In Lecture 2, we found that h* = Median (y1, 2, ..., Yn) is the constant...

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In Lecture 2, we found that h* = Median (y1, 2, ..., Yn) is the constant prediction that minimizes mean absolute error: n 1 Rabs(h) = 2 |ih| z=1 Suppose that we have a dataset of numbers y1, 2, ..., Yn such that n is odd and the values are arranged in increasing order. That is, y Y2, < ... < Yn. Note: Parts (a) and (b) are independent of each other. a) Suppose that Rabs () = V, where V is the minimum value of Rabs(h) and a is one of the numbers in our dataset. Let a be the smallest value greater than a in our dataset, where > 0. Another way of thinking about this is that = (smallest value greater than a) - a. Suppose we modify our dataset by replacing the value a with the value a++1. In our new dataset of n values, what is the new minimum value of Rabs (h) and at what value of h is it minimized? Your answers to both parts should only involve the variables V, a, , n, and/or one or more constants. Solution: Write your solution here. b) Let ya and y be two values in our dataset such that y < y and that the slope of Rabs(h) is the same between h = ya and h = y. Specifically, let d be the slope of Rabs (h) between ya and yb. Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1 values, the slope of Rabs(h) is still the same between h = ya and h = y, but it's no longer equal to d. What is the slope of Rabs (h) between h = ya and h = y in our new dataset? Your answer should depend on d, n, q, and/or one or more constants. Solution: Write your solution here. In Lecture 2, we found that h* = Median (y1, 2, ..., Yn) is the constant prediction that minimizes mean absolute error: n 1 Rabs(h) = 2 |ih| z=1 Suppose that we have a dataset of numbers y1, 2, ..., Yn such that n is odd and the values are arranged in increasing order. That is, y Y2, < ... < Yn. Note: Parts (a) and (b) are independent of each other. a) Suppose that Rabs () = V, where V is the minimum value of Rabs(h) and a is one of the numbers in our dataset. Let a be the smallest value greater than a in our dataset, where > 0. Another way of thinking about this is that = (smallest value greater than a) - a. Suppose we modify our dataset by replacing the value a with the value a++1. In our new dataset of n values, what is the new minimum value of Rabs (h) and at what value of h is it minimized? Your answers to both parts should only involve the variables V, a, , n, and/or one or more constants. Solution: Write your solution here. b) Let ya and y be two values in our dataset such that y < y and that the slope of Rabs(h) is the same between h = ya and h = y. Specifically, let d be the slope of Rabs (h) between ya and yb. Suppose we introduce a new value q to our dataset such that q> y. In our new dataset of n + 1 values, the slope of Rabs(h) is still the same between h = ya and h = y, but it's no longer equal to d. What is the slope of Rabs (h) between h = ya and h = y in our new dataset? Your answer should depend on d, n, q, and/or one or more constants. Solution: Write your solution here.