Calculating and Interpreting Expected and Chi-Square TablesDate: 04/10/98 at 22:31:30 From: Lola Kelly Subject: Stats- chi squared How do you get the frequency expected when doing chi squared? I have a 3-by-3 table. Help. Please. Date: 04/11/98 at 08:06:14 From: Doctor Anthony Subject: Re: Stats- chi squared I will work through a 3x3 contingency table. The method will be directly applicable to any similar problem. The following table refers to the performance of a baseball team on various pitches classified as good, medium and bad. The null hypothesis is that the state of the pitch does not affect the performance of the team. Wins Draws Losses Total ---------|--------------------------|---------- Good | 11 6 4 | 21 Medium | 12 7 7 | 26 Bad | 7 7 14 | 28 ---------|--------------------------|----------- Total | 30 20 25 | 75 To find the expected frequencies, we assume independence of the rows and columns. To get the expected frequency corresponding to the 11 at top left, we look at row total (21) and column total (30), multiply them, and then divide by the overall total (75). So the expected frequency is: 21*30 ------- = 8.4 75 So to complete the expected table, draw up another table similar to that above and having the same row and column totals. For each entry in this table, we simply calculate (row total*column total)/75. The completed table is: Wins Draws Losses Total -----------|--------------------------|----------- Good | 8.4 5.6 7.0 | 21 Medium |10.4 6.9 8.7 | 26 Bad |11.2 7.5 9.3 | 28 -----------|--------------------------|------------ Total | 30 20 25 | 75 The number of degrees of freedom is calculated for an m-by-n table as (m-1)(n-1), so in this case (3-1)(3-1) = 2*2 = 4. To calculate X^2, we then have a further table: O E |O-E| |O-E|^2/E ---------------------------------------- 11 8.4 2.6 0.805 6 5.6 0.4 0.0286 4 7 3 1.2857 12 10.4 1.6 0.246 7 6.9 0.1 0.0014 7 8.7 1.7 0.332 7 11.2 4.2 1.575 7 7.5 0.5 0.033 14 9.3 4.7 2.375 ------------------------------------------- Total = 6.70 = X^2 The tabular 95% value of X^2 (degrees of freedom = 4) is 9.49, so the value of X^2 that we obtained (6.70) is not significant at the 5% level. We conclude that the state of the pitch does not affect the performance of the team. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 04/11/98 at 13:00:10 From: Coolbabe29 Subject: Re: Stats- chi squared I cannot believe what a big help you were. I appreciate your speedy reply. My Stats prof. had me so flustered, a simple straightforward answer helped so much! THANKS! |
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