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I'm currently teaching basic probability and after that I would like to do some simple hypothesis testing. Frankly, probability theory and statistics have never really interested me but I feel like my students like how it can very often be applied to everyday life, so I want to teach them as much as possible.

I've only learned about hypothesis testing at university level, and there it was just at the end of a very theoretical course on probability. I know some parametric and non-parametric tests from my studies (and from talking with friends), for example $t$-tests, sign-tests, $\chi^2$-tests. I usually have to look up what they are about, when I can apply them and how they are applied. This is a huge disadvantage when trying to answer the question:

Which statistical tests should I thoroughly teach?

Also, it is very important to me that my students do not only blindly follow some algorithm and have no idea why (as seems to be the rule for high schoolers doing statistics). I want to teach them why exactly a certain test works the way it does. This obviously requires myself to understand why certain tests work the way they do. The problem is, most sources I've found have rather little detail on "why" and much more on "how".

Even worse, some reasonings behind certain tests seem to be out of scope for most high schoolers (within a normal curriculum). I looked up what the idea for the $t$-test is and I think it would be rather confusing to point out that by $z$-transforming the sample mean, the random variable is no longer normally distributed and then to introduce the Student's $t$-distribution. So maybe, a more accurate question would be:

Which statistical tests can I thoroughly teach while not skipping the idea behind the test?

This is high school level, 12th grade. They already know single-variable calculus, so at least integrals can be understood properly.

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  • $\begingroup$ When you say "basic probability", what do you mean? $\endgroup$ – DavidButlerUofA Nov 22 '15 at 22:13
  • $\begingroup$ @DavidButlerUofA: I'm mostly focussing on discretely distributed random variables and cover some conditional probability (also Bayes' theorem), the binomial distribution and then I plan to proceed with the normal distribution. Nothing too fancy, just what I believe should really be known and can really be understood. $\endgroup$ – Huy Nov 23 '15 at 7:28
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Consider developing the $z$-test first. This takes as a simplifying assumption that the population standard deviation $\sigma$ is known; and supports sketching the sampling distribution of the sample mean as a standard normal curve for visual intuition and double-check purposes. Afterward we can bootstrap to the more practical case of using the sample standard deviation $s$ to estimate $\sigma$, which then gives rise to Student's $t$-distribution.

I think this is the pedagogical standard in any statistics textbook that I've looked at. It only requires very minimal probability content, and some kind of rude understanding of density curves and the normal distribution. In my community college statistics courses, I cut out most of the optional probability content (it's not a probability course, after all), so we can get directly to inferential statistics including hypothesis tests. I develop the $z$-test pretty carefully, and in fact only test on that, with lots of drills about when it's appropriate and when it isn't. Then on the last day I pitch the $t$-test as an example the myriad other statistical tests out there, and note that the human user will always be responsible for checking assumptions, requirements, and interpretations. Of course, your mileage may vary.

For a pretty good resource that basically follows the standard progression, consider OpenStax Introductory Statistics (Ch. 7-9).

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    $\begingroup$ Yes this is usually good enough for high-school, and it helps a lot when one moves to more complicated tests such as χ^2-test because even then the test statistic is 'normalized' in the same manner you would expect, subtract the 'mean' and divide by the 'standard deviation', where of course 'mean' and 'standard deviation' are estimated in some way. Kind of related to ensuring invariance under change of units. Similarly the F-test makes sense in the same way. $\endgroup$ – user21820 Nov 25 '15 at 5:42

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