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In my textbook i found the note that for a Chi-square test ($\chi = \frac{(n-1) \cdot s'^2}{\sigma^2}$) the assumption of a normal distribution in the population is very important - and much more important that for a t-Test. I am wondering why. According to the central-limit-theorem variances should (as means) be normally distributed if the n is large. In fact the chi-square-distribution converges towards a normal distribution for large n. The expected value of the term $\frac{(n-1) \cdot s'^2}{\sigma^2}$ should be $(n-1)$ no matter what population i am drawing the sample from, shouldn`t it?

I did some simulations and found (as my textbook says) that type-1-error was overestimated when drawing samples for a uniform distribution even with a sample size of 1000.

Can someone please give me some intuition about what makes the normality assumption so important for a Chi-square test, and why this is so different from testing for means with a t test.

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