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quiz00_3

# quiz00_3 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering 1.017 Computing and Data Analysis for Environmental Applications Quiz 3 Tuesday December 12, 2000 Please answer any 5 of the following 6 problems (maximum score = 100 points): Problem 1 (20 points) Consider a random sample consisting of n pairs of concentrations ( X 1 , Y 1 ), ( X 2 , Y 2 ),..., ( X n , Y n ) for two solutes X and Y . Suppose that you wish to fit the following trend function to the data: y ( x ) = a 1 x Assume that the measurements can be described by: Y i = a 1 X i + V i ; i = 1 ,... n where the V i are a set of independent, identically distributed random variables with mean 0 and variance 1.0. a) Derive an estimator for the parameter a 1 . b) Show that this estimator is unbiased Problem 2 (20 points) Derive a two-sided 90% confidence interval for the mean of a normally distributed random variable X given the following random sample of X . State any assumptions that you need to make.

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