Suppose we observe (Xi, Yi) ∼ f (x, y) independently for i = 1, ..., n. Let ̄X = ∑n i=1 Xi/n, and ̄Y = ∑n i=1 Yi/n. Let ̃Xi = Xi − ̄X, and ̃Yi = Yi − ̄Y . Let X be the vector formed by ( ̃Xi, i = 1, ..., n), and Y be the vector formed by ( ̃Yi, i = 1, ..., n). For the following scatterplots of (Xi, Yi), i = 1, ..., n, where each (Xi, Yi) is a point, (1) Write down the possible value of correlation for each scatterplot. You do not need to be precise. (2) Plot the vectors of X and Y for each scatterplot. (3) Plot the regression line ̃Y = β ̃X on each scatterplot. Let ei = ̃Yi − β ̃Xi, and let e be the vector formed by (ei, i = 1, ..., n). Suppose β is obtained by minimizing |e|2 = ∑n i=1 e2 i (the so-called least squares estimation, which geometrically amounts to a projection of Y on X). Plot βX and e for each vector plot in (2).
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Suppose we observe (Xi, Yi) ∼ f (x, y) independently for i = 1, ..., n. Let ̄X = ∑n i=1 Xi/n, and ̄Y = ∑n i=1 Yi/n. Let ̃Xi = Xi − ̄X, and ̃Yi = Yi − ̄Y . Let X be the vector formed by ( ̃Xi, i = 1, ..., n), and Y be the vector formed by ( ̃Yi, i = 1, ..., n). For the following scatterplots of (Xi, Yi), i = 1, ..., n, where each (Xi, Yi) is a point, (1) Write down the possible value of
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