Weighted lest square fit

Homework Statement

I have a project for one of my class and I have been given a sheet to do the statistical analyst of my data. I am not convince this sheet is proper and I need someone to look over it it. I don't understand where my Delta R goes...

Homework Equations

$$\chi^2 =-\frac{1}{2} \sum_{i=1}^{N}{\left(\frac{y_i-ax_i}{\sigma_i^2}\right)^2} \\ \frac{\partial \chi^2}{\partial a}$$
$$= \sum_{i=1}^{N}{\frac{x_i}{\sigma_i^2}(y_i-ax_i)} = 0 \\ \sum_{i=1}^{N}{\frac{x_i y_i}{\sigma_i^2}}$$
$$= a\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} \\ a= \sum_{i=1}^{N}{\frac{x_iy_i}{x_i^2}} \\ \sigma_a^2 = \sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} S^2 \\ S^2$$

$$= \frac{1}{N-1}\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}} \\ \sigma_a^2 =$$
$$\frac{1}{N-1}\left(\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}}\right)\left(\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}}\right) \\ r_n^2 = \left(n-\frac{1}{2}\right)\lambda R \\ a= \sum_{n=1}^{15}{\frac{r_n^2\left(n-\frac{1}{2}\right)}{(r_n^2)^2}} \\ \sigma_a^2$$
$$= \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2}{\sigma_r^2}}\right)\left(\sum_{i=1}^{N}{\frac{(r_n^2-a\left(n-\frac{1}{2}\right))^2}{\sigma_r^2}}\right) \\ \Psi(x) = x^2 \\ \sigma_\Psi^2 = (\frac{\partial}{\partial x}(x^2)\Delta x)^2$$$$= 4x^2(\Delta x)^2 \\ \sigma_{r_n^2} = 4r_n^2(\Delta r)^2 \\ \sigma_a^2 = \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2}{4r_n^2}}\right)/\left(\sum_{i=1}^{N}{\frac{(r_n^2-a\left(n-\frac{1}{2}\right))^2}{4r_n^2}}\right)$$

The Attempt at a Solution

The error on r comes from the fact it's a measurement and we square it. I think Delta R should be there somewhere even though it is constant...

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PDF of latex file.

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