Level of significance and acceptance and rejection of the null hypothesis

[Tyto alba]

Why do we reject the null hypothesis in Goodness of fit when the Chi square statistic is less than the tabulated value of chi-square at say 5% level of significance and accept when it is more?

What does it mean to have a Chi-square value more or less than the value assigned to a certain level of significance?

Edited: The homogeneity thing was wrong, so I removed it.

 

[MarneMath]

1. We don't "accept" the null hypothesis we simply fail to reject it.
2. Basically, you're simply saying given some assumptions regarding the data, if the null hypothesis were true and I had a p-value of .00001, then it would be highly unlikely (but not impossible) for that data to come from the same distribution. The 5% is rather arbitrary and depending your requirements may be shifted to the left or right.

 

[Tyto alba]

1. We don't "accept" the null hypothesis we simply fail to reject it.

Thank you.
How do we comment or conclude at the end of a goodness of fit, when the observed X² i.e. less than the expected X², the X² corresponding to a particular significance level?

2. Basically, you're simply saying given some assumptions regarding the data, if the null hypothesis were true and I had a p-value of .00001, then it would be highly unlikely (but not impossible) for that data to come from the same distribution. The 5% is rather arbitrary and depending your requirements may be shifted to the left or right.

I just realized what I wasn't getting for so long-
If suppose the level of significance is constant (can't be changed) then a less than expected X² means increased possiblility that a true null hypothesis can be rejected and is indeed rejected and if larger than expected X², it means lesser possibility of not being rejected and we say we fail to reject?

 

[FactChecker]

Why do we reject the null hypothesis in Goodness of fit when the Chi square statistic is less than the tabulated value of chi-square at say 5% level of significance and accept when it is more?

What does it mean to have a Chi-square value more or less than the value assigned to a certain level of significance?

Edited: The homogeneity thing was wrong, so I removed it.

Do you mean the statistic or the P-value?
The null hypothesis is typically that the data comes from a hypothesized distribution.
The statistic of the Chi-square is small when there is a good fit between the data and the hypothesized distribution. So you should not reject the null hypothesis if the statistic is small.

On the other hand, the P-value is large when there is a good fit between the data and the hypothesized distribution. So you should reject the null hypothesis if the P-value is small.

 

[Stephen Tashi]

What does it mean to have a Chi-square value more or less than the value assigned to a certain level of significance?

It's difficult to tell whether your questions are about the chi-square statistic in particular or whether you need to know about the general procedure of hypothesis testing. As to the general procedure of hypothesis testing, it is important to know that it is not a form of mathematical deduction. There are no mathematical theorems that say you must do this-or-that when you apply hypothesis testing and there are no mathematical theorems that say you will make a correct decision by using hypothesis testing. Hypothesis testing is simply a procedure that has some intuitive appeal and is thought to be empirically effective in certain fields of study.