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midterm02 - STATISTICS 174 APPLIED STATISTICS MIDTERM EXAM...

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STATISTICS 174: APPLIED STATISTICS MIDTERM EXAM OCTOBER 15, 2002 Time allowed: 75 minutes. This is an open book exam: all course notes and the text are allowed, and you are expected to use your own calculator. Answers should preferably be written in a blue book. The exam is expected to be your own work and no consultation during the exam is allowed. You are allowed to ask the instructor for clarification if you feel the question is ambiguous. Answer both Parts, A and B, as far as you are able to within the time allowed. Credit will be given for all partially correct answers. Show all working. In questions requiring a numerical solution, it is more important to demonstrate the method correctly than to obtain correct numerical answers. Even if your calculator has the power to perform high-level operations such as matrix inversion, you are expected to demonstrate the method from first principles. Solutions containing unresolved numerical expressions will be accepted provided the method of numerical calculation is clearly demonstrated. Statistical tables are not provided: you do not need to know precise values for any distributions to be able to answer the following questions. Part A . This is about a variant of the weighing problem (Section 3.2.4, page 117) in which the scale is subject to an unknown bias, i.e. if an object of weight β is placed in the pan, the mean of the measured weight is not β but β + γ , with γ common to all observations, but unknown. As in Section 3.2.4, all observations are independent and have an additional random error with variance σ 2 . For this scenario, the weighing designs proposed in Section 3.2.4 do not work directly because it is not possible to separate the estimation of γ from the actual weights of the objects. (You are not asked to prove this; just take it as given.) However, some alternative estimation schemes do make it possible to estimate all the weights. 1. Consider a scheme with just two objects to weigh, and three weighings set out as follows: (a) Weigh object 1 (b) Weigh object 2 (c) Weigh objects 1 and 2 together 1
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Write the model in the form y i = β 0 + β 1 ( x i 1 - ¯ x · 1 ) + β 2 ( x i 2 - ¯ x · 2 ) + ² i , (1) where β 1 and β 2 are the weights of the two objects; x i 1 is 1 if object 1 is in the pan on weighing i , 0 otherwise; x i 2
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