I am trying to understand the concept of p-value.   mentions:     The p-value is defined as the probability, under the assumption of hypothesis $H$, of obtaining a result equal to or more extreme than what was actually observed.   Though, I found quite vogue the term extreme, I'm taking it as the values beyond the observed value of the test statistic; by that I could somewhat conceived the definition.  Then, like my book, the wiki article writes:     The smaller the p-value, the larger the significance because it tells the investigator that the hypothesis under consideration may not adequately explain the observation. The hypothesis $H$ is rejected if any of these probabilities is less than or equal to a small, fixed but arbitrarily pre-defined threshold value $\alpha,$ which is referred to as the level of significance [...]   Hmmm... I couldn't get that point; why does the smaller p-value means a higher significance viz, the null hypothesis is on the verge of extinction?  Also, as this  summarises:           High P values: your data are likely with a true null.   Low P values: your data are unlikely with a true null.      I'm not getting why it is so- why does the lower p-values means the the significance is high?  Wikipedia seems to reason saying this: 'the hypothesis under consideration may not adequately explain the observation'. Can anyone tell me why it is so?  Please help me explaining why lower the p-value, higher is the significance.
After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value?
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