Statistics 496, Applied Statistics for Industry II Name: _________________ Exam 2, Spring 2009 Site: ___________________ INSTRUCTIONS: You will have 1 hour and 30 minutes to complete the exam. There are 4 questions worth a total of 100 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. [30] The effects of three process variables; A: Seal Temperature (o F), B: Cooling Bar Temperature (o F), and C: % Polyethylene Additive on the seal strength (g/in) of a bread wrapper stock are studied with a 23 factorial experiment with 4 center points. The data are given below. A B C Strength 225 46 0.5 6.6 285 46 0.5 6.9 225 64 0.5 7.9 285 64 0.5 6.1 225 46 1.7 9.2 285 46 1.7 6.8 225 64 1.7 10.4 285 64 1.7 7.3 255 55 1.1 10.4 255 55 1.1 9.5 255 55 1.1 9.8 255 55 1.1 9.9 The estimated full effects for this experiment are as follows: A: –1.75, B: 0.55, AB: –0.70, C: 1.55, AC: –1.00, BC: 0.30, and ABC: 0.35. The variance at the center points is 0.14. a) [6] Compute the standard error of an estimated full effect and the critical effect size (use t=3). Which of the estimated full effects are statistically significant? Explain briefly. 1b) [5] Using only those effects that are statistically significant, give the prediction equation for seal strength. Be sure to explicitly define the variables in the prediction equation. c) [3] Using the prediction equation in (b), predict the seal strength for a seal temperature of 225oF, cooling bar temperature of 64oF and polyethylene additive of 1.7%. d) [5] Construct a prediction interval for your prediction in (c). e) [7] Is there evidence of curvature in the response? Be sure to include the mean of the factorial experiment and the mean for the center points as well as the value and interpretation of an appropriate test statistic. f) [4] In light of your result in (e), would you say your prediction in (c) is accurate? Explain briefly. 22. [20 pts] Golf ball manufacturers appeal to golfers with claims that their product goes farther, flies straighter or lands softer. Of particular interest is the driving distance of a golf ball. Some of the factors that may affect the distance a golf ball travels are: brand, compression, core and cover. Compression is a measure of the resiliency of the ball when it is struck by the golf club. Golf balls have either a one-piece core or a two-piece core (rubberized string wound around a solid or liquid inner core). The two types of cover commonly used are a hard cover (Surlyn) and a softer cover (Balata). Besides the characteristics of the golf ball, the golfer and weather/course conditions have a big effect on the distance a golf ball will travel. Better golfers can hit the ball straighter and farther than beginners. A 23 factorial experiment is run with a single golfer in a completely randomized design with 5 replications. All golf balls have two-piece cores. The factors are: Factor Low (–1) High (+1) A: Brand Titleist Maxfli B: Compression 90 100 C: Cover Balata Surlyn The summarized data are: A B C Mean, iY in Variance, 2is–1 –1 –1 260.8 5 136.7 +1 –1 –1 259.8 5 81.7 –1 +1 –1 258.8 5 120.2 +1 +1 –1 257.4 5 170.3 –1 –1 +1 266.0 5 166.0 +1 –1 +1 276.2 5 113.7 –1 +1 +1 275.8 5 171.7 +1 +1 +1 272.6 5 104.3 a) [6] Complete the following list of estimated full effects. Effect Name Estimate Y 265.925 A B 0.45 AB C 13.45 AC 2.35 BC 2.65 ABC –3.25 3b) [2] Compute the MSrepError for this problem. c) [5] Calculate the standard error of an estimated effect and give the critical effect size, use t=3. What effect(s) is(are) significant? d) [7] In a brief paragraph, summarize your findings. This paragraph should be written so that it could be understood by someone with little or no knowledge of statistics. Be sure to comment on the limitations of this experiment. 43. [20 pts] An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. The levels of each of the 3 factors are given below. low (–1) high (+1) A: cutting speed 50 rpm 100 rpm B: tool geometry Type 1 Type 2 C: cutting angle 5o 10o Since the hardness of the raw material being tooled may vary, heats of raw material are used as blocks. Below are the data. Block A B C I II III Mean –1 –1 –1 22 31 25 26 +1 –1 –1 32 43 30 35 –1 +1 –1 35 50 35 40 +1 +1 –1 45 55 47 49 –1 –1 +1 43 45 38 42 +1 –1 +1 36 38 40 38 –1 +1 +1 50 61 54 55 +1 +1 +1 47 43 39 43 Block Mean 38.75 45.75 38.5 41.0 For this experiment SSrepError = 474.0 with 16 degrees of freedom. Then estimated full effects are: A: 0.5, B: 11.5, AB: –2.0, C: 7.0, AC: –8.5, BC: –2.5, ABC: –2.0. a) [4] Below is a plot of the main effects. Comment on the apparent effects of the three factors. 5b) [5] Below are plots of the A: Speed by C: Angle interactions for Type 1 and Type 2 Geometries. What do these plots indicate about the possible interaction between A and C? What do they indicate about the 3-way interaction? c) [8] What is the standard error of an estimated full effect? d) [3] What effects are statistically significant? Use t=3. 64. [30 pts] A 24 un-replicated factorial experiment was conducted on the polymer resin, PMR-II-50 to see the effect of four factors on weight loss. The four factors and their levels are given below. Factor Low Level (–1) High Level (+1) Oven Position Position 1 Position 2 Kapton no kapton (0) with kapton (1) Preprocessing Time 15 minutes 120 minutes Dianhydride Type Polymer grade (1) Electronic grade (2) All specimens, which initially had about the same mass, are baked at 600oF for 936 hours. The percent weight loss (wghtloss) is the measured response. The output of the analysis of the 16 observations is given below. Fractional Factorial Fit Estimated Full Effects and Parameter Estimates for wghtloss Term Full Effect Parameter Estimate Constant 4.3375 Position 0.2750 0.1375 Kapton 0.0750 0.0375 Time –0.0500 –0.0250 Type –1.0500 –0.5250