# Reading 8 Practice Question #3B- Multiple Regressions

_{0}. If H_{0} is X=0; the p-value is 0.00236 and the t-statistic of X is -3.0565, then why do we not reject H_{0}? n=500. Its a 2 tailed test. So do we reject H_{0 } because -3.0565 is less than -0.00236 and it is a 2-tailed test?

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That’s better.

You might reject H

_{0}, and you might not. You have to compareptoα. What’sαhere?You’re trying to compare a

t-statistic to a level of significance. They’re completely different things.You either compare

ptoα, or else you compare the calculatedt-statistic to the criticalt-value.Simplify the complicated side; don't complify the simplicated side.

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Thank you! The question did not give a though. So in that case should I use the p-value and the df to find the critical t-value? The T Table has columns for p=0.005 and p=0.01, so I guess the critical t-value would fall somewhere between?

No, you cannot use

pto get the critical values. If you did, you’d essentially be comparingpto itself.The question doesn’t ask whether you will reject H

_{0}or not; it simply asks you to interpret thep-value of 0.00236. The answer is that that is the lowest level of significance (i.e., the lowestα) at which you would reject H_{0}.Simplify the complicated side; don't complify the simplicated side.

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Classic case of overthinking. Thank you. You have been a great help

My pleasure.

Simplify the complicated side; don't complify the simplicated side.

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Agreed, and agreed.

Simplify the complicated side; don't complify the simplicated side.

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Alpha is an arbitrary threshold. What if I change my threshold to 0.00236 this time instead of 0.05 or 0.005 or whatever is the common threshold for the specified study. Wouldn’t I still able to reject H0? I think yes.

So, under the most common threshold of 0.05 or whatever, having a p-value of 0.00236 indeed gives me the right to say that lowest threshold possible to choose and still be able to reject H0 would be an alpha = 0.00236.

Sincerely, didn’t understand the mistake about mixing “functional definition” vs “interpretation”. Perhaps you can share more light over it.

Las almas de todos los hombres son inmortales, pero las almas de los justos son inmortales y divinas.

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The interpretation of a p-value is NOT saying it’s the lowest alpha that you could choose to reject Ho. A p-value is a probability– what does it mean? It’s the probability of obtaining a summary measure at least as extreme as the observed one if we assume the null hypothesis is true. This tells you it’s a probability, what it’s the probability of, and what assumptions are entailed in this p-value.

The functional definition isn’t an interpretation; it’s like saying “I hear noise” when someone asks for an interpretation of a jazz music piece.

Not jazz.

Brahms:

Simplify the complicated side; don't complify the simplicated side.

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Ok, so, the problem is not saying that the p-value is the lowest possible alpha in order to still be able to reject H0. The mistake resides in that this is not the correct interpretation of a p-value.

I think the problem of mistaking the interpretation of a p-value resides (unluckily) in the definition of alpha and its use with p-value (we compare both to take decisions). As alpha threshold is considered a probability of error type 1, then, as we compare p-values with alpha we mistakenly consider p-values also as probabilities of committing type 1 errors. However, they are not.

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Do statistics book used in undergrad cover p-values adequately? I mean I have the impression p-values are just

mentionedwhen we practice regressions, and are not properly explained / defined until the student get advanced statistics books. Perhaps we need to bring down p-values for the pedestrians before keep falling in the mistake of confusing p-values’ conceptual definition.Quoting Wikipedia’s p-value definition:

“…thep-valueorprobability valueorasymptotic significanceis the probability for a given statistical model that, when the null hypothesis is true, the statistical summary (such as the sample mean difference between two compared groups) would be greater than or equal to the actual observed results.”This definition is hard to be understood when the student have not achieved yet a reasonable level of knowledge :/

I agree that CFAI, at least, shouldn’t use the incorrect interpretation we talked about.

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Wikipedia is a correct definition, and it’s much easier to understand with a picture. But this also is easy to understand when you know how to calculate a tail probability (say, the probability that some random variable is at least as large in magnitude of a z score of 1.2). This idea transfers easily when you show that in order to calculate the z score you need to assume a true mean value (null hypothesis) and that the probability comes from the assumption that chose true mean value is correct and the distribution (and all other assumptions) is correct. Once you do this for a single observation, you transfer this idea to sampling distributions (making the move to a p-value) which are single observations of sample statistics. The previous example is literally something covered in intro to stat classes. A good teacher will make these connections but also the teacher’s education is important, too (i.e. learning stats from a nonstatistician is more likely to be wrong than learning from a statistician).

Long story short, the deficit is in the fundamental knowledge; the basic teaching is inadequate, so the subsequent topics are much harder for the student. CFAI could rectify this in the QM book for level I, but I double that will happen. People aren’t avoiding a “hard definition” it’s that they don’t know or don’t understand it themselves.