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Hi Nathan. These are big questions that don't have a 30 second answer so I may have trouble getting to them. But for this part: "But the thing is, we only have one dataset (one sample of size 60). How are we able to use CLT? " This is a very common point of confusion, and many, many, many highly viewed videos on RUclips really botch this and perpetuate the misunderstanding. We don't need to repeatedly sample for the CLT to have meaning. The CLT has implications for a single samples in general. The "Suppose we repeatedly sample..." argument is supposed to be illustrative, and I use it myself, but the more decades I put behind me the more I think this just screws people up. The CLT tells us about the probability distribution of the random variable that is the mean (or related things, like a sum). It absolutely applies even though we have a single sample, since it's just speaking to the probability distribution of that statistic. The repeated sampling argument is just an illustrative argument, not something that has to happen for the CLT to have practical meaning. The LLN and CLT are very fundamental things to probability and statistics, and they have implications for hypothesis testing, but asking how they relate doesn't really work as they are very different things. The LLN and CLT are fundamental mathematical theorems, whereas hypothesis testing is a technique that helps us answer questions in certain spots. The biggest thing the CLT brings to the table in applied statistics is that it allows us to use well-developed methods based on the normal distribution, when the distribution of our data is not normal or is unknown, provided we have a large enough sample size.

While I personally lean towards two-tailed alternatives in the vast majority of situations, I think the use of a one-tailed procedure is reasonable in this situation. Before collecting the data, there was a strong belief (based on previous studies and information) that puerarin would have a tendency to reduce alcohol consumption. So it wasn't just a "ooooh, I think my new drug is better" sort of argument. (Or even worse, using the data to inform the choice of alternative.) Abusing the use of a one-tailed test can be a form of p-hacking, sure, but I don't think that happened in this study.

Yes, in the sense that for any random variable X, Var(X) = E[(X - mu)^2].

I think it's fine to teach continuous probability distributions without explicitly discussing integration. e.g. Using vague statements like "Probabilities are areas under curves, and we can find those areas with mathematical techniques, and software has incorporated those mathematical techniques for us." I think someone can develop a very good understanding of statistics from that perspective. Keep in mind that most of the continuous distributions we use in statistical inference (e.g. normal, t, F, chi-square) do not have closed-form cumulative distribution functions and must be integrated numerically. So whether someone knows integral calculus or not, in the end areas are found using software. For a full and deeper understanding of what's going on? Sure, knowledge of integral calculus is meaningful. At a level to achieve an understanding of applied statistics? I don't think knowledge of integral calculus is necessary.

@@jbstatistics Thanks for the quick reply🙏 I'm actually preparing for an exam, and when I got to the topic of continuous probability distribution, I got a little confused because of the integral. I meant more to solve exam questions related to continuous probability distribution than to have a theoretical understanding. It seems that to reach the final answer, you need to know integral calculus (or have an advanced calculator).

@@Didier-cu6cb For any of those distributions I named, and many others, there is no closed form solution for the integral. It does not matter how great of an integrator one is, the integrals can solved only by numerical techniques. We do that using software. The world's greatest integrator and a person knowing no calculus whatsoever solve the problem in the same way: By asking software for the appropriate area.

@@jbstatistics Right. Thank you for your response & content.

I suggest you limit the "should haves" until you have a deeper understanding of the situation. If your logic here was valid, then there would be no such thing as a two-sided alternative. The value of the statistic will always be on one side or the other, so if we let that determine things we'd always have a one-sided alternative. So your logic can't be valid, or nobody would ever speak of a two-sided alternative. The choice of alternative ******never never never never never ever ever ever ever ever ever**** depends on the value that we see in the sample that we're using to carry out the test. If we do that we're distorting the math and then making false statements in the end. The choice of alternative depends on the nature of the situation at hand. One should be able to write out the appropriate hypotheses without ever looking at the sample data. If the sample data has influenced your choice of hypothesis, then you've violated a fundamental requirement of hypothesis testing.

I think bringing that up in an introductory video on the geometric distribution does more harm than good. It's an extra layer of confusion that people don't need at first. The difference in the random variables, the difference in the means, describing why the variances are the same...it just takes away from the big picture of what the geometric distribution does for us. Sure, I bring it up elsewhere, especially as I use R in my courses and R uses the other definition of the r.v., but I think it would cause more confusion than it's worth in an intro video. Once one is understood, the other comes naturally.