You wish to establish a linear relationship between the temperature in San Diego, x, and the temperature in Osaka,y.

San Diego(F)	66	66	67	69	69	72	76	77	77	74	70	66
Osaka(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 andCelsius for Osaka. Youd like the relationship that you find to be in degrees Celsius. One way to do this isto convert all the San Diego temperatures to Celsius before performing least squares regression.You friend thinks you can skip some of that work: Why dont 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 dothe conversion twice instead of for each data point.

1: Is Skip correct that youll get the same regression coefficients either way? Show yourwork. Recall that if a temperature t is measured in degrees Fahrenheit, the equivalent temperature in degrees Celsius is given by g(t) =5/9(t32). 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?

2:More generally, suppose we want to do least squares regression for a linear relationship:y=w₁x+w₀. How do the slope w₁ and the intercept w₀ of the regression line change if we replace x with a linear transformation f(x) = ax+b?

3:Prove in general that the mean squared error does not change if we use as a predictor anylinear transformation ofx. For an arbitrary data set y₁,, y_n, show that if z=ax+b for some constants a, b != 0, then MSEx=MSEz.

Hint: Start by using the result in (b) and expressing d₀, d₁ from the relation ship y=d₁z+d₀ in terms of w₀, w₁ ,a, b, where w₀, w₁ are coefficients inferred for the linear relationship:y=w₁x+w₀.