# Hypothesis testing for non-normal data

I'm measuring the error produced by operators executing planar cuts under different guiding systems. I define a target plane and then measure the euclidean distance from uniformly sampled points in the target plane to the executed plane. Since the target plane is my coordinate system reference, I define it as the plane $xz$ in $\mathbb{R}^3$. The executed plane $E$ is defined as $ax + by + cz + d=0$ with $b \neq 0$ and, given that I'm sampling points from $xz$ and projecting them into $E$, then $b$ could never be $0$. Then $E$ could be rewriten as $y = -\frac{a}{b}x - -\frac{c}{b}z -\frac{d}{b}$ with $x \sim \mathcal{U}(x_{min}, x_{max})$ and $z \sim \mathcal{U}(z_{min}, z_{max})$. Of course, the cutting tool also produces some (maybe normal) noise that affects $y$. The random variable $y$ is not normally distributed (I've tested it and I've plotted some qq-plots of it).

I have 2 different guiding systems and I have to test if they are significantly different wrt the above described measurement method. Which method do you recommend me to use to test my hypothesis?

Thanks!

• What hypothesis? – Glen_b -Reinstate Monica May 15 '14 at 19:01
• @Glen_b Presumably that the executed plane coincides with the target plane. Federico, what can you tell us about the nature of the discrepancy? You have mentioned measurement error of the distances, but could you tell us more specifically how the guiding system might fail to miss the target plane? For instance, perhaps it is designed in a way that can create only a translational error (causing the two planes to be parallel) or maybe only a rotational error (guaranteeing the planes will intersect). Perhaps (alternatively) they are constrained to have a fixed intersection? – whuber May 15 '14 at 19:06
• @whuber but then wouldn't the population value of $b$ be 0? – Glen_b -Reinstate Monica May 15 '14 at 19:08
• @whuber actually the options you give (translational error, rotational error, intersection or not) may all appear at the same time (except for a rotation that makes the executed plane perpendicular to xz; in that case b = 0 and the method does not work). This complexity is why we are trying to assess the effectiveness of the guiding system by measuring the error represented by the random variable y. The guiding system is just helping the human operator, that's why we want to know which one has a better effect on the outcome. Thanks! – Federico May 15 '14 at 19:57
• @Glen_b the hyphotesis that the guiding systems produce different outcomes, in a statistically significant way; then I think we would be able to keep the one with the smallest dispersion and whose expectation is closest to 0 (the result of an ideal guiding system would be one in which y is always zero, that is, plane $E$ is plane xz). Thanks – Federico May 15 '14 at 20:02