Question

John and Jack found a coin on the sidewalk. They argued about the fairness of the coin. John claimed 40% to have Heads according to his careful observation of the coin. Jack doubted and in order to infer the fairness of the coin, he tossed the coin for 50 times and got the results as shown below with 1 representing as heads and 0 as tails:

## [1] 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

## [36] 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0

Let p∗ as the true probability to have heads for the coin. Note that p∗ is a characteristic of the coin, and we want to make some inference about this unknown parameter. And denote X as the random variable which takes 0 if tails show up or 1 if heads show up for tossing the coin.

Intuitively, if the guess, say p0, differs a lot with Jack’s sample proportion pˆ∗, then the guess is probably not acceptable. Based on this intuition, let’s propose the following rejection rule: reject the guess p0 if | pˆ∗ − p0 | > 2 sqrt( p0(1 − p0)/n ). Please try out the sequence: p0 ∈ {0, 0.01, 0.02, · · · , 0.99, 1} to get the maximum and minimum p0’s which

are acceptable by using this rule.

Answer 1