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Unformatted text preview: MTH4106 Introduction to Statistics Notes 6 Spring 2012 Testing Hypotheses about a Proportion Example Petes Pizza Palace offers a choice of three toppings. Pete has noticed that rather few customers ask for anchovy topping. He thinks that if fewer than 1 / 5 want anchovy topping then he should scrap it and replace it by another topping. How should he proceed? Let p be the proportion of the pizza-eating population who like anchovy topping. We do not know what p is. Pete does not need to know the exact value of p . He needs to know whether p > 1 / 5 or not. So there are two hypotheses: p > . 2 p < . 2 . Mathematically and logically, the roles of these two hypotheses are symmetric: each is the negation of the other. In practical terms, they have different roles: if p > . 2 then Pete doesnt change anything; if p < . 2 then he has to go to the trouble and expense of replacing anchovy topping by a new flavour, with the risk that the new one is even less popular. If we could definitely say which hypothesis is true, then this practical difference would not matter too much. But unless Pete asks all of his customers for their preference, all that he can do is to gather evidence in favour of one hypothesis or the other; he can never be quite sure. Because of the expense and risk involved in changing the flavour, Pete will want fairly strong evidence that p < . 2 before he decides to change. Here p > . 2 is called the null hypothesis (written H ) p < . 2 is called the alternative hypothesis (written H 1 ). 1 type of investigation null hypothesis alternative hypothesis to see whether a procedure should be changed the hypothesis that supports the status quo the hypothesis that supports a change to find out if an out- rageous claim is true claim is false claim is true to explain phenom- ena simpler explanation more complicated explanation Example To investigate whether girls are as good at mathematics as boys are. 200 years ago, this would have been considered an absurd suggestion, so the null hypothesis would have been boys are better; nowadays, we dont think that it is absurd, and it is simpler to assume that gender has no effect on mathematical ability, so the null hypothesis would be gender makes no difference. Example In a criminal court in England and Wales, the accused must be assumed innocent unless there is overwhelming evidence to the contrary, so the null hypothesis is the accused is innocent....
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- Spring '12