Suppose we have a coin with a probability p of coming up heads and q = 1 − p of coming up tails on any given toss (so the coin is biased unless p = 0.5), but we are not given what p or q are. We know, from class or the textbook, that the expected value of the number of heads in n tosses is E(X) = np. If we repeatedly flip the coin and record the results, the number of heads that actually turn up, call it Eˆ, can be divided by the number of tosses, n, to give an estimate, ˆp = E/n ˆ , of p. The problem is that we’re likely to have to be pretty lucky for ˆp = p, so the real question is how likely it is that ˆp is close to p. (This sort of thing is what
Suppose we have a coin with a probability p of coming up heads and q = 1 − p of coming up tails on any given toss (so the coin is biased unless p = 0.5), but we are not given what p or q are. We know, from class or the textbook, that the
1. How many times do we need to toss the coin, keeping track of the number of tosses and the number of success, to ensure that P(0.95p ≤ pˆ ≤ 1.05p) ≥ 0.95? Please justify your answer as fully as you can
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