Statistics in Engineering - Assignment Example

STUDENT’S NAME: ADMISSION NO: NAME OF THE INSTITUTION: Question one From experiment one, a closer look of the data from the first experiment one can easily notice that the values of the numbers are not the same, even though they were done under the same condition (Go, et al 2013). Variance can be defined as a statistical principle, which measures the variability within the data. In experiment 1, some sources of variability include, Systematic variability: - this is a variability which comes as result of manipulations of an independent variables and the treatment variability in the experiment condition (Go, et al 2013). Error Variability: - this is attributed to the effect of extraneous variables in the data. Ideally, this error is randomly distributed across conditions in an experiment. Error variability is normally evaluated and controlled within the statistical analysis of the data. There are individual differences within the data condition (Go, et al 2013). Confounded variable: - these are variables resulting from extraneous variables that are correlated with manipulations of an independent variable in an experiment (Cressie 2015) The similarities in 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. In summary, the variability in the experiment 1 can be established using F-statistics in investigating the sources of variability (Cressie 2015). For experiment 1, there are two major sources of variation. First, there are 20 numbers in the first experiment and all are different hence the first source of variability within the table. Secondly, there are three different variations which can be calculated, they include; Variation between groups Variation within groups Total variation Question two In situation where the information about the population is completely known by means of its parameters then that kind of statistical test is called parametric test (Huber 2011). Example of parametric test includes t-test, f-test, z-test and ANOVA. ANOVA test In conducting ANOVA test, one of the assumptions is that the populations tested have the same variance It compares means between three or more distinct/independent groups SUMMARY Groups Count Sum Average Variance Low Speed 20 1730.4 86.52 87.39957895 High Speed 20 3029.7 151.485 113.8518684 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 42204.51225 1 42204.51225 419.4207078 3.9531E-22 4.098171661 Within Groups 3823.7775 38 100.6257237 Total 46028.28975 39         From the result, p-value is3.9531E-22 < 0.05 hence the effect is statistically significant. This implies that the claim that the uniformity in coating thickness is not the same for each speed is not true hence rejecting the null hypothesis and accepting alternative hypothesis which can now be stated as uniformity in coating thickness is the same at the speed (Huber 2011). This can be further be explained using F-test the SPSS output is given below 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. 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. 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 (Huber 2011). 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. Parametric statistics incorporate data that is on a scale or set as a ratio, which makes mathematical manipulation of the data possible. Due to the requirements to use parametric statistics, researchers who employ these techniques retain more validity within the results. 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 (Venables & Ripley 2013). Nonparametric If there is no knowledge about the population or parameters, but still it is required to test the hypothesis of the population, then in such circumstances, it is called non-parametric test. Some examples include mann-whitney, rank sum test, Kruskal-Wallist test (Huber 2011). We analyze the one sample One-Sample Kolmogorov-Smirnov Test One-Sample Kolmogorov-Smirnov Test Low_Speed High_Speed N 20 20 Normal Parametersa,b Mean 86.5200 151.4850 Std. Deviation 9.34877 10.67014 Most Extreme Differences Absolute .133 .147 Positive .120 .147 Negative -.133 -.072 Kolmogorov-Smirnov Z .596 .659 Asymp. Sig. (2-tailed) .870 .779 a. Test distribution is Normal. b. Calculated from data. One-Sample Kolmogorov-Smirnov Test 2 Low_Speed High_Speed N 20 20 Uniform Parametersa,b Minimum 69.20 135.90 Maximum 104.90 178.90 Most Extreme Differences Absolute .192 .313 Positive .192 .313 Negative -.107 -.050 Kolmogorov-Smirnov Z .859 1.399 Asymp. Sig. (2-tailed) .451 .040 a. Test distribution is Uniform. b. Calculated from data. From the result, the two samples were tested with similar population. Since p0.870> 0.05 we conclude that the two groups were sampled from population with different distribution with second experiment having the largest distribution compared with the first experiment. The advantages of nonparametric test are; They are simple and easy to understand It will not involve complicated sampling theory No assumption is made regarding the parent population and lastly This method is only available for nominal scale data Demerits include It can be applied for nominal or ordinal scale For any problem, if any parametric test exists it is highly powerful Nonparametric methods are not so efficient as of parametric test No nonparametric test available for testing the interaction in analysis of variance model Question three Multiple least squares, estimate the  parameters of the following second order response surface model From the multiple regressions, the results are shown below; Descriptive Statistics Mean Std. Deviation N Standard_Deviation_in_coating_thickness_Y 314.6704 227.59537 27 Speed_X1 .0000 .83205 27 Pressure_X2 .0000 .83205 27 Distance_X3 .0000 .83205 27 The multiple least squares, descriptive statistics shows that the mean value of the standard deviation in coating thickness is 314.6704 with deviation of 227.59537. SpeedX1, PressureX2 and DistanceX3 have similar mean of 0.000 and similar standard deviation of 0.83205 and same sample size of 27. Correlation Standard_Deviation_in_coating_thickness_Y Speed_X1 Pressure_X2 Pearson Correlation Standard_Deviation_in_coating_thickness_Y 1.000 .647 .400 Speed_X1 .647 1.000 .000 Pressure_X2 .400 .000 1.000 Distance_X3 .481 .000 .000 Sig. (1-tailed) Standard_Deviation_in_coating_thickness_Y . .000 .019 Speed_X1 .000 . .500 Pressure_X2 .019 .500 . Distance_X3 .006 .500 .500 N Standard_Deviation_in_coating_thickness_Y 27 27 27 Speed_X1 27 27 27 Pressure_X2 27 27 27 Distance_X3 27 27 27 Distance_X3 Pearson Correlation Standard_Deviation_in_coating_thickness_Y .481 Speed_X1 .000 Pressure_X2 .000 Distance_X3 1.000 Sig. (1-tailed) Standard_Deviation_in_coating_thickness_Y .006 Speed_X1 .500 Pressure_X2 .500 Distance_X3 . N Standard_Deviation_in_coating_thickness_Y 27 Speed_X1 27 Pressure_X2 27 Distance_X3 27 The “descriptive” command also gives you a correlation matrix, showing you the Pearson rs between the variables (in the top part of this table). From the analysis, correlation coefficient of distance is 1 which shows positive relationship between distance and coating thickness. Pressure and speed also gives coefficient of 1 showing strong positive correlation between speed, pressure and coating thickness. 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 Question 4 The new model will be similar with the first model due to the fact that all variables are statistically significance hence all will be included in the linear regression the model summary 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

Pressure and speed also gives coefficient of 1 showing strong positive correlation between speed, pressure and coating thickness. 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 Question 4 The new model will be similar with the first model due to the fact that all variables are statistically significance hence all will be included in the linear regression the model summary 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

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