OT basic statistics question

[bighomage]

Hey, this isn't poker related but I'm taking statistics in high school and I am confused by the double-significance test. It seems like all you do is double the tail that is found in a single significance test, but that doesn't make sense to me.
If we use the .05 cutoff for the significance level, and used the stats taken from a simple random sample to get a p-value of .04, that is enough to dispute the null hypothesis. For sake of example, let's assume that the population mean was thought to be 200, but that our p-value suggests that the mean is actually lower than 200. This all makes sense to me.
However, with the same data, it seems to me based on what I was taught, that to find data that the mean simply didn't equal 200 (to show that the true population was either under or over 200), we would double the tail found in the above paragraph and get a p-value of .08. Therefore, with the .05 cutoff, we do not have enough evidence to dispute that the population mean is equal to 200.
So, to sum up my long, probably incoherent question, how can we, with the same data, say that there's strong evidence that the mean is less than 200, while we don't have enough evidence to dispute that the mean IS 200?
I hope I made my question clear enough. I would appreciate any help, because statistics has made a lot of sense to me thus far and I would like to get over this hump.

[gaming_mouse]

Think about like this.

If your null hypothesis is "not equal" (as opposed to "less than"), then there are two possible ways to disprove it. With a very large sample mean, or with a very small sample mean. You are essentially being given two "chances," so to speak, to disprove your hypothesis.

Because of this, you are going to need twice as much evidence to give your argument the same strength it would have if you were only given one chance, which would correspond to a 1-sided test.

Does that makes sense?

If not, I can give you a more detailed answer.

HTH,
gm

[kmvenne]

GM,

Your answers and helpful demeanor are a true asset to this board. Just felt like thanking you.

[gaming_mouse]

[ QUOTE ]
GM,

Your answers and helpful demeanor are a true asset to this board. Just felt like thanking you.

[/ QUOTE ] /images/graemlins/blush.gif

[bighomage]

Thanks a lot for responding. I guess the part I still don't get is why the probabilities above and below the mean are the same. If we find a sample mean that is lower than the population mean, doesn't this suggest that there's a better chance of the actual mean being lower than the suggested one rather than higher?

[gaming_mouse]

[ QUOTE ]
Thanks a lot for responding. I guess the part I still don't get is why the probabilities above and below the mean are the same. If we find a sample mean that is lower than the population mean, doesn't this suggest that there's a better chance of the actual mean being lower than the suggested one rather than higher?

[/ QUOTE ]

You are correct. A sample mean which is lower than the hypothesized mean does suggest that the true mean is lower.

However, when you do an experiment, you must decide on you hypothesis before you do the experiment. If your hypothesis is that the mean is lower than 200, and your sample mean turns out to be 250, you cannot just change your mind and decide to do one-sided test to now prove that the true mean is greater than 250.

Because if you want evidence in either direction to count, the correct hypothesis is that the true mean does not equal 200. And this corresponds to a two-sided test. And in a two-sided test, before we gather any data, we are saying that evidence in either direction will count.

gm

[bighomage]

Thanks a lot. My teacher didn't really explain very well, but it makes perfect sense now.