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?Problem 3. [3 {3 {3 (3 Suppose you have a data set of six data points, with two data points at each of three different as values, a: = 5,2: = 10, and as = 15 (That is, we have 9:;- 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 (5,51), (10,?2), (15,33) where 31, g, 33 represent the means of the two y values at each of the a: values