Quiz Chapter 06.04: Nonlinear Models for Regression

1. When using the transformed data model to find the constants of the regression model $y = ae^{bx}$ to best fit $\left( x_{1}, y_{1} \right), \, \left( x_{2}, y_{2} \right), \, ... \, , \, \left( x_{n}, y_{n} \right)$ , the sum of the square of the residuals that is minimized is

\, \displaystyle\sum_{i = 1}^{n} \left( y_{i} - ae^{bx_{i}} \right)^{2}

\, \displaystyle\sum_{i = 1}^{n} \left( \ln \left( y_{i} \right) - \ln a - bx_{i} \right)^{2}

\, \displaystyle\sum_{i = 1}^{n} \left( y_{i} - \ln a - bx_{i} \right)^{2}

\, \displaystyle\sum_{i = 1}^{n} \left( \ln \left( y_{i} \right) - \ln a - b \ln \left( x_{i} \right) \right)^{2}

2. It is suspected from theoretical considerations that the rate of water flow from a firehouse is proportional to some power of the nozzle pressure. Assume pressure data is more accurate. You are transforming the data

Flow rate, $F$ (gallons/min)	$96$	$129$	$135$	$145$	$168$	$235$
Pressure, $p$ (psi)	$11$	$17$	$20$	$25$	$40$	$55$

The exponent of the power of the nozzle pressure in the regression model $F = ap^{b}$ is most nearly

0.497

0.556

0.578

0.678

3. The transformed data model for the stress-strain curve $\sigma = K_{1} \varepsilon e^{-K_{2} \varepsilon}$ for concrete in compression, where $\sigma$ is the stress and $\sigma$ is the strain is

\ln \sigma = \ln k_{1} + \ln \varepsilon - k_{2} \varepsilon

\ln \dfrac{\sigma}{\varepsilon} = \ln k_{1} - k_{2} \varepsilon

\ln \dfrac{\sigma}{\varepsilon} = \ln k_{1} + k_{2} \varepsilon

\ln \sigma = \ln \left(k_{1} \varepsilon \right) - k_{2} \varepsilon

4. In nonlinear regression, finding the constants of the model requires solving simultaneous nonlinear equations. However in the exponential model $y = ae^{bx}$ that is best fit to $\left( x_{1}, y_{1} \right), \, \left( x_{2}, y_{2} \right), \, ... \, , \, \left( x_{n}, y_{n} \right)$ , the value of $b$ can be found as a solution of a nonlinear equation. That equation is given by

\, \displaystyle\sum_{i =1}^{n} y_{i} x_{i} e^{bx_{i}} - \displaystyle\sum_{i = 1}^{n} y_{i} e^{bx_{i}} \displaystyle\sum_{i = 1}^{n} x_{i} = 0

\, \displaystyle\sum_{i = 1}^{n} y_{i} x_{i} e^{bx_{i}} - \dfrac{\displaystyle\sum_{i = 1}^{n} y_{i} e^{bx_{i}}}{\displaystyle\sum_{i = 1}^{n} e^{2bx_{i}}} \displaystyle\sum_{i = 1}^{n} x_{i} e^{2bx_{i}} = 0

\, \displaystyle\sum_{i = 1}^{n} y_{i} x_{i} e^{bx_{i}} - \dfrac{\displaystyle\sum_{i = 1}^{n} y_{i} e^{bx_{i}}}{\displaystyle\sum_{i = 1}^{n} e^{2bx_{i}}} \displaystyle\sum_{i = 1}^{n} e^{bx_{i}} = 0

\, \displaystyle\sum_{i = 1}^{n} y_{i} e^{bx_{i}} - \dfrac{\displaystyle\sum_{i = 1}^{n} y_{i} e^{bx_{i}}}{\displaystyle\sum_{i = 1}^{n} e^{2bx_{i}}} \displaystyle\sum_{i = 1}^{n} x_{i} e^{2bx_{i}} = 0

5. There is a functional relationship between the mass density $\rho$ of air and the altitude $h$ above the sea level

Altitude above sea level, $h$ (km)	$0.32$	$0.64$	$1.28$	$1.60$
Mass Density, $\rho$ , (kg/m³)	$1.15$	$1.10$	$1.05$	$0.95$

In the regression model $\rho = k_{1} e^{k_{2}h}$ , the constant $k_{2}$ is found as $k_{2} = 0.1315$ . Assuming the mass density of air at the top of the atmosphere is $1/1000$ ^th of the mass density of air at sea level. The altitude in kilometers of the top of the atmosphere is most nearly

46.2

46.6

49.7

52.5

6. A steel cylinder at $80^{\circ}$ F of length $12"$ is placed in a commercially available liquid nitrogen bath at $315^{\circ}$ F. If the thermal expansion coefficient of steel behaves as a second order polynomial of temperature and the polynomial is found by regressing the data below

Temperature ( $^{\circ}$ F)	Thermal expansion coefficient ( $\mu$ in/in/ $^{\circ}$ F)
$-320$	$2.76$
$-240$	$3.83$
$-160$	$4.72$
$-80$	$5.43$
$0$	$6.00$
$80$	$6.47$

The reduction in length of the cylinder in inches is most nearly

0.0219

0.0231

0.0235

0.0307