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The first formula is the general ttest formula. The second is the correlation ttest formula which is less wellknown (at least to me). If i understood it correctly, the second formula’s application is based on a null hypothesis that the correlation between two samples is equal to zero and the alternative hypothesis is there is some forms of correlation.


It seems like you’ve got it, but just in case (plus it helps me to talk about it haha)….
The General Ttest is just a subsititute for the Ztest when your sample size is small and you are not given a population standard deviation. I think the thresold is 30 for sample size? These two are the go to ones we all know about and use for basically everything, the reason being standard deviation of the variable is either given or calculable/estimable.
The correlation Ttest, though I’m not certain on the exact reasoning, but I would dare to claim that this is the equation for seeing if the variable(r) represent the “avg” because correlation coefficients aren’t based on a single point in time, as B’ would be. Given you can “see” a B’, you can know how far off it is from the average(B). But how can you calculate r’ if it is already a variable using multiple points in time? Even worse how could one calculate a standard deviation of a correlation coefficient. You’ve already inputted time as a variable into establishing our correlation coefficient. The standard deviation would have to be based on another observation that the average coefficient incorporates, but the alreadyincalculable r’ does not(just like Bavg incorporates time while B’ does not). This new equation(without me being able to explain the how), adjusts for this idea that you cannot calculate an r’ or standard deviation of r.
Basically if the world was nice and dandy, and information were abundant, we would be able to just memorize the one Zscore equation, but these two versions provide for when either
1. we dont enough have info
2. And/or our variable wasn’t calculable by one observation alone in the first place.One thing that may be worth noting: The general test is usually used for looking for changes; i.e. the average SAT score for a school is 2000. Technically speaking, the test could just be looking for a “onetailed” approach. Either way, these tests don’t usually have an answer you “want, but rather you will adjust for whatever the answer ends up being. (eg. Harvard recruiters want to see if their high school feederschool is still feederworthy. The one tailed ttest says the avg SAT score is now 1800. Recruiters adjust and stop recruiting so many people from that school). They didn’t really care if the average was worse or the same, they just used the output to adjust their efforts.
On the other hand with the correlation, you’re almost always looking to have a huge Tcalc (two tailed test), because that means the variable that you as an analyst chose, actually does correlate to your dependent variable(returns). If it ends up being small i.e. you “accept the null,” well, that test has just proved that you’ve wasted your time using variable that has no correlation to your end of goal of getting higher returns.
To get really complicated…the harvard recruiter could do a connection between the best students and SAT scores haha. This would fall under the correlation ttest. But for the example above, the recruiters assume the SATs do assess “greatness.”
Hope someone doesn’t pick this all apart =)


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