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Can someone explain in simple words and with many good down-to-earth examples what is the p-value and how do we find it?

Is it true that it shows what is the probability that the sample we have tested is true not simply by chance? Isn't that the point of the alpha level to guarantee that for an instance, for 95% of the time, it is not a chance that our hypothesis is true? For the rest of the 5% if it is true, it is by chance?

I will greatly appreciate if you can explain the difference between p-value and alpha level(significance level), how are they connected, what do they tell me for the "real world" and give me examples! (Because any mathematical explanation seems to confuse me more)

Thanks in advance!

PS: I am aware that there are questions about p-value with good explanations out there, would then just like to have answers for the other questions.

Mumima
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    If you would just like to have answers to the other questions, please ask only those questions you want answers to. – Glen_b Apr 05 '16 at 16:58
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    See e.g. [What is the meaning of p values and t values in statistical tests?](http://stats.stackexchange.com/q/31/17230) & [Comparing and contrasting, p-values, significance levels and type I error](http://stats.stackexchange.com/q/33453/17230). I'll eat my hat if you can ask anything about the interpretation of p-values that hasn't already been asked on this site. – Scortchi - Reinstate Monica Apr 05 '16 at 17:13

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