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In Chapter 2 we have seen how a 00 plot or a probability plot could be used to assess visually whether the quakes dataset could be modelled using an Exponential(0) or not. (For this homework, you can also assume the quakes dataset is drawn from an Exponential(9) distribution.) Although useful we would like to complete this graphical method with a statistical hypothesis test which would lead to a more objective and principled decision in order to answer the question whether the Exponential(9) distribution is appropriate or not. Numerous tests have been proposed in the literature (in particular in order to test normality) and we focus here on the Anderson-Darling test. We assume an observed random sample x1 , , xn from a probability density fX(y; 6) (and corresponding cumulative distribution function F X(y; 9)), i.e.x1, , xn are \"D samples of the random variable X. The Anderson-Darling (AD) test statistic is given by + (y) FX()';9))2 TAD(x1,...,xn) =nm mfx0dya where [301): #{ie {1,...n,n} 2x,- Sy}, is the empirical distribution function of the sample x1, . . . , x". This test can be used for any hypothesised F X(y; 6) (e.g. a normal, exponential etc. distribution). 1. (10 marks) State, in words and at most two sentences, the null and alternative hypotheses corresponding to the question above. 2. (10 marks) Briefly explain why the statistic suggested by Anderson and Darling may be useful to answer the question we are trying to solve? You may find it useful to refer to a probability plot and you should comment briefly on the roles played by the three terms (a) (y) Fx(y;9))2 (blFx(y;9)(1-Fx0';9)) (cum; 9). 3. (10 marks) For an observed statistic to)\" state the definition of the p-value for this test. You should state precisely the probability distribution of any random variable you may use and can assume 9* to be known for the moment