Data Analysis Homework

Spring 2004

Assignment

1. Assuming a model of $y=A+Bcost$, find the best fit values for A and B using $\chi 2$ minimization. Set up the algebra and solve the resulting linear equations. You can solve the equations ``by hand,'' but I would suggest using a symbolic algebra program such as maxima, maple, or mathmatica.

   $t$ (deg)	$cost$	y	$\sigma$
   22.5	0.924	0.490	0.01
   112.5	-0.383	0.465	0.01
   202.5	-0.924	0.435	0.01
   292.5	0.383	0.450	0.01

2. Minimize $\chi 2$ for a non-linear set of data.
   • Take the data set (1000 points) from the web page. The first column corresponds to an energy in MeV, and the second column is a number of counts found at that energy (hint: the data is Poisson distributed).
   • Using the model $f(E)\; =\; f$₀ + f₁ exp((E-E₀)²/2σ²) :
     • Estimate value of $f$₀ , $f$₁ , $E$₀ , and $\sigma$.
     • Estimate the uncertainty for $f$₀ , $f$₁ , $E$₀ , and $\sigma$.
     • How well does the model fit the data?

The purpose of this assignment is to understand how to fit data using $\chi 2$ minimization. You will need to find an analysis package, or write a program to perform the fit in part 2. You may find the lecture notes helpful.