I am trying to understand the concept of p-value.
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 site 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.