How do I interpret a ChiSquare analysis? Please help explain.?

Question

I need to know if 6-9-1-3 (observed) are significantly different from expected (5-5-5-4) I do not think that they are, and I get a ChiSq of 6.85 and a p=0.077, with a df of 3, but I am not sure what that really means. Are my results correct, and if so, will you please explain.

Answer 1

The problem with your problem is that Chi Square analysis is typically inaccurate with just a few data points (four in your case). I suggest you look up the Chi Square limitations in your textbook. (I've forgotten what the limits are, but I do know a good Chi Square test needs more than just a few data points.)

If your results were valid, look up the table critical value (CV) under df = 3 and between .100<=p<=.050. If 6.85 exceeds the table CV, then you can posit that the differences between the expected values and the actual values are significant. If you have a value p = .077, in between .100 and .050, you can find the CV for your problem between the table's CV at p = .100 and at p = .05 by ordinary arithmetic.

The degree of freedom (df) simply means you have N - 1 options on selecting your data points, where N = the number of data points (e.g., four in you case). There are N - 1 options (degrees of freedom) because, for example, if you select 6, 9, 1...3 has to be the next one. This results from the fact that the total (sum) for your data points is 19, and if you preselect 6, 9, and 1, for the sum to stay at 19, 3 has to be the fourth data point.

Answer 2

(6-5)^2/5 + (9-5)^2/5 + (1-5)^2/5 + (3-4)^2/4 = 6.85

So, what this sum represents is the magnitude of the difference between what you expect and what you see.

The numerators are squared to make the sum always be positive.

Take for example if what you saw was exactly what you expected, then the sum would be zero.

But if for each item what you saw was far away from what was expected then the chi square value would be large, and would correspond to the tail region of the probability distribution.

The small amount of area under the tail indicates either you have seen a rare event (given the null hypothesis is true) or that the null is false. How small is too small, that depends on the question being asked and the individual's viewpoint.

Let me craft a question to suit this data, suppose you had a friend that told you he could with a computer program and given data in the sports pages could predict the scores a sport team would make. The friend wants you to invest your savings into his new growing company to promote the product. Based on this data would you invest?

Answer 3

I don't feel like doing too much work but maybe I can help.

The Chi-square test is a 'goodness of fit' test. The basic idea is that if you think that you know the distribution that describes your data then you can compare your observed values to those expected (calculated) from you distribution. It's obtained by dividing your range of x values (for random variable X) into a number of bins and using the concept of multinomial distribution of observed values into those bins.

i, x, E(x), del, del^2, del^2/E(x)
---------
1, 6, 5, 1, 1, 1/5
2, 9, 5, 4, 16, 16/5
3, 1, 5, -4, 16, 16/5
4, 3, 4, -1, 1, 1/5
--------------
sum del^2/E(x) = 34/5 = 6.8

df = n-1 = 3

(I'm going to make coffee, brb)
The p-value that you have, 0.077, is the probability of getting a test statistic larger than the one that you have. If you have a level of significance of 5% (typical) then this is larger and so fails the test. So you're data does not support your distribution to this level of significance.