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,

• Because this is a homework/self-study question, you should add the self-study tag to your question. See stats.stackexchange.com/tags/self-study/info Jul 19, 2014 at 17:55
• Sure thing, my mistake
– Jim
Jul 19, 2014 at 17:58

Without more context is difficult to be sure. There are many models that could be fit, and many criteria used to fit those models.

Part (b) suggests that the model is linear ("slope") which suggests that the criterion is Least Squares.

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).

• Yes a linear model fits the data set, the slope is around 1 and the y-intercept is 0. Not sure what other context to give? I put a value of a 5% higher slope into the chi-sqd equation and got a number like 90, which is untenable.
– Jim
Jul 19, 2014 at 22:41
• 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 $H_o:\sigma = \sigma_o$. Jul 20, 2014 at 23:13

Probably, in the 5% slope it is asking what is the probability to have a slope that is 5% higher than the estimated one.

For example, let us assume that you have a slope of 1.00, then what is the probability to have a a slope that is 1.05 or above?

You can estimate using the Standard Error for the slope .When you do a regression the software should provide you the Standard Error for the slope. For example, let us assume Standard Error to be 0.003, you could reframe the question in : What is the probability for a Normal Distribution with mean 1.00 and standard deviation 0.003 to have a slope of 1.05 or higher.

If this probability is very low then you can say that 1.05 is not consistent with the data