L'i Problem C.1 Modelling with the x--statistics (30 Marks) We have N = measurements (eg: observations) of a quantity y with an uncertainty o as a function of x, which is known exactly. i 1 2 3 4 5 8.0 11.0 15.0 19.0 20.5 Yi 2.10 2.25 2.50 2.25 2.60 0 0.15 0.20 0.20 0.35 0.15 Note that i labels the measurements, and o is the uncertainty (standard deviation) on y. In other words, quoting the 1 -o confidence level, the observation for the measurement i is yi toi. We want to model these data, find the parameters of the model and their uncertainty. We also want to assess the quality of the model. To this aim, we use the x distribution, which we define as x = (Yi - m;) i=5 i=1 where m is a chosen model (so here m; might be a function of ri). Questions i) Explain the denominator in the definition of x given above. (2 marks) ii) Given some data and model, explain the purpose(s) of the x? distribution (in a few sentences). (4 marks) iii) Given the data above: (a) give two sensible models, (4 marks) (b) give the best values of the parameters (for both models), (2 marks) (c) give an estimate of the uncertainties on the parameters. (3 marks) iv) Give the corresponding values of x?, and then normalise by the number of degrees of freedom. (5 marks) v) Discuss the models that have you chosen. (10 marks) Notes: The problem can be solved numerically using a computer software of your choice. If you decide to do so, you are requested to submit a copy of your code(s) and some plots together with your numerical results. If you prefer to solve it by hand, you are required to include the details of your computation and to show a graph (or a sketch) of your findings. a