binomial/hypothesis tests

#1
hello

for the following question i was able to answer part A and B but i dont know how to answer part C

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You have two coins, one that comes up heads 50% of the time and one that comes up heads 70% of the time. A friend of yours who wants to win some bets borrows what he thinks is your 70% coin to use on some unsuspecting people. Before using the coin, however, he decides to try it out on himself, by tossing it 10 times. He decides that if heads come up 8 or more times out of 10, he will conclude that you really did give him the biased (70% coin).

a) If in fact, you gave him the fair (50%) coin, what is the probability that he will incorrectly conclude that you agve him the biased coin? answer: 0.055

b) if, in fact, you gave him the biased coin, what is the probability that he will incorrectly conclude that you gave him the fair coin? answer: 0.617

c) if he now tosses the coin 25 times and uses alpha ≤ 0.05 to create a new decision rule. What is the probability that he would now incorrectly conclude that you have him the fair coin if, in fact you gave him the biased coin? answer: 0.488
 

JohnM

TS Contributor
#2
If you are flipping the fair coin 25 times, and are looking for a decision rule with alpha = .05, then you would reject Ho (the coin is fair) if you got 18 or more heads.

However, if you are in fact given the biased coin, but you use the same decision rule (18 or more heads), the probability of getting 18 or more heads with P(H) = 0.7 and P(T) = 0.30 is 0.512.

Therefore, the probability of getting 17 or fewer heads (and thus failing to reject Ho: the coin is fair) is 1 - 0.512 = 0.488.