Pearson chi-squared test (χ2): differences?

[Drudge]

So, as far as I know, there are two χ2-tests: "test for fit of a distribution" & "Test of independence"

How big of a mistake is it to use the one instead of the other in an exam for example (of course all exams are all different to some degree, but generally)?

The only differences I can really find out is how each test counts the theoretical value(s) and the way in which the degrees of freedom are counted

For example a problem might be as follows:
a random sample from population X is, as a function of age, distributed as follows

10-20
5
21-30
4
31-40
3
40-41
9

And the equivalent theoretical values are: 6, 5, 4, 5

Question:

"Does the sample represent the theoretical distribution?"

So, you would use a "test for fit of distribution", but how much of a difference is it to use a "test of independence"?

 

[Stephen Tashi]

So, you would use a "test for fit of distribution", but how much of a difference is it to use a "test of independence"?

That would depend on what theoretical distribution was stated in the problem.

 

[Drudge]

That would depend on what theoretical distribution was stated in the problem.

Non normal distribution.

 

[Stephen Tashi]

"Non-normal" is not a specific distribution. If you really want to know "how much" difference it would make you must be specific about the distribution - and if you are specific then you can calculate the difference yourself.

 

[blue_raver22]

I would say that it is a significant mistake to use the wrong χ2-test in an exam or any research setting. The two tests, "test for fit of a distribution" and "test of independence", have different purposes and assumptions, and using the wrong test could lead to incorrect conclusions and potentially impact the validity of the research.

The "test for fit of a distribution" is used to determine whether a sample follows a specific theoretical distribution, while the "test of independence" is used to determine whether there is a relationship between two variables. In the given problem, the question asks if the sample represents the theoretical distribution, which would require the use of the "test for fit of distribution". If the "test of independence" was used instead, the results would not accurately reflect the question being asked.

Additionally, as stated in the content, the two tests have different methods for calculating the expected values and degrees of freedom. Using the wrong method could also lead to incorrect results and conclusions.

As a scientist, it is important to carefully select the appropriate statistical test based on the research question and data being analyzed. Using the wrong test could compromise the validity and reliability of the research findings. Therefore, it is important to understand the differences between tests, such as the χ2-tests, and use them correctly to ensure accurate and meaningful results.