For Pete's sake. I offered you the chance to just let it go. But you insist on somehow trying to bring your complete ignorance of the subject matter to light.
The
entire point of a t-test is that the statistic is NOT normally distributed. Do you even know where the t-value in "t-test" comes from? I'm sure you could google how it's calculated, but do you actually understand it? There would be NO NEED for it if we assumed all variable values were normally distributed.
Now you're saying "the data need to be uncorrelated." Here's a news flash:
a correlation requires more than one variable. There is
no such thing as a correlation within ONE variable. The fact that you would even bring this up - again, only after *I* mentioned correlation - exposes your lack of understanding.
As if that weren't enough:
1. It requires the data to conform to some probability distribution the general form of which is known a priori. "simple t-test" and "student t-test" usually mean that we assume a normal distribution. This is obviously incorrect.
If this is incorrect, then
every t-test ever performed in any context is incorrect. The test does NOT assume the individual
observations in the sample are normally distributed - it assumes that over
repeated samplings, the
statistic we are measuring will assume a t-distribution (which approximates a normal distribution at very large sample sizes) if the samples are drawn from the
same population. It's the Central Limit Theorem, the most basic freakin' mathematical
proof that exists in statistics. The reason for Student's t's
existence is to address sample size.
Seriously - do you think that when we calculate the inflation rate, personal income changes, the incidence of cancer, etc. that we assume that prices, income, cancer cases etc. conform to a normal distribution? This is a misunderstanding that student interns get over in their first week!
2. The simple t-test assumes that each measurement of the same type has the same a priori distribution.
Exactly - in essence, that the
population does not change between samplings. If you had bothered to read my response to Rainy-Day, you'd actually understand what "population" means, and if you had the ability to comprehend "
in the same situations from which the sample was drawn," you wouldn't be posting utter nonsense as if it had any relevance to what we're talking about.
3. The simple t-test requires the data to be uncorrelated, which we know to be false.
Again - YOU NEED AT LEAST TWO VARIABLES to produce a correlation. Running a correlation between a single variable and itself *always* results in perfect correlation, which is utterly meaningless because it is ONE VARIABLE.
A case can certainly be made that a t-test is not the most appropriate test for the data. But the reasons have nothing to do, even remotely, with the ones you posted above.
Honestly, I had started to feel bad after my last post. I was beginning to think you did have some background at one point, but were rusty and had maybe made some honest mistakes, exacerbated by your reaction to my poor tone in my third post (for which I now apologize). But after this last post, it's becoming clear that you are doing nothing but looking up terms I put up there, trying to grasp their meaning (and failing), and coming back to post even more gibberish in the hopes that it will help you save face. In fact, it's doing the opposite.
More?
According to chi-squared (as utilized by you), this proves (likelihood of error a smidge more than 1%, well within the 5% that you expressed comfort with)
a) "Error" has nothing to do with that probability, and "error" has a completely different (and important) meaning, even in this very test. I even explained to you what the probability figure represents, and you STILL misunderstood it.
b) What "I express comfort with" is not important. What
the vast majority of researchers use as convention is. If you'd like to propose a different target alpha, feel free - the results will be similar.
Which of the following would explain this result:
1. The Patriots are more likely to score a TD on drives starting within their own 20 than when they have better field position
2. The use of chi square in this instance (with your methodology) produces an erroneous result.
I could understand Joe FootballFan presenting a false dichotomy. For someone who claims to know how to apply statistics in any remotely credible way to commit a fallacy like that - it completely strains credulity.
The figures we are looking at explain what happened, and what can be expected to happen
under similar conditions. They don't explain why. There are literally dozens of possible explanations out there, probably the
least credible of which is the idea that "math is wrong."
Do opponents with crappy goal-line defenses tend to try to pin kicks deep, rather than going for touchbacks? Are the Patriots better at wearing down defenses over long drives than other teams? Do the Patriots fair-catch more on deep punts when they feel the opposition's goal-line defense is crappy? I don't know. The numbers don't tell us why they are the way they are - no single stat does - but there are MANY possible explanations, not all of which may be in the Patriots' control.
I think we can safely choose option number 2, but if you believe otherwise feel free to send BB and email and tell him to stop trying to return kickoff returns and punts beyond the 19.
Another fallacy. If you understood what a significance test meant, you'd understand why. Or would you like to assert that any of the tests I've proposed is directional? If so, write up a proof - you'll be hailed as the greatest statistical genius of the past 50 years.
On the other hand, publish a research article containing either of the fallacies above, and you're the laughingstock of the research community.
It is perfectly possible for a person with a firm grip on the underlying subject matter to determine that a number has been artificially inflated by a small number of trials WITHOUT doing any calculations. That is what I did. The absurd number of 78.5% results primarily from the fact that only 14 trials occurred (and that the results of these trials were highly correlated, not independent at all).
You have presented no evidence that you have anything remotely approaching a "firm grip" on the subject matter. You made a statement asserting that the sample size was too small, which I *thoroughly* addressed above. I even addressed the (large) probability that the conversion rate would be lower after repeated trials, and I gave you the 95% confidence interval - which you could easily have done, even on the first page, if you had any idea what you were talking about.
You could even have proposed a binomial test, which would reveal that the 11-for-14 is not significantly different from 50-50. You could have pointed out that, given the t-test results, the "true" value even has a chance of being below 54%. But you didn't, which speaks volumes, considering it would be a fact that might support your case.
Instead of addressing the mathematical
facts I posted, you come back with inaccurate and completely ignorant claims about the validity of assumptions that you clearly don't understand. And you continue to maintain not only that a) you have a good grasp of statistical methods - which, by the way, would not have been brought up in this thread had you not insisted that I "back it up" - but b) that your original statement, now disproven in multiple ways, is true?
On top of that, you have the gall to call
my "statistical knowledge," for which I am professionally employed, "deficient"? Unbelievable.
Finally:
That said, I did not mean calling you an ass to be an ad hominem attack. I called you an ass because you behaved as one, but it is perfectly possible for somebody to be an ass and correct at the same time. I provided ample information as to why your arguments were wrong, and it has nothing to do with you being an ass.
Here are a couple of things you can do in the future to avoid being called an ass:
A. Don't respond to a thread about football statistics with a math question.
B. Don't accuse people of being ignorant if they don't immediately respond to your math question.
C. If you ask a math question, and call people ignorant for not immediately responding, make sure that you have provided all of the information and assumptions necessary to provide a single, indisputable answer.
Point taken. I can see how one could view my early responses as "baiting," and for that I apologize. I had hoped to get into an intelligent debate on the merits of such an analysis, but obviously it didn't turn out that way. I take as much responsibility for this devolving into flaming as anyone else.
And here's my advice to you: Just say "I don't know" if you don't know.