Problem 1.70 Suppose each observation xi is bounded above and below by M, m, respectively; that is, assume that m ≤ xi ≤ M, i = 1, . . . n.

(a) Show that m ≤ x ≤ M.

(b) The writer Garrison Keillor has described the mythical town of Lake Wobegon as a place “where all the women are strong, the men are good looking and the children above average.” Let xi denote the grade point average (GPA) of the ith student and x denote the average GPA for all the students in Lake Wobegon high school. Explain why it is impossible for every student to have a GPA that is above the average. In other words, show that it is mathematically impossible to have xi > x for every xi. Hint: Use the result stated in part (a).