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I am trying to read the significance tests. However, I am a bit confused what exactly is the difference between chisquared test and z test. I read that chisquared is non parametric and it doesn't make any assumption about the distribution of the population. However, z test assumes that the statistic parameter forms a normal distribution. Using central limit theorem, if the number of samples are very high we can assume that the statistic is from a normal distribution

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    $\begingroup$ You ought to define what z test you mean. There's at least 3 well known totally different statistical tests sharing the name: z-test of means (a t-test when population variances are known); z-test of proportions (equal to chi-sq test for 2x2 table); z-test of signs. $\endgroup$ – ttnphns Dec 16 '12 at 8:09
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    $\begingroup$ Please define which specific z- and chi-square tests you mean, and if possible show us what was actually said about the chi-square. I believe there are generally distributional assumptions for most chi-square tests (having derived a number of them as exercises). $\endgroup$ – Glen_b -Reinstate Monica Dec 16 '12 at 10:52
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A starting point is that they are usually testing completely different things.

Note the various uses of words such as usually in the following, which is only a very general guide, as you haven't given us a specific example.

A Chi square test - which crops up in many situations - is usually to test in quite a general sense whether it is plausible a given set of data or the results of a model are generated from a particular distribution. One very common application is whether the two variables in a two-way contingency table are related to each-other, or they are plausibly generated under a null hypothesis of no relation between the two variables. The second most common application is to test whether an observed distribution, once binned (so we have counts of observations in various subsets of the value space) is close enough to what you would expect if a a distribution expected under the null hypothesis generated it.

A Z test is usually for whether there is significant evidence that the mean value from a population differs from some set value. It is usually looking at a more precise problem than a Chi Square test - is this estimate of the mean plausible (Z test), versus is this overall distribution of values (Chi square test) plausible, under the null hypothesis.

A caveat on this is that statistics that have a Chi square distribution under a null hypothesis crop up in all sorts of situations, so generalizations are fraught with challenges.

An interesting fact for you to think through - if you explore statistical inference further - is that if you multiply a variable with a normal distribution by itself, you get a variable with a Chi square distribution.

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