A researcher suggests the profit a small and medium business enterprise should make is \$46000 to be sustainable. The standard deviation is assumed to be \$2000. The mean profit from a sample of 20 businesses is \$45700. Based on this sample would you conclude that small and medium businesses are sustainable, testing at a 5% level of significance?
My questions are as follows:
(1) Is this a one-sided test? If so, is this the correct null and alternative hypothesis?
$H_0: \mu=46000$ or $\mu\lt46000$
$H_A:\mu\geq46000$
(2) Should the z-test or the t-test be used? I'm confused because the standard deviation is given (implying the z-test) but the sample size seems small (implying the t-test).
(3) Is the conclusion that we fail to reject the null hypothesis meaning that small and medium businesses are not sustainable?
Yes, this is a one-sided test, because in order to make a claim of statistical significance that, on average, small/medium businesses are sustainable at the 5% level, the sample mean would need to be greater than the hypothesized mean.
Your statement of the hypothesis is correct--some people write $H_0 : \mu = 46000$, and others write $H_0 : \mu < 46000$, but in your case, I personally prefer the latter.
Technically, the $z$-test is appropriate because you are given an assumption about the true standard deviation of the profit a randomly chosen small/medium business makes. However, in this case, no calculation is even needed--the statistic points in the wrong direction and a failure to reject is assured by the simple fact that the sample mean fails to exceed the hypothesized mean.
If you fail to reject $H_0$, that does not mean that small/medium businesses are not sustainable; it simply means that the data provided in this sample of businesses does not furnish sufficient evidence to make a claim that they are sustainable at the required level of significance. To help you understand the subtlety involved here, note that in order to make a claim that small/medium businesses are not sustainable, you would need to perform a one-sided test in the opposite direction, and gather data to the contrary by seeing if the profits are consistently lower than the hypothesized mean.
Failure to detect significance does not mean that the claim is false--it simply means that the evidence is lacking. To use a familiar legal analogy, it is a bit like a jury trial where a defendant is presumed innocent until proven guilty--failure to prove guilt does not necessarily mean the defendant did not commit the crime; it only means the high legal standard to prove guilt was not met.