Hi, Graduation A is over graduated. Therefore, it failed Chi square test as given in Page 31 example. I am satisfied with this answer. However, Graduation B is under graduated and following data very closely. Therefore, it should pass Chi square test. However, it failed Chi square test as given in answer to Que. 12.9. If for undergraduation also test fails, than what is use of this test? Your explanation will be very helpful. Regards, Ashok
Ashok, I think you've got a pretty good point here. A set of undergraduated rates should pass Chi-Squared test. I think what we're seeing with graduation B is a graduation that isn't smooth (which would tend to suggest undergraduation) but then isn't a good fit either! So, it hasn't been graduated much and the graduation that has been done doesn't seem to be very good! The Chi-square test is a good test for overall fit. Good luck! John
Was the test done on a one-sided or two-sided basis? Overfitted data could fail a two-sided test chi^2, and if you got a test statistic buried in the left tail, I for one would be suspicious.
I'm not sure I've heard of a two-sided Chi-squared test? Calum, do you mean two-sided in the sense that whether the graduated rates are too high or too low, the test would still be failed? I would tend to view the Chi-squared test as being a one-tailed test on whether the graduated rates adhere to the crude rates... If the rates are undergraduated (ie not enough graduation has been done), they would pass the Chi-squared test. If the rates are overgraduated (ie too much graduation has been done), they would fail the Chi-squared test. So, I can understand Ashok's confusion given this, John
By two-sided I mean that if the sum of the squared differences were very low or very high the test would reject. At least in theory, there's nothing stopping you from defining a chi^2 test as one sided or two sided, although in practice I think it's unusual as typically the errors would have to be tiny to reject in the left hand tail. I think it's a reasonable test to make, though: if you expect the differences between your raw and graduated rates to have a normal distribution, then you should reject, or at least closely investigate, a graduation where the error distribution does not appear to be normal. There are other useful tests for issues that a chi^2 does not catch, like rates being too high or too low or consistent runs above or below the graduated line.
Hi Calum, Very interesting idea to have a two-tailed Chi-squared test and I like your reasoning. The actual and expected are so close that it's too good to be true, something fishy is going on, in this case, not enough graduation. However, I would recommend steering clear of talking about the test being two-tailed for the CT4 exam. It might be misinterpretted as a lack of understanding. Yes, of course, you're right that there are other useful tests for issues that a chi^2 does not catch. That's why we have the other tests! Good luck! John
