Interpreting Chi-Squared Test Results
Date: 10/26/2004 at 07:21:21 From: Joseph Subject: Chi-Squared Question The critical values for chi-squared increase as the confidence level increases. My question is why does a distribution "pass" this test if the calculated value is less than the critical value and not vice- versa? For example, let's say for 9 degrees of freedom, you calculate a value of 20. Using a 97.5% confidence test, since 20 > 16.919, there is a significant difference from your expectation (horrible match). Using a 99% confidence test, since 20 < 99%, the probability distribution cannot be rejected with 99% confidence (excellent match). Isn't this a contradiction? I've been trying to understand how chi-squared tests work. I've been taught different things and am quite confused. From how I understand the test, if a test "passes" for "x" confidence then it will automatically "pass" for all confidence levels greater than x and less than 100% (since the critical values increase as the confidence increases). This would also indicate that a test could "pass" for x confidence and "fail" for a confidence less than x. Based on my understanding it would be more logical if this test were "passed" when the calculated value is greater than (not less than) the critical value. I know there's something that I'm not understanding about all of this, but I can't seem to find what it is. Any help would be greatly appreciated. Thanks.
Date: 10/26/2004 at 09:38:27 From: Doctor George Subject: Re: Chi-Squared Question Hi Joseph, Thanks for writing to Doctor Math. It looks like you may have misread the chi-squared table, but this is a minor point. For 9 degrees of freedom we have Area (1-p) critical value ------------- -------------- 0.95 16.919 0.99 21.666 I am here assuming that your test only rejects high values. The area remaining in the tail is often called p or alpha. It will also help to clarify some terminology. When constructing a confidence interval for some statistic we usually speak of a large "level of confidence" such as 95%. When performing a hypothesis test we usually speak of a "level of significance" such as 0.05. This term can be confusing. It is a low number because we do not want a large percentage of errors in the test. The main issue seems to be in understanding what is being tested in relationship to the null hypothesis. The critical values mark the worst condition of the data. For good data the test statistic will be very small compared to the critical value. The hypothesis test asks the question, "Assuming that the null hypothesis is true, what is the interval within which 95% of the test statistic samples will fall?" Looking at it this way you can see that an interval that contains 99% of the samples would have to be larger that the interval that contains only 95% of the samples. Now let's turn things around. The test also means that there is only a 5% chance that the test statistic will be greater than 16.919 if the null hypothesis is true, and only a 1% chance that it will be greater than 21.666. The test does not prove that the null hypothesis is true or false. It only indicates whether or not the data is consistent with it, at some level of signifiance. As the critical value increases we are more confident that the test statistic will be in the interval, but there is less significance that can be inferred from that fact. At the extreme, virtually any set of data will have a test statistic less than the critical value at 99.99999% of the area, but it no longer means much of anything. There are two kinds of errors that occur. A type I error occurs when the null hypothesis is true and the test fails. A type II error occurs when the null hypothesis is false and the test passes. As the level of significance decreases, the probablility of a type I error decreases, but the probablility of a type II error increases. The probability of a type I error is the level of significance. Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
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