Why can we not accept the alternate hypothesis in Chi Squared Testing?

Question

I'm a math teacher, but this aspect of stats is not my strong point. I've asked several other teachers as to why, and their responses was just "don't do it" the why was not very compelling, so I come here. $H_\text{null}$ = $m$ and $n$ are independent. $H_\text{alt}$ = $m$ and $n$ are NOT independent.

If condition $p$ is met, we accept the null hypothesis.

If condition $p$ is not met, we reject the null hypothesis.

Isn't the rejection of the null hypothesis logically equivalent to the alt hypothesis? Isn't the negation of ($m$ and $n$ are independent) = ($m$ and $n$ are Not independent)?

Thank you kindly for your response.

Answer 1

I don't think the answer to this question is specific to the chi-squared distribution or the chi-squared test. In general, the alternative hypothesis is not associated with a specific distribution of the test statistic.

We reject the null hypothesis if the probability that we would observe values of the test statistic as extreme or more extreme than the value of the test statistic that we observed is less than or equal to $\alpha$.

This means that if the null hypothesis is actually true, we will reject it with probability $\alpha$. That's what we're picking when we pick $\alpha$.

The distribution in question for the test statistic is the Chi-squared distribution. This distribution is well-defined because there's really only one way for standard normal random variables to be independent of each other.

However, if the standard normal random variables are not independent, then how would we construct a distribution for the test statistic in principle?

Answer 2

The evidence can persuade you to reject the null hypothesis (though you may be wrong and make a Type I error).

Or it can persuade you to fail to reject the null hypothesis (which is not quite the same as accepting the null hypothesis, particularly if your test used little data: if I flip a coin only once, I am not going to reject the null hypothesis that it is a fair coin, but this is not really the same as accepting it).

But all you have done is test the null hypothesis by seeing whether the data was a reasonably likely outcome given the null hypothesis. You have not tested the alternative hypothesis (which in your case is very unspecific), except perhaps to decide the location of the critical region for the test of the null hypothesis. What you could do next is develop a specific new null hypothesis, perhaps based on the the data you have observed, collect new data, and test the new null hypothesis (with a new alternative hypothesis) with the new data.

This is a sort of Popper argument that a theory in the empirical sciences can never be proven, but it can be falsified. Some people find it unsatisfactory, and so take a confidence interval approach, to try to end up with a more positive conclusion than merely rejecting or failing to reject a null hypothesis.

Answer 3

As @Gregory Nisbet mentioned this isn't exclusive to the Chi Squared Test but applicable to hypothesis tests in general.

I shall draw parallels from the criminal justice system. If a person is arrested for a murder case and goes on trial, they are presumed innocent until proven guilty. The prosecution gathers evidence to prove that the person is guilty but if the prosecution fails to do so it doesn't mean that the person is actually innocent its just that there was insufficient evidence to say that they were guilty of the crime.

A lack of evidence does not prove something does not exist, its just that you did not find it in your specific investigation. Therefore, you never accept the null hypothesis but instead fail to reject the null hypothesis or reject the null hypothesis.