Agent Smith said:
So, a null hypothesis is neither true nor false.
We can either reject it or fail to reject it
What exactly is the terminology we use here.
We're proffering ##2## hypotheses: The Null ##H_0## and The Alternative ##H_a##.
Then we compute the P-value. If ##\text{P-value} \leq \alpha##, we reject ##H_0## and accept ##H_a##. If ##\text{P-value} > \alpha## we fail to reject ##H_0##
Perhaps I should've said
Type I Error: Rejecting ##H_0## when ##H_0## shouldn't be rejected. ##\text{P-value} \geq \alpha##
Type II Error: Failing to reject ##H_0## when ##H_0## should be rejected. ##\text{P-value} < \alpha##
Imagine you want to test H0: mu = 98.6F versus Ha: mu < 98.6F
where the quantity we're concerned with is body temp of a "normal healthy" adult.
Concerning your "neither true nor false" question, think about it this way: are we trying to say that the mean temperature is EXACTLY 98.6 degrees F? Certainly not -- we're saying that the true mean temp is close enough to that value to make it a very usable description, so is H0 true? Not with that interpretation. Is H0 false? Not in the sense that we want to know if 98.6 is a good usable reference value. The purpose of this hypothesis test is this: determining whether the true mean temp is close enough to 98.6 that we can continue to use it or whether it is enough smaller than 98.6 that we need to move on to a new value. Remember that hypothesis testing is ALWAYS about examining the evidence against H0, not the evidence it its favor.
The possible decisions of the test are:
- Reject H0: this means the data indicates to us that the true mean is noticeably smaller than 98.6. We might have a sample mean of 98.48, but after sample size and sample standard deviation are taken into account we decide that is not far enough from 98.6 to convince us Ha makes more sense than H0
- Do not reject H0 -- here the data indicates that the true mean is smaller than 98.6
Why not say "Accept H0" in the first case? Because the word "Accept" indicates we've proven H0 to be true.
A final comment: imagine how the US justice system is supposed to work -- the philosophy of it. Please don't comment on views of how people believe it actually works: I won't respond and it isn't appropriate.
To avoid language awkwardness I'll assume the verdict comes from a judge instead of a jury.
There are two possibilities about the defendant: The defendant is Guilty: DG, or the defendant is innocent: DI
One tenet of the US judicial system is to assume DNG. The goal of the prosecutor is to convince the judge, that DG is the correct assumption: in other words, the prosecutor has to convince the judge to reject the assumption of DNG
There are two possible verdicts: Guilty, G, or Not Guilty, NG. (Note that there is no such thing as a verdict of "Innocent".) Again, as noted above, to obtain a verdict of G the prosecutor has to convince the judge, beyond a reasonable doubt, to drop the assumption that the defendant is guilty. The prosector's job is not to present evidence showing the defendant is innocent, it's to present evidence of the defendant's guilt
There are four possibilities for what can happen at the end of the trial:
The verdict is G and the defendant is Guilty: thus G and DG is a correct outcome
The verdict is G and the defendant is Innocent: thus G and DI is an incorrect output. Since this corresponds to the prosecutor convincing the judge to reject a basic assumption when that assumption should not be rejected this can be considered a Type I Error
The verdict is NG and the defendant is innocent: here NG and DI is a correct outcome
The verdict is NG and the defendant is guilty: here NG and DG can be considered a Type II error: the judge should have rejected the assumption about the defendant but did not
In hypothesis testing:
- the null hypothesis corresponds to DI
-the alternative hypothesis corresponds to DG
- rejecting H0 corresponds to a verdict of guilty
- failing to reject H0 corresponds to a verdict of not guilty