Chapter 2, Problem 2.49P

Interpretation Introduction

(a)

Interpretation:

The given data for system temperature versus time is to be fit into a straight-line using the method of least squares. Also, the formula for $\text{T}\left(t\right)$ is to be derived and converted to $t\left(\text{T}\right)$.

Concept introduction:

Method of least squares:

If $\left(x_i,y_i\right)$ are the points plotted for $\left(i=1,2,\dots ,n\right)$, then $R_i$ is the residual for each point corresponding to $i$.

To fir the data into the straight-line plot, the value of $\varphi \left(a,b\right)$ should be minimized.

$\varphi \left(a,b\right)=\sum _{i=1}^n{R}_i{}^2=\sum _{i=1}^n{\left(y_i-\text{a}{x}_i-\text{b}\right)}^2$

Perform differentiation on equation (1) to calculate the values of $\text{a and b}$, thus values defined are:

$\begin{array}{c}{S}_x=\frac{1}{n}\sum _{i=1}^n{x}_i\\ S_{xx}=\frac{1}{n}\sum _{i=1}^n{x}_i{}^2\\ S_y=\frac{1}{n}\sum _{i=1}^n{y}_i\\ S_{xy}=\frac{1}{n}\sum _{i=1}^n{x}_i{y}_i\end{array}$ ......... (1)

Now, use these values to calculate $\text{a and b}$ as:

$\text{a}=\frac{S_{xy}-S_x{S}_y}{S_{xx}-{\left(S_x\right)}^2}$ ......... (2)

And,

$\text{b}=\frac{S_{xx}{S}_y-S_{xy}{S}_x}{S_{xx}-{\left(S_x\right)}^2}$ ......... (3)

The straight-line that fits into the given data is now,

$y=\text{a}x+\text{b}$

With $\text{a and b}$ being substituted as calculated from equations (2) and (3).

Interpretation Introduction

(b)

Interpretation:

The given data is to be plotted on the rectangular axes and the expression for $\text{T}\left(t\right)$ is to be determined. Also, the value of $R^2$ is to be checked, if it is closer to 1.

Concept introduction:

Given an equation, $y=\text{a}x+\text{b}$, it represents a straight-line equation for the expression $y\left(x\right)$.

Here, $\text{a}$ is the slope of the line and $\text{b}$ is the y- intercept.

According to the method of least square used to fit data into a straight-line, $R^2$ is the square of the residual. For the best fit, $R^2$ should be closer to 1.