Making sense of confidence intervals and P-values from visual examples

P-values determine whether a research proposal will be funded, whether a publication will be accepted, or at least whether an experiment is interesting or not. To start with, let me give you some bullet points about P-values' properties:

• The P-value is a magical probability, but it is not the probability that the null hypothesis will be accepted. Statisticians tend to search for supportive evidence for the alternative hypothesis because the null hypothesis is boring. Nobody wants to hear that there is nothing interesting going on.
• The P-value is the probability of making mistakes if you reject the null hypothesis. If the P-value is very small, it means that you can safely reject the null hypothesis without worrying too much that you made mistakes because randomness tricked you. If the P-value is 1, it means that you have absolutely no reason to reject the null hypothesis, because what you get from...