Setting up a hypothesis test when you don't know the distribution

Question

So far in class we've been setting up hypothesis tests using confidence intervals, which we have been able to calculate because we're given a distribution of the data.

My question is about how I would approach this:

The true radius of a piece of wire is $x$, which is known from a very accurate but slow test.

A new test is created, one which is much faster but does not give as accurate results. We have $n$ samples drawn from this test to an identical wire. The results were: sample mean = $\bar x$, sample variance = $\sigma^2$.

We want a hypothesis test which can determine whether the new, faster test gives as average value different from the true value.

Answer 1

If $n$ is large enough, you might assume that $\bar X$ is nearly normal and use a Z-test. Otherwise you could use a nonparametric test to judge whether the median of the $X_i$ matches the known radius $\mu$. (Maybe the data themselves are nearly normal; you could check that with some test of normality like Shapiro-Wilk or Anderson-Darling, or see if a Q-Q plot is nearly linear.)

The simplest nonparametric test would be a sign test, taking determinations $X_i$ above $\mu$ to be Successes and those below to be Failures. Test whether $P(\text{Success}) = 1/2.$

A second nonparametric possibility is a Wilcoxon signed rank test. A third is a permutation test.