The following table gives data on median salaries of full professors in statistics in research universities in the United States for the academic year 2007.

a. Plot median salaries against years in rank (as a measure of years of experience). For the plotting purposes, assume that the median salaries refer to the midpoint of years in rank. Thus, the salary $124,578 in the range 4€“5 refers to 4.5 years in the rank, and so on. For the last group, assume that the range is 31€“33.

b. Consider the following regression models:

Y_i = α₁ + α₂X_i + u_i €¦€¦€¦€¦€¦€¦€¦. (1)

Y_i = β₁ + β₂X_i + β₃X²_i + ν_i €¦€¦€¦€¦ (2)

where Y = median salary, X = years in rank (measured at midpoint of the range), and u and v are the error terms. Can you argue why model (2) might be preferable to model (1)? From the data given, estimate both the models.

c. If you observe heteroscedasticity in model (1) but not in model (2), what conclusion would you draw? Show the necessary computations.

d. If heteroscedasticity is observed in model (2), how would you transform the data so that in the transformed model there is no heteroscedasticity?

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Median Years in Rank Count O to 1 $101,478 102,400 124,578 122,850 116,900 119,465 114,900 129,072 131,704 143,000 40 2 to 3 24 4 to 5 35 6 to 7 34 8 to 9 33 10 to 14 73 15 to 19 69 20 to 24 54 25 to 30 44 31 or more 25