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I can't remember which statistical tests are needed for the following two tests. I thought maybe the null hypothesis test but wanted to check on here first... Maybe something Bayesian?

Consider you have a plot of y against x with several data points on it, each with y error bars. I would like to calculate the probability that:

a) These points are randomly distributed

b) These points have the same y-value = constant, i.e. y is not a function of x.

Thanks for any advise.

Natski