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--- chi-square_test [2016/05/16 07:52] – hkimscil
+++ chi-square_test [2024/12/09 08:20] (current) – hkimscil
@@ Line 175: / Line 175: @@
 I do not know exactly why the degree of freedom is important in a conceptual way -- so, having a difficulty explaining it. But, the idea behind it is that if you know totals of column and row, and the values of two cells (as a minimum requirement; called degree of freedom), you would be able to obtain the values of the other four cells without consulting the actual observed values.
-{{07-419434-a-tshirt-camp-scils-rutgers.jpg?202 |RUSURE campaign SCILS Rutgers 2000}} Anyway, you just obtained the chi-square value (37.58) and the degrees of freedom (2). You can look up the text book (for chi-square test): (1) find your degrees of freedom (2), that is, the second row of the table; (2) decide the probability you want to employ (usually .05 or .01); (3) write down the numbers; and (4) compare them to your chi-square value.
+{{07-419434-a-tshirt-camp-scils-rutgers.jpg?202 |RUSURE campaign SCILS Rutgers 2000}} Anyway, you just obtained the chi-square value (37.58) and the degrees of freedom (2). You can look up the text book (for chi-square test): (1) find your degrees of freedom (2), that is, the second row of the table; (2) decide the probability you want to employ (usually .05 or .01); (3) write down the numbers; and (4) compare them to your chi-square value. (see [[:chi-square distribution table]])
 The numbers you obtain from the book are 5.991 for the 0.05 probability and 9.210 for the 0.01 probability. They are called critical values. So the critical values are:
@@ Line 181: / Line 181: @@
 .991 at 0.05 probability
 .210 at 0.01 probability
+<code>
+> qchisq(0.95, 2)
+[1] 5.991465
+> qchisq(0.99, 2)
+[1] 9.21034
+>
+</code>
 These critical values do not exceed the chi-square value you obtained from your table -- 37.58. How do you want to relate them together? Think about the expected values -- the ideal types. Suppose you obtained the same values (observed values) as those of expected values, what would be your chi-square value? --Yes, it is going to be zero. Why? If you look at the formula
@@ Line 238: / Line 245: @@
 | yes  | 18  | 25  | 12  | 55  |    |
 | Expected Value  | (22)  | (22)  | (11)  | (55)  |    |
-| (O-T)2 / T  | (-4)2/22=0.73  | (3)2/22=0.41  | (1)2/11=0.09  |    | 1.23  |
+| (O-T)<sup>2</sup> / T  | (-4)<sup>2</sup>/22=0.73  | (3)<sup>2</sup>/22=0.41  | (1)<sup>2</sup>/11=0.09  |    | 1.23  |
 | no  | 22  | 15  | 8  | 45  |    |
 | Expected Value  | (18)  | (18)  | (9)  | (45)  |    |
-| (O-T)2 / T  | (4)2/18=0.89  | (-3)2/18=0.5  | (-1)2/9=0.11  |    | 1.5  |
+| (O-T)<sup>2</sup> / T  | (4)<sup>2</sup>/18=0.89  | (-3)<sup>2</sup>/18=0.5  | (-1)<sup>2</sup>/9=0.11  |    | 1.5  |
 | Total  | 40  | 40  | 20  | 100  | 2.73  |
-Chi-square value = The sum of the entire 6 yellow cells = 2.73.
+**Chi-square value = The sum of the entire 6 yellow cells = 2.73.**  \\
-Degrees of Freedom (df) = (the # of columns-1) x (the # of rows-1)= (3-1) x (2-1) = 2 x 1 = 2.
+**Degrees of Freedom (df) = (the # of columns-1) x (the # of rows-1)= (3-1) x (2-1) = 2 x 1 = 2.** \\
+\\
 Look up the values in your textbook -- which is called "critical values."
+\\
 They are:
 .991 (0.05 probability)
 .210 (0.01 probability)
+OR
+<code>
+> pchisq(2.73, df=2)
+[1] 0.7446193
+</code>
 Now the rest of what you need to do is to compare the numbers (chi-square value and the critical values).
@@ Line 261: / Line 274: @@
 In the first place, you assumed that there would be no differences in the abortion issue among the religious groups to get the expected values. And you compared the expected values to the observed values. In other words, you tested your survey result (the observed values) against the idea of "no difference." Your plans were: If the some of the comparison (chi-square) is big enough, you'd say that the idea of "no difference" was not likely true. If the some of the comparison (chi-square) is small enough, you'd say that there seems to be no reason to reject the idea of "no difference." In other words, in the first place, you assumed that there would be no difference, and you tested your survey result against this idea. What you conclude from this testing was you failed to disapprove the idea -- the idea of no differences.
-{{raritan-river-01.jpg?132|Princeton Park river}} __Why null?__ -- Someone in the class cleverly asked why we should use null hypothesis in the first place. As you see the above, it would be harder to test whether the researcher is right. Most statistic methods (chi-square, t-test, ANOVA, and others) test against the idea of 0 (zero -- no difference).
+{{raritan-river-01.jpg?132 |Princeton Park river}} __Why null?__ -- Someone in the class cleverly asked why we should use null hypothesis in the first place. As you see the above, it would be harder to test whether the researcher is right. Most statistic methods (chi-square, t-test, ANOVA, and others) test against the idea of 0 (zero -- no difference). Therefore, it would not have been safe, had you ever said, "Sure 45% and 62.5% are different."
-Therefore, it would not have been safe, had you ever said, "Sure 45% and 62.5% are different."
 __Another note:__ You might have a question... Hey, wait a minute... If I pick up some other numbers from the chi-square distribution table, the result would be totally different!
+<WRAP clear />
-*** For your information, the table looks as follows. And the chi-square value you got from your data was 2.73.
+For your information, the table looks as follows. And the chi-square value you got from your data was 2.73 (see [[:Chi-square distribution table]]).
 | df  | .30  | .20  | .10  | .05  | .02  | .01  | .001  |
 | 1  | 1.074  | 1.642  | 2.706  | 3.841  | 5.412  | 6.635  | 10.827  |