Chi squared test for goodness of fit
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Example of modelling using the chi squared test for goodness of fit
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Classes with lower frequencies should be combined to yield a class with frequency greater than 5. Example (2) The number of horror movies released per month in the USA is thought to follow a Poisson distribution. Releases of horror movies per month in the USA, X, were recorded by a fanatic over 100 months and she obtained the following observed frequencies. [Table goes here &- download the original to see it.] Determine an appropriate Poisson distribution for this data and test the goodness of fit at the 5% significance level. Solution [Equation goes here &- download the original to see it.] Then [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] Expected frequencies are given by multiplying probabilities by the sample size. Hence[Table goes here &- download the original to see it.] Expected frequencies less than 5 cannot be used, so we combined the classes for 4, 5, 6 and 7 to obtain the following contingency table, from which the contributions to the test statistic can be directly computed. [Table goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] To calculate the critical value we need to know the degrees of freedom. The number of rows is 5. There are two constraints&:1. Because frequencies in the last row are determined by the other frequencies; 2.Because the expected frequencies are calculated from a statistic determined from the observed frequencies. [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] We now illustrate the application of the [Equation goes here &- download the original to see it.] for goodness of fit to a hypothesis concerning a normal distribution. Example (3) The total weight of fish, in kilograms, caught by an experienced angler in a Russian lake in one day during the summer of 1964 is denoted by the random variable X. 90 anglers were sampled and the results obtained and summarised below[Table goes here &- download the original to see it. It is thought that X is normally distributed by N(30, 122). Calculate the expected frequencies for each of the five classes. Carry out a[Equation goes here &- download the original to see it.] goodness of fit test at the 5% level to test this hypothesis. Answer Let X~N(30, 122) Total number of observations = 90 To determine the z value corresponding to the class boundaries&: [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] Determining the contingency table and contributions to the test statistic together&:&- [Table goes here &- download the original to see it.] [Equation goes here &- download the original to see it.]
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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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