Thanks pi r squared. That is most helpful. I think I have now got my head around it. Feel free to jump in (anyone) and disagree:

1) **A hypothesis test for the population mean of a normal distribution cannot test if there is evidence FOR a claim that the population mean is less than (or more than) some amount.** As has been pointed out, a critical region of <27.4 for the "less than 30 seconds" example I discussed is not reasonable, because the population mean could be 29.999 and it would be absurd to demand most values be below this. Nor would a critical region of >32.6 be reasonable, because you would not want to conclude the mean is less than 30 if a value of less than 30 had never been experimentally obtained! The idea of trying to create a cutoff for concluding that there is evidence **for **the hypothesis that the mean is below (or above) some value is a non-starter.

2) **A hypothesis test for the population mean of a normal distribution **can only test if there is evidence AGAINST the claim that the population mean is less than (or more than, or equal to) some amount. So if a claim is made that the population mean is less than (or more than, or equal to) some amount, that claim must be the null hypothesis. If the null hypothesis is a claim that the population mean is more than/less than some amount, it is the awkward case where the sign on the alternate hypothesis is the "wrong" way around.

3) **Any straightforward question can be reworded to make it into one of the awkward cases. Examples...**

**Straightforward:**

Experience shows that the heights of certain plants have a mean of 70cm and a variance of 36.

A random sample of 49 plants is measured. Construct a critical region, at the 5% level, to test the claim that these plants are not as tall as expected.

H0: mu = 70

H1: mu <70

[Note that the claim is not merely that the measured plants come from a population with a mean of less than 70cm (e.g. 69.999). The claim is that they are so short that that it is unlikely they would have been obtained if they came from a population with a mean of 70]

**Awkward:**

Experience shows that the heights of certain plants have a mean of 70cm and a variance of 36.

A random sample of 49 plants is measured. Construct a critical region, at the 5% level, to test whether the measured plants come from a different normally distributed population, one with the same variance, but with a mean height of less than 70cm.

H0: mu = 70

H1: mu > 70

Last edited: Apr 10, 2019