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.

For the calculation of the above relative frequency, we are actually usingsimulation to approximate the probability of pˆ50 = (x1+x2+..+x50)/50 pˆ∗ given that {Xi, i = 1, 2, · · · , 50} are independent and identically distributed as X ∼ Ber(0.4). We could use the following theoretical ways to re-calculate it. (Note that pˆ50 is the random variable notation from which we got the above 1000 observations pˆk . Again pˆ∗ is just a particu- lar value as the sample proportion for Jack’s particular sample.)

(a) Use Pr(pˆ50 < pˆ∗) = Pr(X1 + X2 + · · · + X50 < 50pˆ∗) to calculate the probability. Hint: X1 +X2 +· · · +X50 follows Binomial distribution.

(b) Use central limit theorem for pˆ50 = (x1+x2+..+x50)/50 to approximate the probability.

Answer 1

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Answer:

a)

b)