A straight line is drawn through the points $\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{87}\mathbf{\times }$${\mathbf{10}}^{\mathbf{4}}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{-}\mathbf{12}\mathbf{.}\mathbf{99}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{-}\mathbf{25}\mathbf{.}\mathbf{93}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{-}\mathbf{38}\mathbf{.}\mathbf{89}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{)}$, and $\mathbf{(}\mathbf{40}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{-}\mathbf{51}\mathbf{.}\mathbf{96}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{)}$ to give $\mathbf{m}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{29872}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}$, $\mathbf{b}\mathbf{=}\mathbf{256}\mathbf{.}\mathbf{695}\mathbf{,}{\mathbf{u}}_{\mathbf{m}}\mathbf{=}\mathbf{13}\mathbf{.}\mathbf{190}\mathbf{,}{\mathbf{u}}_{\mathbf{b}}\mathbf{=}\mathbf{323}\mathbf{.}\mathbf{57}$, and ${\mathbf{s}}_{\mathbf{y}}\mathbf{=}\mathbf{392}\mathbf{.}\mathbf{9}$. Express the slope and intercept and their uncertainties with reasonable significant figures.

The standard deviation of $\mathbf{y}\mathbf{:}{\mathbf{s}}_{\mathbf{y}}\mathbf{:}$

The standard deviation of y is mathematically represented as

${\mathbf{\sigma }}_{\mathbf{y}}\mathbf{\approx }{\mathbf{s}}_{\mathbf{y}}\mathbf{=}\sqrt{\frac{\mathbf{\sum }\mathbf{\left(}{{\mathbf{d}}_{\mathbf{i}}}^{\mathbf{2}}\mathbf{\right)}}{\mathbf{n}\mathbf{-}\mathbf{2}}}$

The standard uncertainty of a slope and an intercept:

The standard uncertainty of a slope is mathematically presented as${\mathbf{u}}_{\mathbf{m}}^{\mathbf{2}}\mathbf{=}\frac{{\mathbf{s}}_{\mathbf{y}}^{\mathbf{2}}\mathbf{n}}{\mathbf{D}}$

The standard uncertainty of an intercept is mathematically presented as ${\mathbf{u}}_{\mathbf{b}}^{\mathbf{2}}\mathbf{=}\frac{{\mathbf{s}}_{\mathbf{y}}^{\mathbf{2}}\mathbf{\sum }\mathbf{\left(}{\mathbf{x}}_{\mathbf{i}}^{\mathbf{2}}\mathbf{\right)}}{\mathbf{D}}$

Given data:

The points of the straight line are:

$$\begin{gathered} (3.0,-3.87\times {10}^4),\hspace{0.33em}(10.0,-12.99\times {10}^4),\hspace{0.33em}(20.0,-25.93\times {10}^4),\hspace{0.33em}(30.0,-38.89\times {10}^4), \\ and(40.0,-51.96\times {10}^4). \end{gathered}$$

The slope$m=-1.29872\times {10}^4$

The intercept $b=256.695$

The uncertainty in the slope$u_m=13.190$

The uncertainty in the intercept $u_b=323.57$

The standard deviation of Y is $s_y=392.9$

The slope with its uncertainty is written as:

slope $=-1.29872\times {10}^4(\pm 0.0013190\times {10}^4).$

$=-1.2987(\pm 0.{001}_3)\times {10}^4$(The subscript digits are insignificant figures)

$=-1.299(\pm 0.001)\times {10}^4$(The corrected significant figure).

The intercept with its uncertainty is written as:

intercept$=256.695(\pm 323.57)$

$=3(\pm 3)\times {10}^2$(The corrected significant figure).