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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.

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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.

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