For how many degrees of freedom is the test statistic proved of significance?With which realization probability is this or an even greater effect to be expected when the null hypothesis is valid?

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Carly Yang · Accepted Answer

I assume this is a chi-squared goodness-of-fit test for data in k categories (treated by the test as nominal). The test statistic is  Q  =      ∑          i      =      1        k              (              X        i            =              E        i                    )        2                    E      i        ,where the       X    i   are the observed counts in the categories and the       E    i   are the expected counts (unrounded), computed from the model being fit.Small values of Q represent relative good fit to the model and large values represent bad fit. So you will reject the null hypothesis that the data match the model for sufficiently large values of Q.Provided that the       E    i   are all above 5 (some authors say most are above 3), Q is approximately distributed as       C    h    i    s    q    (  ν  )  ,, where the number of degrees of freedom is   ν  =  k  −  1. However, if the model is given only generically, and you have to estimate r parameters in order to find the       E    i    , then   ν  =  k  −  r  =  1.I believe the second part of the question is asking for the critical value of the test. That is, the value c such that Q&amp;gt;c leads to rejection at a specified significance level. Knowing the significance level and   ν  , you can get the value c appropriate for a particular test from printed tables of the chi-squared distribution or by using software.In R statistical software, the critical value for a test at the 5% level with   ν  =  5 is c=11.07. Here is the computation from R statistical software, perhaps you can compare it with a printed table.qchisq(.95, 5)## 11.0705

For how many degrees of freedom is the test statistic proved of significance? With which realization probability is this or an even greater effect to be expected when the null hypothesis is valid?

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