We would like to check whether a given coin is fair. Therefore we flip that coin 500 times, with 230 tails observed. Letbe the probability for a tail and let X Binomial(500,).

1. If we assume that the coin is fair, what is the appropriate null hypothesis H0 in terms of ? What is the alternative hypothesis H1? Consider a two-sided hypothesis test.

2. What is an appropriate discrepancy measure D for the test problem formulated in (a)? What is the observed value d of D?

3. Using the discrepancy measure D in (b), calculate the p-value for your hypothesis H0 in (a) and conclude whether the coin is fair. To this end, you may use an appropriate approximation.