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**Unformatted text preview: **MAT2378 Rafal Kulik Version 2009/Oct/30 Rafal Kulik MAT2378 Probability and Statistics for the Natural Sciences Hypothesis PART 1 Comments • These notes form an introduction to hypotheses testing. They do not correspond to any section in the textbook . The closest section would be 7.10. • I’m planning to spend one lectures on this material. Rafal Kulik 1 MAT2378 Probability and Statistics for the Natural Sciences Hypothesis PART 1 Claims and Suspicions Consider the following scenario: person A claims they have a fair coin, but for some reason, person B is suspicious of the claim, believing the coin to be biased in favour of tails. Person B flips the coin 10 times, expecting a low number of heads which intend to use as evidence against the claim . Suppose that there are 4 heads. This is less than expected for a B (10 , . 5) (i.e. 5) which is what we would have from a fair coin; it is more like a coin with P ( head ) = 0 . 4 . But does this really constitute any evidence against the claim? If the person A claims that the coin is fair, then the number of heads, X , has B (10 , . 5) distribution. The value 4 is still close to the mean of the distribution B (10 , . 5) . Furthermore, P ( X = 4) = 0 . 21 , thus even if X ∼ B (10 , . 5) , the event { X = 4 } is still very likely. Thus, it seems that there is no evidence against the claim that the coin is fair. Rafal Kulik 2 MAT2378 Probability and Statistics for the Natural Sciences Hypothesis PART 1 B(10.0.5) Prob. Histogram x P(X=x) 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 Rafal Kulik 3 MAT2378 Probability and Statistics for the Natural Sciences Hypothesis PART 1 The sentence it seems that there is no evidence against the claim that the coin is fair is very important. We did not reject the claim that p = 0 . 5 (i.e. the coin is symmetric), but it doesn’t mean that in fact p = 0 . 5 . Accepting , or rather non-rejecting the claim is a very weak statement . To see this why, let’s consider a person C, who claims that our coin from the example above has p = 0 . 3 . Under the claimed B (10 , . 3) , P ( X = 4) = 0 . 22 , thus we do not have enough evidence to reject neither p = 0 . 5 nor p = 0 . 3 ....

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