Chi Squared and Null Hypothesis
If a chi-squared value is less than the critical value, does this
reject the null hypothesis? I have calculated a chi-squared value of 1.51,where
I have been told to choose a value of p=0.05 in a table of critical values,
therefore the critical value is 7.81.
In most common applications of chi-square, you reject the null if the chi-squared
value is greater than the critical value (what is called the upper critical value).
So in your case, you would accept the null hypothesis. You do not state exactly
what you are investigating, but your critical value of 7.81 suggests that maybe
you are comparing the occurrences of something falling into 4 categories with
some expected values in the 4 categories. Say you are testing whether your
observations of 100 ice cream flavor choices differs from what you expect.
You expect 25 people to choose chocolate, 25 strawberry, 25 vanilla, and 25
butter pecan. You observe the following choices: 40 chocolate, 30 strawberry,
20 strawberry, and 10 butter pecan. The chi square for these data is: 20.00
(3 degrees of freedom: 4 categories - 1). The critical value at p=.05 is 7.82.
Since 20.00 is larger than 7.82, what you observe differs from your expectations
enough to reject the null. In fact, there are only 2 chances in 10,000 that you
would observe what you did if the 40, 30, 20, 10 values were the norm. I hope
Eric A. Hagedorn, Ph.D.
University of Texas at El Paso
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Update: June 2012