Testing support and resistance

(a) Prove Proposition 4A.2. (b) You will test whether a level-generating algorithm (Definition 4A.4) produces "bounces" more often than chance: at each touch, record whether price next moves $h$ against the approach direction before moving $h$ through the level. Under the null the bounce probability is $\tfrac12$. How many independent touches do you need to detect a true bounce probability of 0.55 with a one-sided 5% test and 80% power? (c) List three reasons why touches are not independent in practice and how Osler (2000) handled the problem with artificial levels.