Question

(11) p. 3 re Earned Run Averages for the indicated Leagues LEAGUE SMcan of laraed Run Averages of Starting Pitehers 8.25 6.52 10.00 Standard Dexiation of Earned Run ,97 .55 Averages of Starting Pitchers 1.15 Sebastian: 1.90 Caruthers: 1.87 Mungo: 2.04 RA Assume that all distributions are normal. Note: the lower an Earned Run Average is, the better (more impressive) it is. (a) Rank Sebastian, Caruthers, and Mungo in impressiveness relative to their respective Leagues. จา ha 1. 15 (b) What would Mungo's Earned Run Average have to have been in 2078 for his performance to have been exactly as impressive as was Caruthers' in 20221? 1.81-10.00 l. 15 二.091 (c) What would a pitcher's Earned Run Average have to have been in 2022 to have been in the lowest 2% of Earned Run Averages?

Answer 1

a)
Using central limit theorem, we find z-score for each

Sebastian, z = (1.9 - 8.25)/0.97 = -6.5464 #Rank 3
Caruthers, z = (1.87 - 6.52)/0.55 = -8.4545 #Rank 1
Mungo, z = (2.04 - 10)/1.15 = -6.9217 #Rank 2

b)
z-value for Caruther -8.4545
xbar = mean + z*sigma
xbar = 10 -8.4545*1.15
xbar = 0.2773

c)
z-value for lowest 2% is -2.0537
xbar = mean + z*sigma
xbar = 6.52 -2.0537*1.87
xbar = 2.6796

score has to be equal to or less than 2.6796 in order to fall in lowest 2%