What exactly is a Type II Error

allnamesaregone

I'm wondering what is the actual definition of a Type 2 Error?

For example:
In a hypothesis test, if I fail to reject the null hypothesis this is clearly not the same as accepting the null hypothesis. I am merely saying that I don't have enough evidence to say the null hypothesis is not true.

"Absence of evidence is not evidence of absence".

The question I ask is when is a type 2 error committed?
Is it committed when I fail to reject the null hypothesis or is it committed when I accept the null hypotheis or could it even be that both situations are true?

Thanks.

gra26

A type II error occurs when you accept your null hypothesis but your null hypothesis is false.

gra26

Which is why you say that you do not reject the null hypothesis instead of saying you accept the null hypothesis.

allnamesaregone

So am I correct in saying that if I my conclusion is to fail to reject the null hypothesis I cannot possibly make a Type 2 Error because I havent accepted the null hypothesis?

Thomas_S_Hunterson

allnamesaregone wrote: »

So am I correct in saying that if I my conclusion is to fail to reject the null hypothesis I cannot possibly make a Type 2 Error because I havent accepted the null hypothesis?

Incorrectly rejecting the null is a type 1 error.

gra26

http://www.intuitor.com/statistics/T1T2Errors.html

There's table bout half way down the page that might help

allnamesaregone

Sean_K wrote: »

Incorrectly rejecting the null is a type 1 error.

I never said that I was rejecting the null, I said fail to reject null.

allnamesaregone

gra26 wrote: »

http://www.intuitor.com/statistics/T1T2Errors.html

There's table bout half way down the page that might help

Thanks.
This is how I understand it.

If the null is true and we reject the null then that is a Type 1 error.
If the null is false and we accept the null then that is a Type 2 error.

But If the null is false and we fail to reject the null what is the error?

I'm presuming that failing to reject the null is not the same as accepting the null.

Thomas_S_Hunterson

allnamesaregone wrote: »

Thanks.
This is how I understand it.

If the null is true and we reject the null then that is a Type 1 error.
If the null is false and we accept the null then that is a Type 2 error.

But If the null is false and we fail to reject the null what is the error?

I'm presuming that failing to reject the null is not the same as accepting the null.

Failing to reject is effectively the same as accepting:pac:

allnamesaregone

Sean_K wrote: »

Failing to reject is effectively the same as accepting:pac:

Yes and no. Failing to reject the null hypothesis could come from the fact of the null hypothesis being true, as you said, or perhaps the experiment was not a valid test of hypotheses.

For example consider the court case example where
Null - Defendent innocent
Alternative - Defendent guilty

If we fail to reject the null we basically say that the defendent is not guilty but does this mean that they are innocent? There are several reasons that could have caused us to fail to reject null and innocence is only one reason.

RoundTower

Sean_K wrote: »

Failing to reject is effectively the same as accepting:pac:

no it's not -- this is the whole point!

Thomas_S_Hunterson

allnamesaregone wrote: »

Yes and no. Failing to reject the null hypothesis could come from the fact of the null hypothesis being true, as you said, or perhaps the experiment was not a valid test of hypotheses.

For example consider the court case example where
Null - Defendent innocent
Alternative - Defendent guilty

If we fail to reject the null we basically say that the defendent is not guilty but does this mean that they are innocent? There are several reasons that could have caused us to fail to reject null and innocence is only one reason.

No your hypotheses aren't correctly formed for the court-case example.

The test would be more along the lines of

Null: Not-guilty
Alternative: Guilty

With the presumption of innocence until the alternative is accepted.

You're going to run into problems like this when hypothesis testing down to the fact that you're pitting one hypothesis against another and seeing which one is better, you've obviously got to use your intuition a little when forming the tests.

allnamesaregone

But now you cannot presume innocence because you haven't stated it in your null hypothesis. You have used not guilty which is not the same as innocence. So know you must seek evidence against the null of "not guilty". You can then have three situations.

1. Reject the Null and conclude that the person is guilty
2. Fail to reject the Null and conclude that there is not enough evidence to say that the person is guilty beyond a reasonable doubt.
3. Accept the Null and conclude that the person is not guilty (which again can mean a number of things not just innocence).

All I want to know is where can the Type 2 error happen above. In conclusion 2 or 3 or both.

If you use another example. Say you where a bouncer in a niteclub and I was a customer trying to get in. You presume I'm under 18 until proved above 18.

Null Hypothesis: - Under 18
Alternative Hypothesis: - Over 18

If you fail to reject the null does that mean I'm under 18? No it just means the bouncer wasn't able to prove I was over 18. I could be 20 for all he knows and just look very young.

You cannot say "not 18" as part of your Null Hypothesis because there are an infinite number of numbers that are "not 18".

Thomas_S_Hunterson

You're trying to mix law and statistics here. Under the law, there is an automatic presumption of innocence up until the point where a person is found guilty beyond reasonable doubt.

The jury return a verdict of either "Guilty" or "Not guilty".

They don't say whether a person is innocent or not.

allnamesaregone wrote: »

You cannot say "not 18" as part of your Null Hypothesis because there are an infinite number of numbers that are "not 18".

These are very poorly formed hypotheses. You're not using any common sense.

Your hypotheses would be "Under 18" and "Not under 18"

Thomas_S_Hunterson

Here is wiki's definition. It's right.

Type I (α): reject the null hypothesis when the null hypothesis is true, and
Type II (β): fail to reject the null hypothesis when the null hypothesis is false

There really is no more possibility. You just have to choose your hypotheses intelligently, whether it be the complement of the null or otherwise.

/edit: to clarify: The null can either be true or false. You can conclude it is either true or false. You can either be mistaken or not mistaken.

If you are mistaken and the null is true then type 1, if you are mistake and the null is false then 2.