Factory inspections on a budget This one goes out to my dad, an EPA inspector for 35 years, whose stories of challenging protocol inspired this question.

A factory inspector is testing the efficiency of $n$ machines. For a factory to pass the inspection, all $n$ machines is required to run at or above a certain fixed standard of efficiency $\alpha \in (0,1)$. Mathematically, we'll encode each machine's efficiency as $X_i\in [0,1]$, and that passing the inspection means that $X_i > \alpha$ for all $i=0,\ldots, n$.
An inspection is conducted through a sequence of tests. In each test, the inspector chooses a subset $\sigma\subset \{1,\ldots, n\}$ of the machines to be running, and measures their average efficiency $\frac{1}{|\sigma|}\sum_{i\in\sigma}X_i$. Each test costs the factory the same amount of money regardless of how many machines are running. Thus, it is in the factory's interest to reduce the number of tests to a minimum. 
Let's discuss some strategies for conducting the inspection.


*

*The most obvious strategy to conduct the inspection is to measure the machines one at a time (i.e. choosing consecutive singleton sets $\{i\}$, thus measuring the value $X_i$). Doing so requires $n$ tests to prove that all machines are in compliance. On the other hand, if one of the machines is not in compliance, the inspection has failed, and the inspector can stop testing.

*In some cases, testing multiple machines at once could allow the inspector to complete the inspection in fewer than $n$ tests. For example, measuring an average efficiency of $1$ while running all machines at once would reveal that all machines were individually running at efficiency $1$, thus completing the inspection in a single test. Similarly, measuring a total average efficiency below $\alpha$ would mean at least one machine's efficiency must be below $\alpha$, so the inspection would fail and terminate after only one test. Therefore, it may be advantageous to construct a strategy around multiple-machine tests, as they allow for the possibility of early termination.
My questions are,

In the general case, is there a strategy to conduct the inspection in fewer than $n$ tests that guarantees a correct answer?

I suspect that the answer to the above is no. In particular, I think that there may be strategies other than one-by-one that guarantee a correct answer in $n$ tests, but none that will guarantee an answer in fewer than $n$ tests.
The more general question, therefore, is...

Suppose that the machine efficiencies $X_i$ are i.i.d. random variables distributed uniformly. What inspection strategy minimizes the expected number of tests required to guarantee a correct answer?

 A: The answer to the first question is indeed no.
To see why, examine the case when you have $n$ machines, only perform $n-1$ tests and every test gives an average value of $\alpha+\epsilon$, where $\epsilon$ is arbitrarily small. The possible values $x_i$ of the machines can be found by examining the resulting linear system of equations $Ax=(\alpha+\epsilon)$, where $A$ represents the tests performed.
$$ \left[
\begin{array}{ccc|c}
  a_{11}&\cdots&a_{1n}&\alpha+\epsilon\\
  \vdots&\ddots&&\vdots\\
  a_{(n-1)1}&&a_{(n-1)n}&\alpha+\epsilon\
\end{array}
\right] $$
Every row represents one test and every column a machine so a restriction on the values $a_{ij}$ is that every row must sum to one and every non-zero value must be equal to the others in their row.
A valid solution for this would be $x_i=\alpha+\epsilon$ and thus all machines are in compliance. But because it is an underdetermined linear system ($n-1 < n$) there is some degree of freedom and so at least one $x_i$ can be equal to $\alpha$ and then the machines wouldn't be in compliance. So with less than $n$ tests, we can't be sure if our machines are in compliance or not.
