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Unformatted text preview: STATISTICS 174: APPLIED STATISTICS MIDTERM EXAM OCTOBER 15, 2003 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 teaching assistant for clarification if you feel the question is ambiguous. Each question is worth 20 points. All answers will be graded, but the max- imum total score is 100. Thus, you can obtain full marks by answering five of the six questions. Show all numerical and algebraic calculations. Statistical tables are not provided: you do not need to know precise values for any distributions to be able to answer the following questions. Smoothing is the generic name for a class of statistical procedures that aim to fit smooth curves through data. One example of a smoothing problem is to take a simple scatterplot of observations ( x 1 ,y 1 ) , ( x 2 ,y 2 ) ,..., ( x n ,y n ) and to construct a smooth curve y = f ( x ) that passes approximately, but not exactly, through the n points of the scatterplot. The purpose of this question is to develop detailed formulas and properties of a specific solution this problem: namely, to estimate the smooth curve for a given value of x , we take the five nearest data points to x and fit a quadratic regression through them. The problem is simplified by assuming that the data values of x are equally spaced. We therefore consider the problem of estimating a smooth function f ( x ), by quadratic regression based on five data points at x- 2 h, x- h, x, x + h and x + 2 h , where h is the spacing between data points. Since the structure of the problem is invariant to location and scale changes, there is no loss of generality in assuming x = 0 and h = 1. This leads therefore to the following formulation of the problem:...
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