Exact interpretration of p-value and significance of test First question:
Let's say we have a hypothesis test:
${ H }_{ 0 }:u=100$
and ${ H }_{ 1 }:u\neq 100$.
The sample has a size of 10 and gives an average $u=103$ and a p-value = 0.08.
The level of significance is 0.05.
I'm asked the following question (exam):
A) We can conclude that $u=100$
B) We cannot conclude that $u\neq100$
The 2 answers are rather similar, but not the same. I would say B) but I'm not so sure given what I've read.
The p-value here indicates that we cannot reject the null hypothesis, so we cannot accept H1 ?
Second question:
What does it mean exactly that a test is significant ?
Does it mean that we can reject the null hypothesis ?
Thanks in advance.
Regards,
 A: A hypothesis test in the frequentist sense is a procedure by which one arrives at a decision about whether the data contains sufficient evidence to accept the alternative hypothesis.  In other words, there are two choices:  either you reject the null $H_0$, or the test is inconclusive.
The reason why you cannot ever "accept the null" under such a test is because the test statistic and the resulting $p$-value are calculated under the distributional assumption that the null is true.  Therefore, a $p$-value that is not sufficiently small (i.e., smaller than the $\alpha$ level) is somewhat tautological:  it essentially says that, assuming the data indeed is drawn from a distribution that follows the null hypothesis, the probability of seeing a sample as extreme as that you obtained is $p$.  If $p$ is "large," that doesn't mean $H_0$ is true, because you assumed that it was true in order to get $p$ in the first place.
All that you can say if you fail to reject $H_0$ is that the data does not furnish enough evidence to suggest with a high degree of confidence that $H_0$ is false.
Another way to think of it is this:  suppose I give you a coin and you wish to test if it is biased.  If you toss it 10 times and observe 5 heads and 5 tails, that does not necessarily mean that it is in fact fair--you could have observed this result from a biased coin purely due to random chance.  All you can say is that the result you obtained does not furnish strong evidence that the coin is biased.
A: Aren't you also given either the population or sample deviation? If any of these are given,  you assume the data is distributed a certain way, e.g., normally, with $\mu=100, \sigma=\sigma_0$. Knowing this, i.e., the (assumed) parameters of the population you are sampling from, then allows you to compute the probability of obtaining the value of 103 under this distribution (say , but not neccesarily, normal) $\mu=100, \sigma=\sigma_0$. If the probability of this value 103 is less than , say,(the significance level) 5% , then you reject ;otherwise you do not reject. 
So rejecting a hypothesis at a significance level of k% given the assumption $\mu=\mu_0$, given some value s for either the sample deviation , when the sample mean is $\mu_2$ just means that  the probability of obtaining a value of $\mu_2$ in a population with mean $\mu_0$ and deviation s is less than $k$%. Of course there are variants of this test for different parameters.
As above poster wrote, if the p-value-- the probability of observing the value you observed-- is less than the significance value/level, then you reject , but you reject at the given significance value/level. The explanation for what is going on is in the above paragraphs ; let me know if it was not clear.
A: A P-value can be reported more formally in terms of a fixed level α test. Here α is a
number selected independently of the data, usually 0.05 or 0.01, more rarely 0.10. We
reject the null hypothesis at level α if the P-value is smaller than α , otherwise we fail to
reject the null hypothesis at level α.
Now figure out what you have to do?
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