Problem 1. Fahrenheit or Celsius? You wish to establish a linear relationship between the temperature in San Diego, x, and the temperature in Osaka, y. The monthly temperature data from Homework 1 is given in the table below. San Diego, US (F) 66 66 67 69 69 72 76 77 77 74 70 66 Osaka, Japan (C) 9 10 14 20 25 28 32 33 29 23 18 12 Recall that the temperatures in these places are measured in different units, Fahrenheit for San Diego and Celsius for Osaka. You'd like the relationship that you find to be in degrees Celsius. One way to do this is to convert all the San Diego temperatures to Celsius before performing least squares regression. You friend Skip from Homework 1 is also back, and he thinks you can skip some of that work: "Why don't we perform least squares regression first, with x in Fahrenheit, and then do the Fahrenheit to Celsius conversion for both the slope and the intercept in the regression coefficients? That way we only need to do the conversion twice instead of for each data point." a) 6060 Is Skip correct that you'll get the same regression coefficients either way? Show your work. Recall that if a temperature t is measured in degrees Fahrenheit, the equivalent temperature in degrees Celsius is given by g(t) = , x (t -32). If Skip is not correct, can you think of a different shortcut that allows you to get the same regression coefficients without converting each data point to Celsius? b) 6060 More generally, suppose we want to do least squares regression for a linear relationship: y = wix + wo. How do the slope w1 and the intercept wo of the regression line change if we replace x with a linear transformation f(x) = ax + b?A few years ago, a millenial decided to grow their own avocados to keep up with their rapid consumption of avocado toast. They quickly noticed that rain is fundamental to avocado production. Over the last four avocado seasons, they have recorded the number of rainy days that season, 2:, and the number of avocados produced that season, y. III- Ila-I- a) (3 Q What linear relationship 3: = bo + 61:17 best describes the number of avocados as a function of the number of rainy days? What is the mean squared error, R3q((bo, ()1); DE) (where bo and (71 are the optimal choices), for this data set? b) (3 {3 The millenial reconsiders their approach to this problem, and decides that the number inches of rain, 2, may be a better predictor than the number of rainy days, :12. Ian-In Ila-I- What linear relationship 3; = do + (112 best describes the number of avocados as a function of the number of inches of rainfall? What is the mean squared error, qu((do, d1); Dz) (where do and d1 are the optimal choices), for this data set? c) (3666(3) In the above example, notice that R3q((bo,b1);Dm) = qu((do,d1);Dz), so the mean squared error is the same if we use the predictor m or the predictor z. This happens because the number of inches of rainfall z is linearly related to the number of rainy days a: by the following formula: 2 = 2:: + 1. Prove in general that the mean squared error does not change if we use as a predictor any linear transformation of m. For an arbitrary data set 311, . . . ,yn, Show that if z = 60 + 61.1: for some constants 60,01 7E 0,1}11611 R34((b01 bl); D517) : RSQ((dUid1)i DZ) Hint: Start by expressing do,d1 from the relationship 3; = do + dlz in terms of co,c1,bo,b1, where y = ()0 + blur