Answer
The linear least-squares fit:
$f\left( x \right) = 1.963x - 1.552$
Work Step by Step
Let $x$ and $y$ represent the current and the laser power, respectively.
So, we list and evaluate the data points in the following table:
$\begin{array}{*{20}{c}}
{{\rm{Sum}}}&{{\rm{Current}}}&{{\rm{Laser{\ }power}}}&{}&{}\\
{}&{{x_j}}&{{y_j}}&{{x_j}^2}&{{x_j}{y_j}}\\
{}&{1.}&{0.52}&{1.}&{0.52}\\
{}&{1.1}&{0.56}&{1.21}&{0.62}\\
{}&{1.2}&{0.82}&{1.44}&{0.98}\\
{}&{1.3}&{0.78}&{1.69}&{1.01}\\
{}&{1.4}&{1.23}&{1.96}&{1.72}\\
{}&{1.5}&{1.5}&{2.25}&{2.25}\\
\sum &{7.5}&{5.41}&{9.55}&{7.106}
\end{array}$
In this exercise $n=6$.
Substituting the values from the table above in the two equation from Exercise 56:
$m\left( {\sum\limits_{j = 1}^n {{x_j}} } \right) + bn = \sum\limits_{j = 1}^n {{y_j}}$
$m\sum\limits_{j = 1}^n {{x_j}^2} + b\sum\limits_{j = 1}^n {{x_j}} = \sum\limits_{j = 1}^n {{x_j}} {y_j}$
we obtain
$7.5m + 6b = 5.41$
$9.55m + 7.5b = 7.106$
Solving the last two equations we obtain $m \simeq 1.963$ and $b \simeq - 1.552$.
So, the linear least-squares fit is
$f\left( x \right) = mx + b$
$f\left( x \right) = 1.963x - 1.552$
