Recitation 20 Mr. President, either p is within 0.03 of 0.53 or something very strange (less than 5-in-100) has happened For each statement answer: (i) Are you justified to claim it?(ii) Is it true? Solution. The first statement is clearly off the mark (i) Since you havent asked all Americans, you can only make probabilistic state ments about p (ii) The statement is also false, since p=0.52. However, with a different choice of voters, it could have been true. Of course, even in that case, you wouldnt be justified to claim it The second statement is also wrong (i) The unknown constant p is either within 0.03 of 0.53 or more than 0.03 away of 0.53. In the first case, the probability of it being as you claim is 1; in the second case,it is 0. The crucial point is that you dont know which case holds. You could make the above claim, only if you knew you were in the first case. Sadly, ou dont (ii) The claim is actually true in this case. Since p=0.52, the unknown constant is indeed within 0.03 of 0.53. So the probability that you talk about is 1, and therefore at least 95%. But as we said it could be o and then the statement would be false The third statement is the correct one (i) You are justified to claim this statement. To see why, start with the statement either|0.53-pl≤0.030r0.53-p>0.03 which is clearly true. Now read it as follows: Either p is within 0.03 of 0.53 or it is not and therefore my randomrandom variable P took a value from a set that is hit only 5 times in 100. So, clearly either p is within 0.03 of 0.53 or something (ii)The statement is true. In the particular case, it is true because the first half of it Is true.Recitation 20 3 • Mr. President, either p is within 0.03 of 0.53 or something very strange (less than 5­in­100) has happened. For each statement answer: (i) Are you justified to claim it? (ii) Is it true? Solution. The first statement is clearly off the mark. (i) Since you haven’t asked all Americans, you can only make probabilistic state￾ments about p. (ii) The statement is also false, since p = 0.52. However, with a different choice of voters, it could have been true. Of course, even in that case, you wouldn’t be justified to claim it. The second statement is also wrong. (i) The unknown constant p is either within 0.03 of 0.53 or more than 0.03 away of 0.53. In the first case, the probability of it being as you claim is 1; in the second case, it is 0. The crucial point is that you don’t know which case holds. You could make the above claim, only if you knew you were in the first case. Sadly, you don’t. (ii) The claim is actually true in this case. Since p = 0.52, the unknown constant is indeed within 0.03 of 0.53. So the probability that you talk about is 1, and therefore at least 95%. But, as we said, it could be 0 and then the statement would be false. The third statement is the correct one. (i) You are justified to claim this statement. To see why, start with the statement either | | 0.53 − p ≤ 0.03 or | | 0.53 − p > 0.03. which is clearly true. Now read it as follows: Either p is within 0.03 of 0.53 or it is not and therefore my random random variable P took a value from a set that is hit only 5 times in 100. So, clearly either p is within 0.03 of 0.53 or something strange has happened. (ii) The statement is true. In the particular case, it is true because the first half of it is true