Chi-Square Tests are used to find out whether distributions of categorical variables differ from one another. There would typically be a set of observed values and those which are expected theoretically, a Chi Square test would tell us the goodness of fit for the same. The chi square statistic shows any discrepancies between the observed values and those expected in theory.
The Chi-square test begins with collecting numbers of the hypothesis which is being tested and then comparing it with the expected numbers. For example if a coin is being tossed 50 times, the expected result would be 25 heads and 25 tails. But the observed value would be 20 and 30. A Chi-square test would let us know if there is a significant discrepancy between these two. If there is a significantly large difference we can reject the null hypothesis which means that there is no interaction between the variables. On the other hand if the scores are close it means that they are the same.
Take another example of Chi- Square tests. Say, for example, we tabulate who stays absent in school more, boys or girls? If we tabulate this result it would be a simple table showing the number of boys absent and present and the number of girls absent and present with the totals. We can even calculate percentages for both given that we have the total and say that boys stay more absent than girls or vice versa based on the actual numbers. However if we want to test the hypothesis we need to deploy statistical tools such as the Chi Square.
The chi-square test is used under two circumstances
- thegoodness-of-fit test: here we estimate how closely an observed distribution matches an expected distribution
Here say if a coin is tossed 60 times, theoretically speaking there will be 30 heads and 30 tails but the actual outcome might differ to something like 25 heads and 35 tails. So Chi-Square will be applied to understand if the discrepancy is significant, what is causing it etc.
Problem solving would typically involve, establishing hypotheses. Then using the formulae, calculate the calculate chi-square statistic. We need to know the number of observations, set of Expected values. This will be compared with the observed values. Degree of Freedom would be required to ascertain the significance level and arrive at the conclusion i.e. accept or reject the hypothesis.
- For estimating whether two random variables have any interaction between them or are independent or not, for e.g. is there any relationship between engineering students and those who get admission in top MBA colleges.
The Application of Chi Square tests with be studied with respect to One Sample Bernoulli Model, Multi-Sample Bernoulli Model, One Sample Multinomial Model, Multi-Sample Multinomial Model, Goodness of Fit Tests, Tests of Independence, Computational and Simulation Exercises
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