hypothesis test_ determine null and alternative

Question

What is the most suitable null and alternative hypothesis for following problem?

It is believed that the average level of Prothrombin in a normal population s 20 mg/100 ml of blood plasma with a standard deviation of 4 mg /100 ml. To verify this, a sample is taken from 40 individuals in whom the average is 18.5 mg/100 ml.

My answer is

H0:average level of prothombrin in a normal population = 20mg/100ml

H1:average level of prothombrin in a normal population != 20mg/100ml

but in the answer sheet they have stated as follows

H0:sample is not taken from 40 individuals in whom the average is 18.5 mg/100 ml.

H1:sample is taken from 40 individuals in whom the average is 18.5 mg/100 ml.

is my answer incorrect?

How do we determine the exact null and alternative hypothesis for given problem?

Answer 1

The answer sheet seems to suggest an answer that I think is wrong for several reason:

1. the tested hypotheses depend on the observed data,
2. the tested hypotheses include the sample size (why?!!) and
3. are seemingly unrelated to hypothesized value that is to be verified (20 mg/100 ml).
4. Rejection of the null hypotheses of the mean not being 18.5 mg/100 ml (or failure to reject it) would not say much about whether the mean is 20mg/100ml.
5. I am also uncertain whether there even exists a sensible test for testing this null hypothesis versus the stated alternative. Point null hypothesis versus point alternative (or interval alternative) is quite normal, interval null hypothesis versus interval alternative is quite normal (see my proposal below), but interval null hypothesis versus point alternative seems difficult to me.

In short, that answer does not seem to make any sense to me, unless I am missing something.

However, your answer is also not suitable to achieve the stated goal of verifying that the mean is 20 mg/100 ml (I assume the standard deviation is not the thing we are trying to verify) or at least reasonably close to it. A failure to reject the null hypothesis of mean = 20mg/100ml (your answer) does not mean that the mean is actually 20mg/100ml.

In practice, I guess one would say that values between 20-$\delta_1$ mg/100ml and 20+$\delta_2$ mg/100ml are considered practically equivalent to 20mg/100ml. The null hypothesis would then be that the true mean lies outside of this interval and the alternative would be that the true mean would lie inside of this interval.