1. A company wants to relate the sales of a product (Y, in million dollars) to the amount of advertizing expenditures (X, in million dollars) and the types of cities (T) in where the product promotion were made. Data of seven cities are available and they are listed below (i = 1, . . . , 7 is the index for the cities). Advertising City Number Sales enditure Type of City 11 Y; X: Tl l 39 8 American 2 12 3 American 3 22 6 Asian 4 24 4 Asian 5 34 8 European 6 32 7 European 7 18 5 European (8.) If there is no difference among the di'erent types of cities, the company would t the regression model Y; = 30 + 131x.- + 3;, where Bo and 31 are model parameters and 3.- ~ iid N(D,02). Write down the model matrix for this regression model; i.e., the matrix X in Y = X5 + e, where Y = (39,12,22,24,34,32,18)', ,6 = (30,31)' and e = (61, . . . ,67)'. (b) Write down the regression model for tting three parallel regression lines for the three types of cities. That is, all types of cities should have the same slope but each will have a separate intercept. Note that you need to dene new indicator variable(s). (c) For your model in Part (b), write down the corresponding model matrix X. Also write down the null and alternative hypotheses for testing if the three types of cities are the \"same". (d) Write down the regression model for tting three di'erent regression lines for the three different city types (i.e., different slopes and intercepts). Also write down the corresponding model matrix X