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Chi square analysis Just when you thought statistics was over!!

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More statistics… Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel's laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the "goodness to fit" between the observed and expected.

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Hmmmmm… Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors? How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected?

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Null hypothesis The chi-square test is always testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result.

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Chi Square x 2 = ( O - E ) 2 E The formula…. Just get it over with already!!

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Sample problem Suppose that a cross between two pea plants yields a population of 880 plants, 639 with green seeds 241 with yellow seeds. You are asked to propose the genotypes of the parents. Your hypothesis is that the allele for green is dominant to the allele for yellow and that the parent plants were both heterozygous for this trait. If your hypothesis is true, then the predicted ratio of offspring from this cross would be 3:1 (based on Mendel's laws) as predicted from the results of the Punnett square

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GreenYellow Observed (o)639241 Expected (e)660220 Deviation (o - e)-2121 Deviation 2 (o - e) 2 441 d 2 /e0.6682 x 2 = d 2 /e = 2.668.. Chi Square x 2 = ( O - E ) 2 E

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So what does 2.688 mean? Figure out your Degree of freedom (dF) Degrees of freedom can be calculated as the number of categories in the problem minus 1. In our example, there are two categories (green and yellow); therefore, there is 1 degree of freedom.

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Now that you know your dF… Determine a relative standard to serve as the basis for accepting or rejecting the hypothesis. The relative standard commonly used in biological research is p > 0.05. The p value is the probability that the deviation of the observed from that expected is due to chance alone (no other forces acting). In this case, using p > 0.05, you would expect any deviation to be due to chance alone 5% of the time or less.

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Conclusion Refer to a chi-square distribution table Using the appropriate degrees of 'freedom, locate the value closest to your calculated chi-square in the table. Determine the closest p (probability) value associated with your chi-square and degrees of freedom. In this case ( X 2 =2.668), the p value is about 0.10, which means that there is a 10% probability that any deviation from expected results is due to chance only.

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Degrees of Freedom (df) Probability (p) 0.950.900.800.700.500.300.200.100.050.010.001 10.0040.020.060.150.461.071.642.713.846.6410.83 20.100.210.450.711.392.413.224.605.999.2113.82 30.350.581.011.422.373.664.646.257.8211.3416.27 40.711.061.652.203.364.885.997.789.4913.2818.47 51.141.612.343.004.356.067.299.2411.0715.0920.52 61.632.203.073.835.357.238.5610.6412.5916.8122.46 72.172.833.824.676.358.389.8012.0214.0718.4824.32 82.733.494.595.537.349.5211.0313.3615.5120.0926.12 93.324.175.386.398.3410.6612.2414.6816.9221.6727.88 103.944.866.187.279.3411.7813.4415.9918.3123.2129.59 NonsignificantSignificant

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Step-by-Step Procedure for Chi-Square 1. State the hypothesis being tested and the predicted results. 2. Determine the expected numbers (not %) for each observational class. 3. Calculate X 2 using the formula. 4. Determine degrees of freedom and locate the value in the appropriate column. 5. Locate the value closest to your calculated X 2 on that degrees of freedom (df) row. 6. Move up the column to determine the p value. 7. State your conclusion in terms of your hypothesis.

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Analysis If the p value for the calculated X 2 is p > 0.05, accept your hypothesis. 'The deviation is small enough that chance alone accounts for it. A p value of 0.6, for example, means that there is a 60% probability that any deviation from expected is due to chance only. This is within the range of acceptable deviation.

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If the p value for the calculated X 2 is p < 0.05, reject your hypothesis, and conclude that some factor other than chance is operating for the deviation to be so great. For example, a p value of 0.01 means that there is only a 1% chance that this deviation is due to chance alone. Therefore, other factors must be involved.

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Chi Square x 2 = ( O - E ) 2 E 100 Flips of a coin Contingency table 40 60 50 100 Heads Tails O E ( 40 - 50 ) 2 50 ( 60 - 50 ) 2 50 + ( 10 ) 2 50 ( 10 ) 2 50 + = = 100 50 100 50 + = = 2 + 2 = 4.00 df = 1

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Time for some M&M’s! http://us.mms. com/us/about/ products/milkc hocolate/

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Distribution of colors….or so they say..… hmmmmmmm

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