# Uncertainty consistency?

1. The problem statement, all variables and given/known data

I am given a set of x and y values x: (1,2,3,etc.) y: (1.2,2.2,3.1,etc.) with a given uncertainty and am asked

a) find the best fit

b) at what prob can you rule out a 5% higher slope

c) is the stated uncertainty consistent with the data?

I can find the best fit relatively easily by minimizing chi-sqd and setting the derivatives to 0. I am confident in my result as it matches with the graph given by excel.

For part 2 I tried putting a higher value for the given slope into the chi-sqd equation and checking a chart but that didn't give me a reasonable answer (not completely sure about the degree of freedom)

I am not sure how to approach the third part. I have computed the errors in the fitting coefficients but that doesn't seem to play into it.

Thanks,

Part (c) reinforces the hint that Least Squares is the criterion, because the MS Residual estimates the uncertainty from the data. This could be used for a $\chi^2$ test of the uncertainty (with $n-2$ df).
• Why are you using $\chi^2$? I'm at a bit of a loss to figure out how you are supposed to test the hypothesis $\beta > 1.05\, \hatP\{\beta}$ This is a data-dependent hypothesis, and so isn't covered by Neyman-Pearson theory that I am aware of. If you let $\sigma_=o$ be the specified variance (uncertainty), then $\frac{SSRes}{\sigma_o^2} \sim \chi^2_{n-2}$ when $H_o:\sigma = \sigma_o$. – Dennis Jul 20 '14 at 23:13