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# 04_09 - STAT 410 Examples for Spring 2008 Pearsons 2 Test...

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STAT 410 Examples for 04/09/2008 Spring 2008 Pearson’s χ 2 Test for Goodness of Fit ( Based on Large n ) A random sample of size n is classified into k categories or cells. Let Y 1 , Y 2 , … , Y k denote the respective cell frequencies. n k i i = ° = 1 Y . Denote the cell probabilities by p 1 , p 2 , … , p k . H 0 : p 1 = p 10 , p 2 = p 20 , … , p k = p k 0 . 1 1 0 = ° = k i i p . 1 2 k Total Observed frequency O Y 1 Y 2 Y k n Probability under H 0 p 10 p 20 p k 0 1 Expected frequency E under H 0 n p 10 n p 20 n p k 0 n Test Statistic: ( ) ( ) ( ) ° ° ° - = - = - = = = - cells k i i i i k i i i i k p n p n E E O E E O 2 1 2 1 0 2 0 1 Y Q Rejection Region: Reject H 0 if Q k – 1 2 α χ , d.f. = k – 1 = (number of cells) – 1 Pearson’s χ 2 test is an approximate test that is valid only for large samples. As a rule of thumb, n should be large enough so that expected frequency of each cell is at least 5.

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