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In Reply to: RE: statistics question posted by mike1127 on June 25, 2009 at 13:31:55
OK, a little on null hypothesis...A null hypothesis is some statement that has can be modeled mathematically, and hence something you can compare experimental results against, specifically against what the mathematical model "says" about the value obtained experimentally.
For example we can mathematically model the probabilities associates with coin tosses, the probability of getting a head (or a tail) on a single independent toss (trial) is 50%, and we can answer questions like:
. what is the probability of getting exactly 12 Heads when we toss the coin 125 times.
. what is the probability of getting at least 23 Tails when we toss the coin 50 times.As it turns out the coin toss experiment for the second question, probability of getting X Heads (or Tails) for n tosses (trials) is modeled by the Binomial Mass Function when p=.5 (the probability of getting a head (or a tail) on a single independent trial).
Hence jumping ahead we see that the traditional DBT/ABX test is similar to coin tosses when modeled mathematically.
But before we get there ...
---
OK, let's say we have a cable test. We want to test if the cables "sound different". So what is our null hypothesis? Is it...
The cables sound different.
So we do our test, say we get 34 out of 50 correct, what does it mean? What is the mathematical model for "The cables sound different"? There isn't one! So forget it, that's not the null hypothesis!
Instead we propose that when it comes to being able to distinguish between the two cables (their "sound") that such is determined by "chance" alone, and that on a single independent trial the chance of correctly identifying X (i.e. in a traditional ABX test) is exactly 50%. Now we are getting somewhere, in fact that is our null hypothesis but we can put it simply as...
Distinguishing between the cables is determined by chance alone with p=.5
Hence now when we run the test and get some result we have something to compare against, namely the Binomial Mass Function with p=.5 (simple BMF hereafter) Aren't we clever!
Then we get to level of significance. Well for any test where the null hypothesis is modeled by the BMF (or by some other function for that matter) we decide in advance what result we require to "reject" the null hypothesis, that is we set a "level of significance" (LOS). A 5% LOS means...
We will reject the null hypothesis for any result for which the probability of obtaining that result due to chance alone is less than 5%.
So we run a test getting X correct identifications and the BMF tell us that probability of getting at least X correct identifications is 7.5%. Well that's greater than our 5% LOS so we *don't* reject the null hypothesis, in other words we accept that X correct identifications could have been due to chance alone... which fails to demonstrate a difference between the cables.
Now if we get Y correct identifications and the BMF tell us that probability of getting at least Y correct identifications is 1.3% then that's less than our 5% LOS so we *do* reject the null hypothesis, in other words we agree that the result *could not have been due to chance alone* (given our LOS) ... which would then imply that there is a difference between the cables.
Hope that helps.
Everything matters, don't forget to tweak your placebos!
Edits: 06/25/09 06/26/09
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