Homework Set 8

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School

University of Florida *

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Course

3344

Subject

Mechanical Engineering

Date

Dec 6, 2023

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pdf

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Problem Set 8 Problems Completed: 14.1, 14.5, 14.8, 14.18, 15.3, 15.6, 15.15

Problem 14.1 Given the data: 0.90 1.42 1.30 1.55 1.63 1.32 1.35 1.47 1.95 1.66 1.96 1.47 1.92 1.35 1.05 1.85 1.74 1.65 1.78 1.71 2.29 1.82 2.06 2.14 1.27 Determine (a) the mean, (e) standard deviation, (f) variance, and (g) coefficient of variation. Solution: a) 1.6244 e) 0.339388 f) 0.115184 g) 20.89%

Problem 14.5 Use least-squares regression to fit a straight line Along with the slope and intercept, compute the standard error of the estimate and the correlation coefficient. Plot the data and the regression line. Then repeat the problem, but regress x versus y — that is, switch the variables. Interpret your results. Solution: ▪ MATLAB Code: see file: EGM3344_HW8_145a.m ▪ Output:

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- MATLAB Code: see file: EGM3344_HW8_145b.m - Output: y vs x x vs y Slope 0.3591455 2.486137 Y- Intercept 4.88812 -11.1349 Best fit equation y=4.88812+0.3591455x x =-11.1349 + 2.486137 y Standard error 0.851097 2.23927 Correlation coefficient 0.9449 0.9449 The x values are greater than y values when matrices are switched, so the standard error is greater for x vs y compared to y vs x.

Problem 14.8 Beyond the examples in Fig. 14.13, there are other models that can be linearized using transformations. For example, 4 4 x x y e   = Linearize this model and use it to estimate a4 and b4 based on the following data. Develop a plot of your fit along with the data. Solution: - MATLAB Code: see file: EGM3344_HW8_148.m - Output: Plot of linearized data with linear regression fit:

Plot of original data with nonlinear model fit: α 4 = 9.661786 β 4 = -2.4733

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Problem 14.11 Determine an equation to predict metabolism rate as a function of mass based on the following data. Use it to predict the metabolism rate of a 200-kg tiger. Solution: - MATLAB Code: see file: EGM3344_HW8_1411.m - Output: Linear Regression:

Nonlinear Regression: Equation using linear regression: y = 0.7266x + 0.5301 Equation using nonlinear regression: y = 3.3893x^0.7266

Problem 14.18 The following data show the relationship between the viscosity of SAE 70 oil and temperature. After taking the log of data, use linear regression to find the equation of the line that best fits the data and the r 2 value. Solution: - MATLAB Code: see file: EGM3344_HW8_1418.m - Output: Linear regression:

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Plot of untransformed data and regression fit: α 2 = 38,147.94 β 2 = -3.01338 r^2 = 0.9757

Problem 15.3 fit a cubic polynomial to the following data: Along with the coefficients, determine r 2 and S y/x . Solution: - MATLAB Code: see file: EGM3344_HW8_153.m - Output: y = -11.4887 + 7.143817x – 1.04121x^2 + 0.046676x^3 r^2 = 0.8290

Problem 15.6 Use multiple linear regression to derive a predictive equation for dissolved oxygen concentration as a function of temperature and chloride based on the data from Table P15.5. Use the equation to estimate the concentration of dissolved oxygen for a chloride concentration of 15 g/L at T = 12 °C. Note that the true value is 9.09 mg/L. Compute the percent relative error for your prediction. Explain possible causes for the discrepancy. Solution: - MATLAB Code: see file: EGM3344_HW8_156.m - Output: The error is considerable. The discrepancy is due to the dependence of oxygen concentration of the unknowns is nonlinear.

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Problem 15.15 Given the data use least-squares regression to fit (a) a straight line, (b) a power equation, (c) a saturation- growth-rate equation, and (d) a parabola. For (b) and (c), employ transformations to linearize the data. Plot the data along with all the curves. Is any one of the curves superior? If so, justify. Solution: (a) Linear Model - MATLAB Code: see file: EGM3344_HW8_1515a.m - Output:

(b) Power Model - MATLAB Code: see file: EGM3344_HW8_1515b.m - Output: (c) Saturation-growth rate model - MATLAB Code: see file: EGM3344_HW8_1515c.m - Output: