# Chi-squared Goodness of Fit Questions

#### theswits

##### New Member
Hi folks,
A request for some basic help with Chi-squared testing:
Say we have two different Likert samples (strongly agree, agree, neutral, disagree, strongly disagree) that I want to do a test fit on. The first sample has a population of 10, its expected frequency is (1,4,2,2,1) and the observed is (2,3,2,2,1). The second sample has a population of 100, its expected frequency is (10,40,20,20,10) and the observed is (20,30,20,20,10), or simply ten times sample one. My significance level is 0.1, and it looks like I have 4 degrees of freedom.

So, question 1 is: using the formula for goodness of fit as (sum of ((O-E)^2)/E), why does sample 1 return a chi-squared of 1.25, and sample two 12.5? It looks like each sample has the same distribution relative to the expected, just one has a greater population. Yet, looking at my critical chi-squared values, one is greatly different than the other. I don't understand why. Can someone explain to me why these apparently similar distributions give different results? Using my critical chi-squared, I can keep Ho for sample one, but must reject for sample two.

Question 2 is the significance factor. I have chosen 0.1 and I think that I must reject Ho (the observed "fits" the expected) if my chi-squared value is less than the critical. So for my example above, I must reject Ho for the 100 sample case with a signifcance level of 0.1, but if I change my significance level to 0.01, then I can keep Ho. I thought that as I make the signifcance level smaller, I am more sure that the fit of observed to expected in in fact "a match". Yet, in this example, I can keep the Ho at 0.01, but must reject at 0.1. How have I misinterpreted the meaning of the significance factor?

E

#### elnaz

##### Guest
Hello
I say you about your first question, statistic for goodness of fit test as (sum of ((O-E)^2)/E) is true, and if you do it then you can decide about their distribution but you attention to The first sample's expected frequency is (1,4,2,2,1) and with this frequancy we cant do goodness of fit test because in every cell this frequancy is less than 5 so you should merge them such that frequancy in every new cell will be greater than or equal 5.
about second question you should attention too that significance level will choice before doing tests in statistics.
i hope you be succeed
Best Regards
Elnaz

#### theswits

##### New Member
Chi squared Goodness of fit

Hi Elnaz,
Thanks for your responses to my questions, but I think you may have missed the point.

In the first question, although I accept your assertion with respect to the frequency counts of less than 5, the question still applies of we take populations of 100 and 1000.

So the question becomes:
The first sample has a population of 100, its expected frequency is (10,40,20,20,10) and the observed is (20,30,20,20,10). The second sample has a population of 1000, its expected frequency is (100,400,200,200,100) and the observed is (200,300,200,200,100), or simply ten times sample one. Using the formula for goodness of fit as (sum of ((O-E)^2)/E), why does sample 1 return a chi-squared of 12.5, and sample two 125? It looks like each sample has the same distribution relative to the expected, just one has a greater population. Yet, looking at my critical chi-squared values, one is greatly different than the other. So, the question still stands.

Regarding the second question, I'm not certain what your response means, or how it applies to the increasing value of critical chi-squared and the ability to accept the null hypothesis for smaller significance values.

Thanks for responding,
E.

E

#### elnaz

##### Guest
Hello
excuse me please you read your question that have represented the first in this forum you can see you said pop in first sample is 10 so you must merge cell.
about your second question you know that if you like increasing value of critical chi-squared you should decrease significance factor before test.
Best regards
Elnaz

#### theswits

##### New Member
Thanks, but .....

Hi Elnaz,
Thanks for your answers, but I am not any closer to understanding the underlying principles .....
Kind regards,
e.

E

#### elnaz

##### Guest
Hello
i said you i think you didnot represent your intent in question clearly
and i said you my opinion.
i hope you be succeed
Best Regards
Elnaz