You first decide on a null hypothesis. Expected frequencies are calculated on the basis of the null hypothesis, that is, assuming that the null hypothesis is true.
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For goodness of fit test using Chisquare test, Expected frequency = Total number of observations * theoretical probability specified or Expected frequency = Total number of observations / Number of categories if theoretical frequencies are not given. For contingency tables (test for independence) Expected frequency = (Row total * Column total) / Grand total for each cell
Goodness of fit test is used to test a single population. The null hypothesis will be that the observed frequencies are equal to expected frequencies (based on computed intrinsic values given the extrinsic values). A good example would be comparing observed phenotype frequencies against expected frequencies calculated from the parental genotypes (Simple dominance gives rise to 1:2:1 ratio with two parental heterozygotes). Contingency test is used to see whether or not different populations are associated. The null hypothesis will be that that different populations are independent of one another. A good example would be comparing the effect of different host plants on different populations of insects. (Effect of Host A on Insect population 1, 2, 3; Effect of Host B on Insect population 1, 2, 3; etc)
It is often a "goodness of fit" test. This is a test of how well the observations match the frequencies that would have been expected on theoretical basis. The theoretical basis may simply be your hypothesis.
The word goodness has two syllables. Good-ness.
A chi-square test is often used as a "goodness-of-fit" test. You have a null hypothesis under which you expect some results. You carry out observations and get a set of results. The expected and observed results are used to calculate the chi-square statistic. This statistic is used to test how well the observations match the values expected under the null hypothesis. In other words, how good the fit between observed and expected values is.