Problem 3. Q (3 {:3 (3 Suppose you have a data set of six data points, with two data points at each of three different a: values, a: = 5,2: = 10, and as = 15 (That is, we have 2:, E {5,5,10,10,15,15} for 2' = 1,...,6). Show that the least squares regression line tted to these six data points is identical to the least squares regression line tted to the three points (5%), (10,?2), (15, $3) where El, @2, p3 represent the means of the two y values at each of the a: values. Problem 4. Suppose that we survey 100 randomly sampled avocado farmers to nd out the number of avocado trees on their farm and the total number of avocados produced by those trees in a given year. In the collected survey data, we nd that the number of avocado trees has a mean of 80 and a standard deviation of 30. Then we use least squares to t a linear function H (:12) = wla: + we, which we will use to help other farmers predict their avocado yield based on the number of trees they have. a) (:3 Is a linear function ideal here, or is there another function form for H (:12) that you think would better model this scenario? Explain. b) [36) (3 {3 Now suppose that one particular farmer from the 100 sampled farmers was a very poor farmer. His 15 avocado trees yielded only 150 avocados, the smallest total number reported by any of the survey participants. The farmer then has a conversation with a millenial, who enlightens him by pointing out that avocado yield can be increased by additional watering. The farmer waters the avocado trees more frequently, and the next year, increases the yield from his 15 trees to 650 avocados. If a new linear predictor H ' (:17) = wim+ wf, is t using the new data (with only this one farmer's yield changed), what is the difference wi w] between the new slope and the old? Q g) 6 Suppose some farmers who were not surveyed plan to use the data from this survey to predict how much yield to expect from their avocado farms. Who would be more affected by changing prediction rules from H (3:) to the new predictor H' (m) someone with 20 avocado trees or someone with 40 avocado trees? Q, g) [3 If we had increased a different farmer's yield instead, would the original line or the new line have a steeper slope? How can you tell, based on the farmer? Is it possible that by increasing a farmer's yield, we keep the slope the same? 636 Q {3 In this problem, since cc represents number of avocado trees and y represents yield, it is reasonable to expect that the regression line should go through the origin. In other words, if there are no avocado trees, there are no avocados. We can force our prediction rule to go through the origin by using a prediction rule of the form wlx instead of the usual we + wlm. Minimize the mean square error to nd the best choice of 1.01, in the case that we force our prediction rule to be of the form 1011