1. ## Hypothesis Testing: p-value

Definition: A p-value is the probability that the test statistic would take a value AS EXTREME OR MORE EXTREME than that ACTUALLY OBSERVED when H_o, the null hypothesis, is true. If the p-value is samll, it gives evidence against H_o.

Consider the test for the pouplation mean μ of a normal population with known population variance.
Case (i):
Ho: μ=μ_o
Ha: μ>μ_o
p-value=P(Z>Z_stat) where Z~N(0,1)
Here I can understand the > here since the "as extreme or more extreme" is the direction away from H_o towards H_a.

Case (ii):
Ho: μ=μ_o
Ha: μ<μ_o
p-value=P(Z<Z_stat)
Here I can understand the < here since the "as extreme or more extreme" is the direction away from H_o towards H_a.

Case (iii):
Ho: μ = μ_o
Ha: μ μ_o
p-value = P(Z<-|Z_stat|) + P(Z>|Z_stat|) = 2 P(Z>|Z_stat|)
This part I don't understand. How does the "as extreme or more extreme" idea above translates into this formula for p-value?

So, for example, if the observed value of the test statistic for a two-tailed test is 1.65, then the direction away from H_o towards H_a should be to the right (I believe), so it looks like p-value = P(Z>1.65), but the above says that p-value = P(Z>1.65) + P(Z<-1.65). How come?
I think I am confused about the meaning of the sentence "a value AS EXTREME OR MORE EXTREME than that ACTUALLY OBSERVED". What does it actually mean?

Thanks for explaining!

2. This definition is quite similar to the one I made up years ago.
I say worse off than the test stat.
The point here is that in a two sided test. if your test stat is 1.65 then -2 is even worse. YOU decide on $\displaystyle H_a$ if your test stat is -2 or 3 or -4. You decide on $\displaystyle H_o$ for z-scores is near 0 and decide on $\displaystyle H_a$ any time you are far away from zero in either direction in the two sided situation.

3. If the test stat is 1.65, then 2 is even wrose since it's larger than 1.65, I got that.
But -2 is also even worse, why?
I am quite confused about the p-value...

4. Because this is a two sided test.
You will agree with the null hypothesis for z scores near zero.
BUT you reject the null hypothesis for z score that are far from zero in EITHER direction. So a z score of -2 or 2 say the same thing.

5. OK!

In case (i), how come there are no absolute value signs around Z_stat? i.e. why is p-value=P(Z>Z_stat) and not P(Z> |Z_stat|)?

So if Z_stat=-1000, then we still have p-value=P(Z>-1000)?

6. Originally Posted by kingwinner
OK!

In case (i), how come there are no absolute value signs around Z_stat? i.e. why is p-value=P(Z>Z_stat) and not P(Z> |Z_stat|)?

So if Z=-1000, then we still have p-value=P(Z>-1000)?

Because the uniformly most powerful test via Neyman-Pearson say to only reject the null hypthesis for large values of your sample mean in case i.
It's 3am and I am going to bed.

7. Ho: μ=μ_o
Ha: μ>μ_o
So in case (i), what if Z_stat=-1000? Do we still have p-value=P(Z>-1000) in this case?

8. yes, and that's approximately one and you will NOT reject $\displaystyle H_0$

9. Originally Posted by matheagle
yes, and that's approximately one and you will NOT reject $\displaystyle H_0$
Ho: μ=μ_o
Ha: μ>μ_o

So if we OBSERVED a value of sample mean WAY WAY smaller than μ_o, we would still accept (not reject) Ho? That looks a bit weird to me...

10. Originally Posted by kingwinner
Ho: μ=μ_o
Ha: μ>μ_o

So if we OBSERVED a value of sample mean WAY WAY smaller than μ_o, we would still accept (not reject) Ho? That looks a bit weird to me...

why?
if we're testing $\displaystyle H_0:\mu=2$ vs. $\displaystyle H_a:\mu>2$
and you observe the sample mean to be 0, you should NOT accept $\displaystyle H_a$.
The goal is whether or not to accept $\displaystyle H_a$.
AND what if the sample mean was 2? Then your p-value is .5 and again you WON'T decide on $\displaystyle H_a$.

And by the way the null hypothesis of $\displaystyle H_0:\mu=2$ is treated as the same as $\displaystyle H_0:\mu\le 2$.
In this second case the $\displaystyle \alpha$ is obtained by taking the supremum over the region $\displaystyle \mu\le 2$, since it's not a simple hypothesis.

11. Originally Posted by matheagle
why?
if we're testing $\displaystyle H_0:\mu=2$ vs. $\displaystyle H_a:\mu>2$
and you observe the sample mean to be 0, you should NOT accept $\displaystyle H_a$.
I agree that you should not accept H_a (since 0 is even further away from H_a).
But we should still reject H_o (since the observed value is much smaller), right? Can we not accept H_a and at the same time reject H_o? Not accepting H_a does not mean that we cannot reject H_o.

And by the way the null hypothesis of $\displaystyle H_0:\mu=2$ is treated as the same as $\displaystyle H_0:\mu\le 2$.
In this second case the $\displaystyle \alpha$ is obtained by taking the supremum over the region $\displaystyle \mu\le 2$, since it's not a simple hypothesis.
But how can a simple hypothesis and a composite hypothesis be regarded as the SAME??

Thanks!

12. YOU have to decide on one or the other, there's no $\displaystyle H_2$.
BUT what I say is that we either accept $\displaystyle H_a$ or we fail to accept $\displaystyle H_a$.
Since we assume accept $\displaystyle H_0$ from the get go, doesn't insure it's correct.
Again, I look at this like a proof by contradiction.

13. Originally Posted by matheagle
And by the way the null hypothesis of $\displaystyle H_0:\mu=2$ is treated as the same as $\displaystyle H_0:\mu\le 2$.
How can they be treated as the SAME? Can you please explain a bit more on this? I think this idea might be important...

Thank you!

14. Now I am having a little trouble translating the word problem into a set of hypotheses Ho & Ha...

A random sample of 26 students who are enrolled in School A was taken and their SAT scores were recorded. The sample mean was 548 with a sample standared derivation s=57. The principal of School A claims that the mean SAT scores of students in her school is higher than the mean SAT scores of all the students in City B which is known to be 530. (City B is the city where School A is located) Does the data support the principal's claim? Include a set of hypotheses. Use alpha=0.05.

Just to confirm...in this case, the test would be Ho: μ=530 v.s. Ha: μ>530 , am I right?

15. Originally Posted by kingwinner
Now I am having a little trouble translating the word problem into a set of hypotheses Ho & Ha...

A random sample of 26 students who are enrolled in School A was taken and their SAT scores were recorded. The sample mean was 548 with a sample standared derivation s=57. The principal of School A claims that the mean SAT scores of students in her school is higher than the mean SAT scores of all the students in City B which is known to be 530. (City B is the city where School A is located) Does the data support the principal's claim? Include a set of hypotheses. Use alpha=0.05.

Just to confirm...in this case, the test would be Ho: μ=530 v.s. Ha: μ>530 , am I right?
I have checked my notes and textbook and I believe the test Ho: μ=530 v.s. Ha: μ>530 is CORRECT because in many similar problems in my notes and textbook, this is the implied test based on the wording "...claims that the mean SAT scores of students in her school is HIGHER THAN..."

Now my question is:
Why is the test NOT Ho: μ530 v.s. Ha: μ>530? In this case, Ho and Ha are complements of one another. In general, do Ho and Ha always have to be complements?

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