To propagate the uncertainty in a linear equation of the form $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept, uncertainties in both parameters can affect the results.

Here, your values are:

• Slope ($y$): $m = 0.006927703$
• Y-intercept ($c$): $c = 0.059646296$

Steps for Uncertainty Propagation

1. Determine the uncertainties in $m$ and $c$:

   • Are these uncertainties provided? If not, you'll need to know them to calculate the propagated uncertainty.

2. Propagating the total uncertainty: The total uncertainty in $y$, for a given $x$, is propagated using:

   $\sigma_y = \sqrt{\left( x^2 \cdot \sigma_m^2 \right) + \sigma_c^2 }$

   where:

   • $\sigma_m$ = uncertainty in slope ($m$)
   • $\sigma_c$ = uncertainty in y-intercept ($c$)
   • $x$ = the input variable

3. Plugging in values: If you provide the uncertainties $\sigma_m$ and $\sigma_c$, we can compute $\sigma_y$.

Example:

Let’s assume:

• Uncertainty in slope $\sigma_m = 0.0001$

• Uncertainty in y-intercept $\sigma_c = 0.001$

• For $x = 5$, uncertainty propagation becomes:

  $\sigma_y = \sqrt{(5^2 \cdot 0.0001^2) + (0.001^2)}$

  Computing this would yield:

  $\sigma_y = \sqrt{(25 \cdot 10^{-8}) + 10^{-6}} = \sqrt{3.5 \cdot 10^{-6}} = 0.00187$

4. The result for $y$ at $x = 5$ would include this uncertainty.

Let me know your specific uncertainties for $m$ and $c$, and I can refine this computation.

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Would you like to explore this further? Below are some related questions:

1. What are the actual uncertainties for $m$ and $c$ in your case?
2. Do you need a detailed derivation for this formula?
3. Is there a