The p-value denotes the amount of unusual sample values we get from a population. It defines how extreme the test statistics is in the direction of alternative hypothesis. Also it defines the consistancy of a test statistics.
Let α be the significance level, then (1-α) is the confidence level.
Assume α = 5% i.e., 95% (1-α) confidence level
Let H0 be the null hypothesis.
If the calculated p-value is less than α i.e., 5%, then H0 will be rejected.
i.e., 5% of the time we would reject null hypothesis H0 wrongly.
i.e., 5% of the time we erroneously conclude that the coin is unfair.
This leads to 5% Type-I error (fail to accept H0 when it is true)
i.e., 95% of the time we accept H0 correctly
i.e., 95% of the time we correctly conclude that the coin is indeed fair.
Suppose we get p-value just above 5%, then H0 is accepted and observed data is a rare event.
Please read this page for p-values & binomial trials.
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