Level of significance and acceptance and rejection of the null hypothesis

In summary, the null hypothesis is typically that the data comes from a hypothesized distribution. If the statistic of the Chi-square is small when there is a good fit between the data and the hypothesized distribution, then you should not reject the null hypothesis. However, if the statistic is large, then you should reject the null hypothesis.
  • #1
Tyto alba
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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.
 
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  • #2
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.
 
  • #3
MarneMath said:
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?
 
  • #4
SanjuktaGhosh said:
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.
 
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  • #5
SanjuktaGhosh said:
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.
 

1. What is a level of significance?

A level of significance, also known as alpha (α), is a threshold value used to determine whether the results of a statistical test are significant or not. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Commonly used values for alpha are 0.05, 0.01, and 0.1.

2. How is the level of significance related to p-value?

The level of significance and p-value are closely related. The p-value is the probability of obtaining a result at least as extreme as the observed data, assuming that the null hypothesis is true. If the p-value is less than or equal to the level of significance, then the result is considered statistically significant and the null hypothesis is rejected. If the p-value is greater than the level of significance, then the result is not significant and the null hypothesis cannot be rejected.

3. What does it mean when the null hypothesis is rejected?

When the null hypothesis is rejected, it means that the observed data is unlikely to have occurred by chance alone. This suggests that the alternative hypothesis is more likely to be true. In other words, there is enough evidence to support the claim that the relationship or effect being tested is real and not just due to random chance.

4. Can the null hypothesis ever be proven?

No, the null hypothesis can never be proven. It can only be rejected or fail to be rejected based on the available evidence. Even if the null hypothesis is rejected, there is always a small possibility that the result was due to chance and not a true effect. Therefore, it is important to carefully interpret the results and consider the limitations of the study.

5. How is the acceptance or rejection of the null hypothesis determined?

The acceptance or rejection of the null hypothesis is determined by comparing the p-value to the level of significance. If the p-value is less than or equal to the level of significance, then the null hypothesis is rejected. If the p-value is greater than the level of significance, then the null hypothesis is not rejected. This decision is based on a predetermined level of confidence, typically 95% or 99%, in the results of the test.

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