$($Taylor $1997,$ pp$.182-8,$ or Barford $1985,$ Section $3\mathrm{.}3).$ The results are:

$c=\frac{{\displaystyle \underset{i}{\overset{?}{}}}{x}_i^2{\displaystyle \underset{i}{\overset{?}{}}}{y}_i-{\displaystyle \underset{i}{\overset{?}{}}}{x}_i{\displaystyle \underset{i}{\overset{?}{}}}{x}_i{y}_i}{}$

and

$m=\frac{N{\displaystyle \underset{i}{\overset{?}{}}}{x}_i{y}_i-{\displaystyle \underset{i}{\overset{?}{}}}{x}_i{\displaystyle \underset{i}{\overset{?}{}}}{y}_i}{}$

with the error in the intercept,

${}_c={}_{CU}\sqrt[\phantom{2}]{\frac{{\displaystyle \underset{i}{\overset{?}{}}}{x}_i^2}{}}$

and the error in the gradient,

${}_m={}_{CU}\sqrt[\phantom{2}]{\frac{N}{}}$

where ${?}^4$

$=N{\displaystyle \underset{i}{\overset{?}{}}}{x}_i^2-({\displaystyle \underset{i}{\overset{?}{}}}{x}_i{)}^2$

and the so$-$called common uncertainty ${}_{CU}$ is defined as

${}_{CU}=\sqrt[\phantom{2}]{\frac{1}{N-2}{\displaystyle \underset{i}{\overset{?}{}}}(y_i-m{x}_i-c{)}^2}.$

Suppose that measurements are made of an object$-$lens$-$image system which should follow

$\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$

rearranging this gives

$\frac{1}{p}=-\frac{1}{q}+\frac{1}{f}$

which means that a graph of $\frac{1}{p}$ vs $\frac{1}{q}$ should have a slope of $-1$ and a y$-$intercept of $\frac{1}{f}.$

Complete the code cell below to calculate the best fit slope and y$-$intercept as well as their uncertainties for an object$-$lensthe equations on page $58.$ Report your answer including units $($if applicable$)$ to the correct number of digits.

import numpy as np

# Object and image distances and uncertainties

$p=np.$array$([0\mathrm{.}8000,0\mathrm{.}5000,0\mathrm{.}3000,0\mathrm{.}2000,0\mathrm{.}1800])$ # meters

punc $=$ np$.$ array$([0\mathrm{.}0005,0\mathrm{.}0005,0\mathrm{.}0005,0\mathrm{.}0005,0\mathrm{.}0005])$ # meters

$q=np.$array$([0\mathrm{.}163,0\mathrm{.}182,0\mathrm{.}242,0\mathrm{.}40,0\mathrm{.}52])$ # meters

qunc $=$ np$.$array $,$ of$.\mathrm{.}01,0\mathrm{.}01$ meters

# Define $x$ and $y$ values

$x=\frac{1}{q}$

$y=\frac{1}{p}$

# Calculate the uncertainty in $x$ and $y$ using the calculus method $dxdq=-\frac{1}{q}****2$

x$\_$err $=$ np$.$sqrt $(($dxdq$*$qunc$)****2$

# Fill in code to calculate the uncertainty in $y$ and assign to a

# variable called y$\_$err

# Calculate the best fit slope and $y-$intercept using the equations

# on pg $58$ of Huges & Hase

# Fill in the code to perform these calculations. I

# Print the results to the screen print$(\text{'}$Slope $=\{$:$\mathrm{.}2f\}$ lu$00$B$1\{$:$\mathrm{.}2f\}\text{'}.$ format $($m$,$ alpham$))$

print $(\text{'}$y$-$int $=\{$:$\mathrm{.}2f\}$ lu$00$B$1\{$:$\mathrm{.}2f\},$ format $,$ alphac$))$

# Modify the print commands to display the correct number of digits.