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STAT 496, Spring 2013 Homework Assignment #2, Due by Friday, February 8 1. The three fundamental principles of a designed experiment are: control of outside variables, randomization, and replication. Consider each of the following scenarios. Does each adhere to all three principles? If so, describe how. If not, indicate how you would change the design to incorporate the missing principle(s). If a principle is partially adhered to but could be improved, comment on what you would do to improve the experiment. Be specific and be sure to comment on all three principles. a) An experiment is performed to evaluate the effect of drill speed on the size of drilled holes. A single 0.25 inch drill bit in a vertical head milling machine is used for the experiment. All drilling is done by the same operator. A single bar steel with a Brinell hardness of 400 is used. Four speeds are evaluated: 200, 250, 300 and 350 rpms. The operator picks a speed at random from the four and will drill 20 holes. The process is repeated, picking a speed at random from those remaining and drilling 20 holes until all 4 speeds have been used. After all of the holes are drilled, each of the holes is measured using a special caliper. The order in which the 80 holes are measured is done at random. b) A chemist wants to compare a new and simpler assay method for determining the concentration of a mixture with the standard method. She prepares a batch of solution, divides it into 30 specimens, and then selects 15 at random. She asks her technician to analyze these 15 specimens using the new method and the remaining 15 using the standard method. The technician finishes analyzing the 15 specimens using the new method in the morning. He takes a break for lunch and does the remaining 15 specimens using the standard method in the afternoon. c) An industrial psychologist does a study to determine whether incentives and the size of the incentive, will improve productivity. Thirty workers, who all do the same assembly job, agree to participate in the study. The thirty workers are divided, at random, into three groups of 10. Each person in one group is told that if he/she increases the number of correctly assembled products by 5% he/she will receive a 5% increase in their pay. Each person in a second group is told that if she/he increases the number of correctly assembled products by 5% she/he will receive a 10% increase in her/his pay. Each person in the third group is told that the company’s goal is to increase the number of correctly assembled parts by 5% but is given no individual pay incentive. 1 2. A manufacturer wishes to compare two foundries that supply parts for the manufacturer’s production lines. An important quality characteristic is the % elongation for the heat of metal that is used to make the parts. a) If the chances of making errors are set at Alpha=0.05 and Beta=0.10 and a difference in mean % elongation of 1 standard deviation is to be detected, how many heats of metal should be evaluated for each foundry? b) If resources limit the number of heats to 10 from each foundry, give two combinations of Alpha and Beta and the size of the difference in means that can be detected. Explain briefly the trade-offs between the two combinations. c) Ten heats are randomly selected from those available at Foundry A and ten heats are randomly selected from those available at Foundry B. All heats are evaluated and % elongation values are obtained. Explain briefly why this is an observational study and not an experiment. Data for this study are given in the table on the next page. Output from JMP is provided at the end of the assignment. 1. Foundry A 44 49 44 47 49 42 51 48 46 44 y1 = 46.4 2. Foundry B 58 53 61 60 60 63 64 61 59 55 y 2 = 59.4 s1 = 2.8752 s 2 = 3.3731 s12 = 8.27 s 22 = 11.38 d) Comment on the side-by-side box plots for the two foundries. Be sure to discuss both center and spread. e) Is there a statistically significant difference between the % elongation means for the two foundries? Support your answer statistically. f) The JMP output reports the observed difference in means (Difference 13.0000) for the 20 heats evaluated. It also reports a 95% Confidence Interval for the difference in foundry means (Upper CL Dif 15.9446, Lower CL Dif 10.0554) which can be interpreted as the likely values for the difference in means if all heats available at Foundry A were compared to all heats available at Foundry B. Explain why this difference and the confidence interval represent enumerative purposes. g) If we wanted to say that in the future that heats produced at Foundry B will have a mean elongation between 10 and 16 percentage points higher than the mean elongation for future heats produced at Foundry A (an analytic purpose) what must be true about the two foundries? 2 JMP Output for Homework 2 Oneway Analysis of % Elongation By Foundry 65 % Elongation 60 55 50 45 40A B Foundry Oneway Anova t Test B-A Assuming equal variances Difference 13.0000 Std Err Dif 1.4016 t Ratio 9.275204 DF 18 Upper CL Dif 15.9446 Prob > |t| <.0001 * Lower CL Dif 10.0554 Prob > t <.0001 * Prob < t 1.0000 0.95 Confidence -15 -10 -5 0 5 10 15 Means and Std Deviations Std Err Level Number Mean Std Dev Mean Lower 95% Upper 95% A 10 46.4000 2.87518 0.9092 44.343 48.457 B 10 59.4000 3.37310 1.0667 56.987 61.813 3