Chi-squared goodness of fit test

[actually ..?]

Hi ... I have a question regarding the Chi-squared goodness of fit test -

I thought that the test-statistic is the following:
X^2 = Sigma[(Actual - Expected)^2]/Expected

However, in Sweeting, I see that he has the following (where Variance replaces the Expected in the denominator) as the test-statistic:
X^2 = Sigma[(Actual - Expected)^2]/Variance
i.e. X^2 = Sigma[(T_n - Tp_n)^2]/[Tp_n * (1 - p_n)] based on pg 156-157.

I would like to understand the difference between the two test-statistics above (or if I'm wrong!). Does anyone have any insights as to the difference between these two ... or alternatively, point me to a good source that distinguishes between the two?

Thanks

 

[mcbainco]

Chi square test

I had also stumbled against that.

I have been unable to understand what the denominator refers to! I am reasonably confident that this is an error.

you are right to state that the test-statistic is:
X^2 = Sigma[(Actual - Expected)^2]/Expected

I shall compare the results from the example in the text book with those obtained from using statistical packages.

 

[td290]

I'm afraid Sweeting's derivation of the goodness-of-fit test is way off! Use the formula you both think you should.

 

[manish.rex]

I'm afraid Sweeting's derivation of the goodness-of-fit test is way off! Use the formula you both think you should.

A Chi square variable is nothing but square of a standard normal variable.

Y= (X-E(X))^2/Var(X).

The Var(X) is replaced with E(X), when either we use binomial distribution with b being very small or when we use apoisson distribution.

Hope this helps

 

[td290]

A Chi square variable is nothing but square of a standard normal variable.

Y= (X-E(X))^2/Var(X).

The Var(X) is replaced with E(X), when either we use binomial distribution with b being very small or when we use apoisson distribution.

Hope this helps

I'm not sure it has a lot to do with p being small and it's quite a bit more complicated than noting that \(\chi^2\) is the square of a standard normal variable. If you really want to know why we put E(X) in the denominator check out:

http://sites.stat.psu.edu/~dhunter/asymp/fall2006/lectures/ANGELchpt07.pdf

Otherwise, just know that you have the right formula for a goodness-of-fit test and use it well!