8. A production engineer has identified four methods by which a worker might perform a job- related task. The engineer's goal is to reduce the time required to complete the task. Four workers are randomly assigned to each of the four methods. The resulting times in seconds) are summarized in the accompanying table. (22 pts) (d) Use Tukey's test to check for significant differences in treatment means using a = 0.05 (Table A.12 is found below). Also, use the visual method from class to denote which differences are not significant. Method 1 27 32 29 30 2 25 28 24 26 3 29 31 29 30 4 31 30 33 32 (a) State the appropriate null and alternative hypotheses the experimenter would be interested in testing (e) Based on your results in part (d), which method (or methods) would you recommend? (b) Construct and Analysis of Variance table for the data (Note: Eyi= 13672). Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic 5.06 Table A. 12 Upper Percentage Points of the Studentized Range Distribution: Values of 9(0.05;k, v) Degrees of Number of Treatments to Freedom, v 2 3 4 5 6 7 8 9 10 1 18.0 27.0 32.8 37.2 40.5 43.1 15.1 47.1 49.1 2 6.09 5.33 9.80 10.89 11.73 12.43 13.03 13.54 13.99 3 4.50 5.91 6.83 7.51 8.04 8.47 8.85 9.18 9.46 4 3.93 5.04 5.76 6.29 6.71 7.06 7.35 7.60 7.83 5 3.64 4.60 5.22 5.67 6:03 6.33 6.58 6.80 6.99 6 3.46 4.34 4.90 5.31 5.63 5.89 6.12 6.32 6.49 7 3.34 4.16 4.68 5.35 5.59 5.80 5.99 6.15 8 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 9 3.20 3.95 4.42 4.76 5.02 5.24 5.43 5.60 5.74 10 3.15 3.88 4.33 4.66 4.91 5.12 5.30 5.46 5.60 11 3.11 4.26 4.58 4.82 5.03 5.20 5.35 5.49 12 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.40 13 3.06 3.73 4.1.5 4.46 4.69 4.88 5.05 5.19 5.32 14 3.03 3.70 4.11 4.41 4.65 4.83 4.99 5.13 5.25 15 3.01 3.67 4.08 4:37 4.59 4.78 4.94 5.08 5.20 16 3.00 3.65 4.05 4.34 4.56 4.74 4.90 5.03 5.05 17 2.98 3.62 4.02 4.31 4.52 4.70 4.86 4.99 5.11 18 2.97 3.61 4.00 4.28 4.49 4.67 4.83 4.96 5.07 19 2.96 3.59 3.98 4.26 4.47 4.64 4.79 4.92 5.04 20 2.95 3.58 3.96 4.24 4.45 4.62 5.01 24 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 30 2.89 3.48 3.84 4.11 4.30 4.46 4.60 4.72 4.83 40 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.74 .60 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 120 2.80 3.36 3.69 3.92 4.10 4.24 4.36 4.47 4.56 oo 2.77 3.32 3.63 3.86 4.03 4.17 4.29 4.39 3.82 3.08 4.77 4.90 (c) Using a = 0.05, what conclusion would you draw from part (b)? 4.47 3 4 (c) Complete the corresponding ANOVA table below and draw conclusions on the significant effects using a=0.05 (2 y’ijk = 6618.14): 9. An appliance manufacturer is interested in determining whether any of three factors (brand of laundry detergent, water temperature, and type of detergent) affects the amount of dirt removed (measured in milligrams). A 2° factorial design with two replicates was employed and the results are presented in the table below: (20 pts) Source of Sum of Degrees of Mean F value from Variation Squares Freedom Squares F Statistic Table (a) Complete the test matrix below, including the labels of the corners that would correspond to a cube representation, resulting signs for the interactions, and calculations for contrast, effect and sum of squares: A Label A B с AB AC BC ABC Obs 1 Obs 2 Total B 27.04 (1) 14.8 15 29.8 + 18.4 18.8 37.2 С + 17.8 18.2 36.0 + 19.9 20.3 40.2 AB 0.49 + + + 20.7 21.1 41.8 + + + 21.7 21.9 43.6 AC + + + 22.6 23.2 45.8 + + + + + --- + 23.4 24.6 48 BC Contrast 20.8 -2.8 0.4 Effect 2.6 -0.35 0.05 ABC 0.01 SS 27.04 0.49 0.01 Error (b) The axes and the origin, (1), for a design cube, are denoted below. Label the corners of the cube with the appropriate letter notation. Total B (1) A 5 6 11. Data collected on a measurements experiment is shown below. (8 pts) 10. Aircraft primer paints are applied to aluminum surfaces by two methods: dipping and spraying. The purpose of the primer is to improve paint adhesion. A two-factor experiment is performed to investigate the effect of paint primer type and application method on paint adhesion. The response variable is adhesion force (Note: higher values represent a stronger adhesion). The collected data is shown below: (16 pts) Part Number Operator 1 Measurements 1 2 11 10 14 13 10 11 17 17 19 18 1 2 3 4 5 Application Method Dipping Spraying 4.0, 4.5, 4.3 5.4, 4.9, 5.6 Cell total = 12.8 Cell total = 15.9 5.6, 4.9, 5.4 5.8,6.1,6.3 Cell total = 15.9 Cell total = 18.2 3.8, 3.7, 4.0 5.5, 5.0, 5.0 Cell total = 11.5 Cell total = 15.5 Primer Type 1 Operator 2 Measurements 1 2 10 10 14 14 12 15 18 16 19 18 2 3 (a) Estimate the variability attributed to the measurement error. (a) State the appropriate hypotheses to be conducted for this experiment. (b) Construct the analysis of variance table for this data. (Note: E yok 458.72) Source of Sum of Degrees of Mean Variation Squares Freedom Squares F statistic Primer Type Appl. Method Interaction 0.241 Error 0.987 Total (b) If the LSL = 5 and the USL = 75, what can you say about the capability of the gage for this study? (c) Using a = 0.05, state your conclusions. For any significant effects, also state which primer type and/or application method should be used. 7 8 12. To monitor the process mean in a bottle manufacturing process, three finished bottles are sampled at 20 points in time. The data (weight, in ounces) for last month's production is used to construct the X and R chart below. (16 pts) (b) What do the charts suggest about the stability of the process? If either or both are out-of- control, which would you focus your attention on first? Xbar-R Chart of x1, x3 6.51 UCL= (c) Outline the logical sequence of steps that the manufacturer should proceed with next. 6.0- Sample Mean