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"why is it..." Because one of the first persons (Karl Pearson?) to calculate the density function of $Z_i^2$ and $\sum_{i=1}^n Z_i^2$ chose to name these random variables $\chi^2$ random variables with $1$ and $n$ degrees of freedom respectively. If he had chosen some other name, say $\Phi^2$ random variables, you would have been asking why $Z_i^2$ ...


7

The Pearson test is popular because it's simple to compute - it's amenable to hand-calculation even without a calculator (or historically, even without log-tables) - and yet generally has good power compared to alternatives; the simplicity means it continues to be taught in the most basic subjects. There might be argued that there's an element of ...


4

scipy uses the continuity correction, statsmodels does not. If you pass correction=False to the scipy test, then the results will be identical.


4

There are several possible explanations. Here is one of them. It should be viewed as partly intuitive rather than entirely rigorous. Suppose you have $K$ categories and your null hypothesis is that the number of occurrences of the $i$th category is $\mathsf{Pois}(\lambda_i).$ Then the count in the $i$th category is $X_i \sim \mathsf{Pois}(\lambda_i).$ For ...


2

There is, of course, no way to say for certain that the $E_i - T_i$ are normally distributed (they could theoretically obey any probability distribution, subject to the experimental setup and underlying theory). However, due to the Central Limit Theorem and many uncontrolled random fluctuations present in any given measurement, it is often a very good ...


2

In finding the distribution under the null hypothesis $H_0,$ you need to include the assumptions of the test and the parameter value specified by $H_0.$ Then you compare a statistic from your data to compare with the 'null' distribution. I agree with @Nuclear03020704's answer (+1), and take a more computational approach here. In your example, the data are ...


2

Why $\sigma^2$ (population variance) is treated as constant?. The fact that $\sigma^2$ is treated as a constant does not imply that it is known. It is there, static, but unknown to us. All we can do is estimate it using sample. To demonstrate it's a static value, suppose that we are able to collect all of population and calculate its variance. The value of $...


2

Yes, scaling the weights to sum to the sample size will mitigate the problems caused by ignoring the sampling design in the analysis. The chi-squared test will still not be correct, and Fisher's Exact Test isn't. (I would recommend against Yates' correction, because it's quite badly conservative) You can do better. Rao & Scott worked out the actual ...


1

If you run 8 separate Chi-square tests you will inflate your type 1 error (chance of false positive) due to the multiple comparisons problem.


1

Here are some fragmentary answers based on what you have told us about your data and analysis. If $X \sim \mathsf{Chisq}(\nu = k),$ then $E(X) = k$ and $Var(X) = 2k.$ [See Wikipedia or your text or class notes for some details of chi-squared distributions.] P-value. If you're doing a chi-squared test for which the null distribution is (approximately) $\...


1

Is grazing method 1 consistently different than grazing method 2? Pretty clearly not: At sites A, B, and C more plants were grazed with method 1, but the reverse is true at site D. If you need a test to show that sites are not homogeneous for grazing counts, then a chi-squared test of homogeneity will suffice. The P-value (whether you do it in R, as below, ...


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