Chi squared test for goodness of fit
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Approximation by the chi squared test
Provided certain assumptions are fulfilled the statistic [Equation goes here &- download the original to see it.] can be approximated by a probability distribution known as the [Equation goes here &- download the original to see it.] &-distribution (pronounced &'&'kye squared&'&'). Standardised values for the [Equation goes here &- download the original to see it.] &-distribution as functions of the critical significance levels are provided in tables. In order to conduct a [Equation goes here &- download the original to see it.] test it is advisable to construct a contingency table. This is a table setting out the expected and observed frequencies side by side; from this the contributions of individual columns to the test statistic can be readily calculated The contingency table for our example is&: [Table goes here &- download the original to see it.] Like other distributions (for example, the t&-distribution) the [Equation goes here &- download the original to see it.] &-distribution is a function of the degrees of freedom. The degrees of freedom are the number of ways in which the contribution of [Equation goes here &- download the original to see it.] can vary. The maximum degrees of freedom is therefore equal to the number of rows in the contingency table. From this number we subtract the number of constraints on the way the data has been constructed. There is always at least one constraint. This is because the last entry in a frequency table is determined by all the others &- given the total frequency. In our current example, since the die is to be thrown 160 times and the sum of the first nine frequencies is153, this constrains the entry for the tenth frequency &- it must be 160&-153=7. Constraints can arise in other ways. In particular, there is one other constraint that can arise in tests of goodness of fit. If the expected values are modelled around a parameter determined from the experimental data themselves this adds one more constraint to the contingency table. We will illustrate this in a further example. For the present, our current example has just one constraint, hence degrees of freedom = v = 10&-1 = 9 We are now in a position to complete the test. [Table goes here &- download the original to see it.] Equation goes here &- download the original to see it.] = 0.5+1.125+1+0.5+2.667+1.125+0.281+0+0.9+0.12 = 8.223 [Equation goes here &- download the original to see it.] Our next example illustrates the application of the test for goodness of fit and the determination of expected values for a Poisson distribution. It also illustrates the procedure for dealing with small expected frequencies. Expected frequencies below 5 should not be used.
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Contents of
Chi squared test for goodness of fit
1 Medians, quartiles and percentiles.
2 Modelling of theoretical distributions to given data and the Chi squared test for goodness of fit
3 Approximation by the chi squared test
4 Example of modelling using the chi squared test for goodness of fit
5 Cumulative distribution function.
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