# Dependent t-test with extra information provided

#### ondansetron

##### TS Contributor
I was trying to perform a one tailed t-test, with Ho: mu <=20 (given that mu can go up to 20 and still be considered normal).
Isn't this correct?
If you fail to reject Ho, will you conclude the measurements are normal, on average?

#### martinl

##### Member
Yes, I would conclude there's NO EVIDENCE that there is a difference between before and after measurements. Since 20 is the upper bound of normalcy.
So what's the problem?

#### ondansetron

##### TS Contributor
Yes, I would conclude there's NO EVIDENCE that there is a difference between before and after measurements. Since 20 is the upper bound of normalcy.
So what's the problem?
The problem is that this is a common misunderstanding, including in biomedical research. A few things:

1. Failing to reject Ho does not allow someone to conclude Ho. The methodology doesn't allow for that. This kind of hypothesis testing cannot provide evidence FOR a null hypothesis, only against it (which is why I suggested setting up your test differently).
2. The only way you could suggest there is "no evidence" is if the p-value were exactly equal to 1. That means the point estimate is identical to the hypothesized value in Ho. If the point estimate is not identical to the Ho value (p-value not precisely 1), there is some evidence in the data to suggest Ho may not be correct.

This is why I suggested you set your test as
Ho: mu >= 20
Ha: mu < 20

If you reject Ho you have sufficient evidence at the selected alpha level to conclude the average after measurements are less than 20 (normal). The only assumption is whether 20 is considered normal, otherwise this isn't totally the answer you need. You could then use the set up I gave for Ho mu <=30 with Ha mu >30 and reject Ho allows you to claim abnormality, on average.

Or you could use something along the lines of equivalence/noninferiority/superiority testing, or a Bayesian method.

#### martinl

##### Member
The problem is that this is a common misunderstanding, including in biomedical research. A few things:

1. Failing to reject Ho does not allow someone to conclude Ho. The methodology doesn't allow for that. This kind of hypothesis testing cannot provide evidence FOR a null hypothesis, only against it (which is why I suggested setting up your test differently).
2. The only way you could suggest there is "no evidence" is if the p-value were exactly equal to 1. That means the point estimate is identical to the hypothesized value in Ho. If the point estimate is not identical to the Ho value (p-value not precisely 1), there is some evidence in the data to suggest Ho may not be correct.

This is why I suggested you set your test as
Ho: mu >= 20
Ha: mu < 20

If you reject Ho you have sufficient evidence at the selected alpha level to conclude the average after measurements are less than 20 (normal). The only assumption is whether 20 is considered normal, otherwise this isn't totally the answer you need. You could then use the set up I gave for Ho mu <=30 with Ha mu >30 and reject Ho allows you to claim abnormality, on average.

Or you could use something along the lines of equivalence/noninferiority/superiority testing, or a Bayesian method.
Oh, I see, now I can clearly see your point.
I didn't know that, stats books don't mention that.

#### ondansetron

##### TS Contributor
Oh, I see, now I can clearly see your point.
I didn't know that, stats books don't mention that.
It's pretty odd that many of them don't! I have noticed that it is more commonly overlooked when the book is not written by a statistician, but even when it is, it isn't necessarily spelled out for the reader in words.

Try running the test with the different hypothesis set up and let's see what you get!