the 2015 major league baseball season expeirenced one of the

the 2015 major league baseball season expeirenced one of the most inpressive pitching performances (collectivley) in the 145 seasons that baseball has been playing.

for a starting pitcher to have an ERA (earned run average) below 2 runs per game is unheard of (starters with at least 25 games started).

Zack Greinke had the best ETA of x_bar= 1.657 for 32 starts in 2015

The major league ERA for 2015 was x_bar=3.95 with a s=.454 stdv for 30 teams and 735 ptichers of which zack was one of them.

The entire historical major league ERA is Mue=3.814 with stdv. of Theta=.768 ( I took out the years where less than 50 pitchers were in the league; all during the dead ball era years)

in 1968, The ERA was x_bar=2.98 (the lowest era of modern baseball era 1920) with a stdv of s=.515.

1. Did zack greinke fall in the 99%ile of all ERAs in major league history? This would be like testing zacks ERAfor an a=.01 what is his %ile?

2. Where would zack fall with respect to percentage with the pitchers who posted the best ERA in 1968 (the best ERA in the modern era)?

3. Where does the 1968 season rank in major league history with respect to ERA?

Solution

1. The entire historical major league ERA is Mean=3.814 with s.d=0.768.....
the lower limit 99% th percentile( the lower 99%) of all time = [ 3.814  - ( 2.326348* 0.768) ] =  2.027365...

Zack Greinke falls even further in percentile.
his time =1.657..
1.657 = 3.814 - ( z-score * 0.768) so, z-score = 2.808594...
his percentile = 99.75121
he belongs to the (lower) 99.75121% percentile.

2)  xbar=2.98 (the lowest era of modern baseball era 1920) with a stdv of s=.515.

1.657 = 2.98 - ( z-score * 0.515)
or, z-score = 2.568932....

percentile = 99.48994 .......
( i haven\'t used t-score here because, the total no. of pitchers in 1968 is not known)

3) 2.98 = 3.814 - ( z-score * 0.768).
so,  z-score = 1.085938....
percentile =86.12467. 1968 season rank in 86.12467% percentage of all time with respect to ERA..