Exercise with known population Consider a linear model of the form y, = 100,000 + 100xI , + 50xz, + E,. Also assume the error terms and independent variables are independently drawn from E,

- N(0;800Z), XIt - N(1,000;200Z), and xz, - N(500;150Z). A random sample of size 15 is !isted below.

a] Suppose you ignore the two independent variabies. What is your best statistieal estimate (using the above data) of the population mean from which variable y, is drawn? Given you know the underlying population, what are its exaet mean and varianee? Test the hypothesis that the true me an is 225,000.
b] Suppose you ignore the second independent variable. What is your best statistical estimate·of the slope and intereept of the underlying population? What is your estimate of the varianee of the population error term? How does this compare to the known varianee of the error term in this misspeeified regression? Test the hypothesis that the slope is 100.
el Now use both independent variables in your regression. Test the hypothesis that 81 is 100 and that 82 is 50. How does your best estimate of the varianee of the population error term compare with the known varianee of E,?

AppendixLO1

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observation t y X11 X21 1234567890 230,487 1,077 434 207,894 831 529 210,231 921 367 229,039 996 580 223,874 1,048 382 180,250 583 423 217,359 936 458 239,155 1,161 483 220,072 983 445 187,771 705 387 11 259,680 1,339 507 12 216,520 878 579 13 218,899 960 467 14 263,649 1,251 742 15 210,586 800 610