The process is called hypothesis testing. If you are inquiring on how to prove or disprove a claim, search for the article on wikipedia.
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Rejecting a true null hypothesis.
Yes, usually.
An alpha error is another name in statistics for a type I error, rejecting the null hypothesis when the null hypothesis is true.
No. Rejecting the Null Hypothesis means that there is a high degree of probability that it is not correct. This degree of probability is the critical level that you choose for the test statistic. However, there is still a small probability that the null hypothesis was correct.
Rejecting a true null hypothesis.