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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.

Calvin Khor
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datadan
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    By the way, it seems to me that you are asking for the philosophical underpinning of hypothesis testing. You may be able to get answers on that front by asking in https://stats.stackexchange.com – Calvin Khor Jun 29 '21 at 07:15
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    I have a coin. My hypothesis is that it is a fair coin. I flipped the coin twice, and got one heads, and one tails. Do you accept my hypothesis? Do you reject it? – 5xum Jun 29 '21 at 08:30

3 Answers3

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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?

Greg Nisbet
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    Thank you for you comment. "This means that if the null hypothesis is actually true, we will reject it with probability α. That's what we're picking when we pick α." I need to understand this better. We we are not actually hard rejecting the null hypothesis, we are say there is alpha probability that it is not true? – datadan Jun 29 '21 at 21:07
  • $\alpha$ is not the probability that the null hypothesis is false. The null hypothesis is true or it isn't. The null hypothesis is a proposition about the world that may be true or false, but is unknowable from the perspective of our framework. However, not everything can be a null hypothesis. "The variables $A$ and $B$ are dependent" is not a good null hypothesis because there isn't a good way to convert it to a CDF of a test statistic. "The variables $A$ and $B$ are independent", however, does not have this drawback. The alternative hypothesis is in some ways badly named, (cont...) – Greg Nisbet Jun 30 '21 at 01:26
  • since it [the alternative hypothesis] does not need to be testable the same way that the null hypothesis does. – Greg Nisbet Jun 30 '21 at 01:27
  • Ok so if the null hypothesis is: Hnull = m and n are independent. And we find this is NOT true. What does that mean? Does that mean that m and n are NOT independent? – datadan Jul 01 '21 at 02:23
  • If the null hypothesis is that $m$ and $n$ are independent (and both come from some known distribution like a standard normal distribution) and you look at some test statistic, then the null hypothesis is associated with a distribution of that test statistic. If you observe the value $w$ for that test statistic and the probability that the test statistic would be $w$ or more extreme given that the null hypothesis is true is 0.03, then you would reject the null hypothesis at the 0.05 significance level. You haven't determined that $m$ and $n$ are independent "in real life" (cont...) – Greg Nisbet Jul 01 '21 at 02:31
  • , instead you have shown that the hypothesis that $m$ and $n$ are independent has made a bad guess, and if we personify the hypothesis a bit, was "more surprised by the real world than you're willing to allow". A hypothesis that consistently assigns low probabilities to things that actually happen is a bad hypothesis. I guess that's the intuition. – Greg Nisbet Jul 01 '21 at 02:33
  • Thank you Gregory Nisbet. I'm going to have to accept non closure on this. A rejection of the null hypothesis meaning as the "a bad guess", is not very satisfying. I think I follow what you are stating. It just means the Alt Hypothesis is meaningless and mis leading, and that this more of a "good enough" scenario rather than a logical system. Thank you. – datadan Jul 01 '21 at 02:48
  • I'm not very knowledgeable on statistics, I'd recommend asking some follow-up questions and reading some more stuff here such as this answer. In my explanation, I meant to say that we can think of a hypothesis as kind of like a scientific theory (in a loose, intuitive sense) ... and scientific theories that make bad predictions or consistently assign low probabilities to stuff that actually happens should be discarded. (There are several things that are technically wrong with this analogy) – Greg Nisbet Jul 01 '21 at 02:54
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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.

Henry
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  • You make some great points Henry. I really like exploring the idea of being convinced if the sample size is too low. That said, a rejection of the null hypothesis means what? Is it a negation of the null hypothesis statement? – datadan Jun 29 '21 at 21:11
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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.

yudhiesh
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