Solved Homework 7 for Statistics for Engineering | STAT 4706, Assignments of Statistics

Download Solved Homework 7 for Statistics for Engineering | STAT 4706 and more Assignments Statistics in PDF only on Docsity! 1 Stat 4706 HW 7 Solutions Total Points Possible:46 Instructions: Answer the questions. Turn in Minitab output along with your work. You must write out answers to the problems – don’t just highlight the output. The following sample data was collected to determine the relationship between processing variables (diffusion time and sheet resistance) and the current gain of a transistor in the integrated circuit. a) Identify the 3 matrices used to generate multiple linear regression coefficients if we were predicting current gain using the other two variables. y = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 6.12 5.6 2.7 1.8 1.9 8.10 8.9 4.7 8.7 3.5 b = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 2 1 0 b b b Diffusion Time (Hours) Sheet Resistance ( Ω - cm) Current Gain 1.5 66 5.3 2.5 87 7.8 0.5 69 7.4 1.2 141 9.8 2.6 93 10.8 0.3 105 9.1 2.4 111 8.1 2.0 78 7.2 0.7 66 6.5 1.6 123 12.6 Value of part a: 6 points; 2 points per matrix 2 X = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 1236.11 667.01 780.21 1114.21 1053.01 936.21 1412.11 695.01 875.21 665.11 b) Using the matrices identified in part a), find the multiple linear regression coefficients. You may use a calculator or software such as Matlab to do this. List the matrices that you generate in the process: (X’X)= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 941319.1458939 9.145885.293.15 9393.1510 (X’X)-1= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −− −− −− −− − 44 4 700050851.897432141.50150657604. 897432141.51572804381.1855438825. 0150657604.1855438825.798557039.1 ee e (X’X)-1X’=> too big to show here …. Matrix with 3 rows and 10 columns (X’X)-1X’y= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 0623044565. 2246501989. 265896727.2 Note that the apostrophe is another way of indicating “transpose”. c) Using Minitab, generate the multiple linear regression output using current gain as the response variable. List the model. Current Gain = 2.27 + 0.225 Diffusion Time + 0.0623 Sheet Resistanc d) Test to see if the model is useful in predicting current gain. Hypotheses: Ho: β1 = β2 = 0 Η1: At least 1 differs from 0. Test Statistic: F=4.72 p-value: .05 Decision: Since p-value equals alpha (.05), we will reject H0. Conclusion: Model is useful in predicting current gain. Value of part b: 2 points; -2 if final matrix is not correct since comparison against Minitab can be made Value of part c: 2 points; Value of part d: 5 points; 1 point per each part of hypothesis test; since pvalue=alpha and critical region approximates F, I will take either fail to reject or reject as correct 5 j) Fit the better of the two simple linear regression models (current gain as response). List the model. Current Gain = 2.53 + 0.0631 Sheet Resistance k) Test to see if the model in (j) is useful in predicting current gain. Hypotheses: Ho: β1 = 0 Η1: β1 does not equal 0 Test Statistic: F=10.46 p-value: .012 Decision: reject H0. Conclusion: Model is useful in predicting current gain. Analysis of Variance Source DF SS MS F P Regression 1 23.758 23.758 10.46 0.012 Residual Error 8 18.166 2.271 Total 9 41.924 l) Generate plots to see if there are problems with the assumptions of normality and constant variance for the model in (j). Indicate any problems. Fitted Value St an da rd iz ed R es id ua l 1211109876 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 Residuals Versus the Fitted Values (response is Current Gain) Value of part j: 2 points; -1 if identified as model 1 but fitted model not listed Value of part k: 3 points; 1/2 point per each part of hypothesis test; Value of part l: 2 points; 1 points per each graph; 6 SRES2 Pe rc en t 3210-1-2-3 99 95 90 80 70 60 50 40 30 20 10 5 1 Mean -0.03076 StDev 1.082 N 10 AD 0.465 P-Value 0.197 Probability Plot of SRES2 Normal m) Fit the other simple linear regression model that you didn’t use in (j). [In other words, fit the worse of the two simple linear regression models.] List the model. Current Gain = 7.79 + 0.440 Diffusion Time n) Conduct a hypothesis test for the strength of the relationship for the model in (m). List the hypotheses, test statistic, pvalue, decision and conclusion. From Minitab: R-Sq = 3.0% Rejection Region: Decision: Conclusion: Value of part m: 2 points; -1 if identified as model 2 but fitted model not listed Value of part n: 5 points; 1 point per each part of hypothesis test; -2.5 points if test for significance of variable is given since question asks for strength of relationship