Suppose that we survey 100 randomly sampled avocado farmers to find 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 find that the number of avocado trees has a mean of 80 and a standard deviation of 30. Then we use least squares to fit a linear function H(x) =w1x+w0, which we will use to help other farmers predict their avocado yield based on the number of trees they have.

1.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(x) =w1x+ w0 is fit using the new data (with only this one farmer's yield changed), what is t she difference w1w1 between the new slope and the old.

2.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(x) to the new predictor H(x): someone with 20 avocado trees or someone with 40 avocado trees?

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?

4.In this problem, since x 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 w1x instead of the usual w0+w1x. Minimize the mean square error to find the best choice of w1, in the case that we force our prediction rule to be of the form w1x.