Statistics 496, Applied Statistics for Industry II Name: _________________ Final Exam, Spring 2009 Site: ___________________ INSTRUCTIONS: You will have 2 hours to complete the exam. There are 5 questions worth a total of 125 points. Not all questions have the same point value so gauge your time appropriately. Read the questions carefully and completely. Answer each question and show work in the space provided on the exam. Turn in the entire exam when you are done or when time is up. For essay questions, think before you write. 1. [35 pts] An experiment on the fuel economy of a particular make and model of an automobile was designed using five factors, each at 2 levels. Factor Low Level High Level A: Engine Size (L) 4.0 4.5 B: Fuel Octane 87 93 C: Tire Pressure (psi) 22 28 D: Driving Speed (mph) 45 55 E: Air Conditioning Off On The experiment was run as a 25–1 Resolution V fractional factorial with the levels of E set according to the plus/minus coding of the ABCD interaction. Below are the data, in standard Yates order, and estimated full effects. Treatment Combination Fuel Economy (mpg) Estimated Full Effect Effect Name Alias e 25 25.50 Mean a 21 –4.75 A b 29 3.50 B abe 23 1.75 AB c 29 2.00 C ace 20 –0.25 AC bce 29 0.50 BC abc 29 0.25 ABC d 27 –0.25 D ade 19 0.00 AD bde 26 –0.25 BD abd 26 0.00 ABD cde 27 –0.25 CD acd 22 –0.50 ACD bcd 31 –0.25 BCD abcde 25 –2.50 ABCD a) [4] Fill in the aliases. 1b) [4] On the normal plot of effects given below identify, by name and alias, those effects that appear to be significant. c) [5] Can you reduce this to a smaller experiment in fewer factors? If so, what are those factors? Will you pick up pseudo-replication? d) [7] Using an MSError = 0.20, which of the estimated effects are significant. Use t=3 to compute the critical effect size. 2e) [5] Give a prediction equation using only those factors that are found to be significant in d). Explicitly define any variables used in your equation. f) [5] What factor levels, do NOT give me plus/minus codes but the actual levels, would you recommend in order to maximize the fuel economy? g) [5] What are the predicted value and prediction interval for the fuel economy at the recommended levels in f)? 2. [15 pts] Statistical thinking consists of the recognition that variability always is, and always will be, present and that learning is an iterative procedure (one must go through several cycles to truly learn about something). How do statistically designed experiments fit into the framework of statistical thinking? Be sure to mention both aspects; variability and the iterative nature of learning. 33. [30] An experiment is performed on a CNC milling machine. The three factors of interest are A: Spindle Speed (1750, 2000, and 2250 rpm), B: Feed Rate (6, 12, and 18 in/sec) and Depth of Cut (0.04, 0.06, and 0.08 in). All 27 treatment combinations are run in a completely random order. A: Spindle Speed B: Feed Rate C: Depth of Cut Level Mean X1,A X2,A Level Mean X1,B X2,B Level Mean X1,C X2,C 1750 63.2 –1 +1 6 54.7 –1 +1 0.04 71.3 –1 +1 2000 86.2 0 –2 12 75.6 0 –2 0.06 84.0 0 –2 2250 78.0 +1 +1 18 97.1 +1 +1 0.08 72.1 +1 +1 a) [13] Complete the following table of estimated slope parameters. Note MSError = 80.74. Term Estimate Std Error t Ratio Intercept 75.8 ******* ******** X1,A 7.4 X1,B 21.2 X1,C X2,A –5.2 1.223 X2,B X2,C –4.1 X1,AX1,B –2.4 2.594 –0.93 X1,AX1,C 2.0 2.594 0.77 X1,BX1,C –2.7 2.594 –1.04 4b) [5] According to the t-Ratios, which terms are statistically significant? Explain briefly? c) [4] Give the reduced model prediction equation that includes only those terms that are statistically significant. d) [6] A low value of the surface roughness is desirable. According to your prediction equation in c) how would you set Spindle Speed, Feed Rate and Depth of Cut to get the lowest surface roughness? e) [2] What is the predicted surface roughness for your settings in d)? 54. [30 pts] The data below are the voltage levels at which failures occurred in two types of electrical cable insulation when specimens were subjected to an increasing voltage stress in a laboratory experiment. The test involved 20 specimens of each type, and the failure voltages (in kilovolts per millimeter) were: Type 1 insulation 32.0 35.4 36.2 39.8 41.2 43.3 45.5 46.0 46.2 46.4 46.5 46.8 47.3 47.3 47.6 49.2 50.4 50.9 52.4 56.3 Type 2 insulation 39.4 45.3 49.2 49.4 51.3 52.0 53.2 53.2 54.9 55.5 57.1 57.2 57.5 59.2 61.0 62.4 63.8 64.3 67.3 67.7 a) [5] Looking at the probability plot on the next page, explain briefly why a Weibull model is an appropriate model for the insulation failure voltage . b) [5] Assuming a Weibull model for Type 2 insulation, graphical estimates of the model parameters are: 1.9ˆ=β and 017.0 . Using these values, estimate the chance that Type 2 insulation will withstand 50 kilovolts per millimeter. ˆ=λ c) [5] Using the Weibull model, estimate the median failure voltage for Type 2 insulation? 6d) [10] For Type 1 insulation, obtain graphical estimates of β andλ. Again assume that a Weibull model is appropriate. e) [5] For Type 1 insulation the chance that it will withstand 50 kilovolts per millimeter is approximately 0.13 and the median voltage to failure is approximately 44.2 kilovolts per millimeter. Which insulation should be purchased if protection against voltage stress is needed? Why? What assumptions are implicit in your decision? 785. [15 pts] The three fundamental principles of a well designed experiment are control of outside variables, randomization and replication within the experiment. a) [4] Why is control of outside variables important? b) [5] If an outside variable is not controlled, how can randomization help? c) [2] What is replication within an experiment? d) [4] What problems can occur when you do not have replication in an experiment? Because I may not receive your final exam before grades are due at the Registrar's Office, you may receive notification from the Registrar's Office, or see on AccessPlus,