignment is due Thursday, April 14 by 11:59 pm.

You may turn in the assignment at lecture or upload it to CourseWeb, in the Assignment 5 folder.

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Solution

Solution