How Distance, Pressure, and Speed Influence Thickness of Uniform Coating - Assignment Example

Surname Name of the institution Date of submission 1.0 Introduction Semiconductors (chips) are produced on a wafer that contains hundreds of chips. The wafer yield is defined to be the proportion of these chips that are acceptable for use and clearly control engineers aim to maximize this yield during manufacturing. This yield is greatest when the thickness of the coating material to the wafer is uniform. Control engineers are having difficulties producing a uniform coating (Y) at their plant. The main process variables that the engineers need to contend with are speed (X1), pressure (X2) and distance (X3). This report explained experimental design which was aimed at determining how these three factors that is distance, pressure and speed influence thickness of uniform coating. 2.0 Results and discussion The first question is to describe and establish the sources of variability present in each data set of Experiment 1. From the experiment 1, there are three sources of variability which can be easily identified. They include; Conditions or factor of interest Measurement process variability Experimental material By manipulating variables in a factor experimenter, is trying to create variability in the response variable (Cohen, Jacob, et al p34). This explains why an experiment takes place. In an experiment, the factor of interest is manipulated to form treatments hence the treatments are assigned to experimental units that brings variability. The experiment always hopes to induce variability in the response variable through manipulation of the factor (Cohen, Jacob, et al p34). The measurement process which can bring variability, the experiment doesn’t need this. The experimental materials resulting from uncontrolled outside factor they are not required by experimenter since they make experimental materials to be more homogeneous. On the similarity and differences in the two experiments, the two experiments are that both experiment used twenty trials in both cases, which is 20 observations in both cases. The degree of freedom in both cases is 19 each that is DF = 19. The two experiments are different in mean and standard deviation. The second question is to construct appropriate parametric and non-parametric tests to assess the claim that the uniformity in coating thickness is not the same for each speed (Seber, George & Alan p73). Parametric test can be defined as where the information about the population is completely known by means of its parameters. While non parametric are test where there is no knowledge about the population or parameters, but still it is required to test the hypothesis of the population. Parametric test T-test One-Sample Statistics N Mean Std. Deviation Std. Error Mean Low_Speed 20 86.5200 9.34877 2.09045 High_Speed 20 151.4850 10.67014 2.38592 One-Sample Test Test Value = 0 t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Low_Speed 41.388 19 .000 86.52000 82.1446 90.8954 High_Speed 63.491 19 .000 151.48500 146.4912 156.4788 The degree of freedom is 19 while significant p-value of 0.00 hence we conclude that the mean difference is statistically significantly different from 0. The second test is F-test F-Test Two-Sample for Variances   Low Speed High Speed Mean 86.52 151.485 Variance 87.39957895 113.8518684 Observations 20 20 df 19 19 F 0.767660471 P(F 87.39957895. If F >F Critical one-tail, we reject the null hypothesis in this case 0.767660471 > 0.461201089. Assumption in Parametric test The assumptions for F-test are that the variances of the compared populations are the same and the estimates of the population variance are independent (Hocking & Ronald p23) The data for parametric statistics follows a normal distribution. This is an indication that the data points have a mean or an average and standard deviation, the standardized dispersion in a data set which is one of the advantages of parametric data. For parametric data, where you can confidently say that the data come from a specified probability model, then parametric statistics will usually give you more information However, they can also lead to significantly biased conclusions if the wrong model is used giving major. Nonlinear data becomes linear through the use of data transformation in parametric statistics, such as using square roots of the data or creating logarithms (Hocking & Ronald p23) Parametric statistics incorporate data that is on a scale or set as a ratio, which makes mathematical manipulation of the data possible. Correlation, regression, t-tests and analysis of the variance are some of the popular parametric statistical techniques. These tests express the relationship between two or more variables On nonparametric analysis, we investigate chi-square Test Chi-Square Test The assumption on Chi-Square test include The sample size is much smaller the population size The sample is a representative for the target population Assumption of independent and identically distributed variable Frequencies Low_Speed Observed N Expected N Residual 69.20 1 1.0 .0 74.00 1 1.0 .0 74.40 1 1.0 .0 76.20 1 1.0 .0 78.90 1 1.0 .0 81.90 1 1.0 .0 82.10 1 1.0 .0 82.20 1 1.0 .0 87.30 1 1.0 .0 87.70 1 1.0 .0 87.80 1 1.0 .0 88.90 1 1.0 .0 89.40 1 1.0 .0 90.20 1 1.0 .0 90.30 1 1.0 .0 90.90 1 1.0 .0 95.50 1 1.0 .0 95.90 1 1.0 .0 102.70 1 1.0 .0 104.90 1 1.0 .0 Total 20 High_Speed Observed N Expected N Residual 135.90 1 1.0 .0 136.80 1 1.0 .0 141.10 1 1.0 .0 142.20 1 1.0 .0 143.10 1 1.0 .0 145.50 1 1.0 .0 146.10 1 1.0 .0 147.00 1 1.0 .0 149.20 1 1.0 .0 150.20 1 1.0 .0 150.30 1 1.0 .0 151.10 1 1.0 .0 152.30 1 1.0 .0 152.90 1 1.0 .0 154.70 1 1.0 .0 160.50 1 1.0 .0 161.40 1 1.0 .0 161.80 1 1.0 .0 168.70 1 1.0 .0 178.90 1 1.0 .0 Total 20 Test Statistics Low_Speed High_Speed Chi-Square .000a .000a df 19 19 Asymp. Sig. 1.000 1.000 Monte Carlo Sig. Sig. 1.000b 1.000b 95% Confidence Interval Lower Bound 1.000 1.000 Upper Bound 1.000 1.000 The p-value, denoted by “Asymp.Sig. (2-tailed)”, is 1.00. This means that there's 100% chance to find the observed (or a larger) degree of association between the variables if they're perfectly independent in the population. Disadvantages of nonparametric test Nonparametric methods are not so efficient as of parametric test No nonparametric test available for testing the interaction in analysis of variance model It can be applied for nominal or ordinal scale For any problem, if any parametric test exists it is highly powerful The advantages of nonparametric test are; It will not involve complicated sampling theory No assumption is made regarding the parent population They are simple and easy to understand This method is only available for nominal scale data The third question investigates the multiple least squares; estimate the  parameters of the following second order response surface model: Where Y is the standard deviation in coating thickness, X1 is the speed, X2 is the pressure and X3 is the distance.  is the prediction error or residual. We run OLS regression on the equation above, the results are shown below starting with the model summary Model summary Model R R Square Adjusted R Square Std. Error of the Estimate Change Statistics R Square Change F Change df1 1 .900a .810 .785 105.53202 .810 32.643 3 Model Change Statistics df2 Sig. F Change 1 23a .000 a. Predictors: (Constant), Distance_X3, Pressure_X2, Speed_X1 From the model, the table tells informs of what % of variability in the Dependent Variable is accounted for by all of the Independent Variable together (it’s a multiple R-square). The footnote on this table tells you which variables were included in this equation (in this case, all three of the ones that we put in). From that the variability of dependent variable that is coating thickness counted in the study by independent variables is 78.5%. The predictors counted include distance, pressure and speed. ANOVA Model Sum of Squares df Mean Square F Sig. 1 Regression 1090639.825 3 363546.608 32.643 .000b Residual 256151.151 23 11137.007 Total 1346790.976 26 a. Dependent Variable: Standard_Deviation_in_coating_thickness_Y b. Predictors: (Constant), Distance_X3, Pressure_X2, Speed_X1 This table gives you an F-test to determine whether the model is a good fit for the data. According to this p-value, it is since 0.000< 0.05 meaning it is perfect good fit. The lower the significance level the fit the model since the significance level is 0.000, it shows that the model is fit. Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 314.670 20.310 15.494 .000 Speed_X1 177.011 24.874 .647 7.116 .000 Pressure_X2 109.422 24.874 .400 4.399 .000 Distance_X3 131.472 24.874 .481 5.285 .000 Model 95.0% Confidence Interval for B Lower Bound Upper Bound 1 (Constant) 272.657 356.684 Speed_X1 125.555 228.467 Pressure_X2 57.966 160.878 Distance_X3 80.016 182.928 From the Unstandardized coefficient, we can derive the following equation Y = 314.670 + 177.011X1 + 109.422X2 +131.472X3 All the variables, are statistically significant since p-value in the three variables that is speed, pressure and distance is 0.00< 0.05. From the equation, assuming other factors constant, a unit increase in speed will cause 177.011 increases in coating thickness, a unit increase in pressure will cause 109.422 increases in coating thickness while a unit increase distance will cause 31.472 increases in coating thickness. With all other factors constant, the coating thickness will be 314.670. In the regression analysis above, we assumed that variables have normal distributions. Furthermore we Assumption of a Linear Relationship between the Independent and Dependent Variable In question four, different type of regression, is not possible since all the variables are statistically significance. Therefore we are going to repeat the same results but with scattered graph. The result is shown below. Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate Change Statistics R Square Change F Change df1 1 .900a .810 .785 105.53202 .810 32.643 3 Model Change Statistics df2 Sig. F Change 1 23a .000 a. Predictors: (Constant), Distance_X3, Pressure_X2, Speed_X1 b. Dependent Variable: Standard_Deviation_in_coating_thickness_Y The significance of the variables still remain constant using the ANOVA test Model Sum of Squares df Mean Square F Sig. 1 Regression 1090639.825 3 363546.608 32.643 .000b Residual 256151.151 23 11137.007 Total 1346790.976 26 Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 314.670 20.310 15.494 .000 Speed_X1 177.011 24.874 .647 7.116 .000 Pressure_X2 109.422 24.874 .400 4.399 .000 Distance_X3 131.472 24.874 .481 5.285 .000 The three variables are still statistically significance as the p-value Read More