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# using t test to compare sample mean to population mean

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Hi
I am new to statistics to please bear with me.

I am using a t test to compare a sample mean to a population mean (it's a question in a book). The t test gives a vaslue of 2.536. This gives a two tailed probability of
0.02 <p <0.05

and a one tailed probability of
0.1 < p< 0.025

The two tailed probability is double the one tailed probability. This doesnt make sense to me. Shouldn't it be the other way around.

Please can you clarify for me

thanks
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Commented:
No it is just right. Each tail is similar to the other. If you have a 5% possibility to be in one tail you have a 5% probability to be in the other tail too. Total 10% probability

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Commented:
Remember that p is the probability of obtaining a t of 2.536 or greater just by chance alone (i.e., not because your sample is actually different from the population).

If you're doing a one-tailed test, p is the probability of getting a result just in the one tail you're interested in. If you're only interested in really high scores, you care about the p of t >= 2.536, and if it's really low scores, it's p of t <= -2.536. Either way, in your case, p is between 0.01 and 0.025.

If you're doing a two-tailed test, p is the probability of getting EITHER t >= 2.536 OR t <= -2.536 just by chance alone. If your sample truly is random and there's nothing special about it, there's an equal chance of it being in the extreme of either tail. Thus the probability of it being in one tail OR the other is double the probability of it being in one particular tail.
Commented:
Maybe you're confused because you're thinking of the critical region, which is distinct from p.

First, define alpha as the risk of a type I error. This is something you decide. Say you will accept alpha = .05.

Next you decide if you want a 1 tailed or 2 tailed test. If it's one tailed, the entire .05 is in one side of the distribution. If it's two tailed, alpha is split in 2, with .025  on the left and .025 on the right. But alpha is still .05.

Thus you have an idea in your mind that a two tailed test has a smaller "p-value" but what's small is the size of the rejection region which is alpha/2. A two tailed test has a smaller critical rejection region on each side of the distribution.

The other commenters have it right when they're explaining why the p-value is larger for 2 tailed.

What I'm pointing out is why you are surprised by the fact that a 2-tailed p is larger. I think you're surprised because you're imaging the distribution with alpha/2 critical regions on each side of the curve, and you're thinking "smaller." But those are not p-values. Those are components of alpha.

Commented:
thanks

Commented:
thanks

Commented:
by the way jtm11, you were right in seeing how I had confused myself. Very intuitive
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