Question

Major League Baseball now records information about every pitch thrown in every game of every season. Statistician Jim Albert compiled data about every pitch thrown by 20 starting pitchers during the 2009 MLB season. The data set included the type of pitch thrown (curveball, changeup, slider, etc.) as well as the speed of the ball as it left the pitcher’s hand. A histogram of speeds for all 30,740 four-seam fastballs thrown by these pitchers during the 2009 season is shown below, from which we can see that the speeds of these fastballs follow a Normal model with mean μ = 92.12 mph and a standard deviation of σ = 2.43 mph.

Compute the z-score of pitch with speed 92.2 mph. (Round your answer to 2 decimal places.)

Approximately what proportion of these four-seam fastballs would you expect to have speeds between 85.6 mph and 94.4 mph? (Express your answer as a decimal and round to 4 decimal places.)

Approximately what proportion of these four-seam fastballs would you expect to have speeds above 94.4 mph? (Express your answer as a decimal and round to 4 decimal places.)

A baseball fan wishes to identify the four-seam fastballs among the fastest 12% of all such pitches. Above what speed must a four-seam fastball be in order to be included in the fastest 12%? (Round your answer to the nearest 0.1 mph.)

Answer 1

Population mean, µ = 92.12

Population standard deviation, σ = 2.43

(a) z = (X-µ)/σ = (92.2-92.12)/2.43 = 0.03

(b) Proportion between 85.6 mph and 94.4 mph,  P(85.6 < X < 94.4) =

= P( (85.6-92.12)/2.43 < (X-µ)/σ < (94.4-92.12)/2.43 )

= P(-2.68 < z < 0.94)

= P(z < 0.94) - P(z < -2.68)

Using excel function:

= NORM.S.DIST(0.94, 1) - NORM.S.DIST(-2.68, 1)

= 0.8227

(c) Proportion that have speeds above 94.4 mph, P(X > 94.4) =

= P( (X-µ)/σ > (94.4-92.12)/2.43)

= P(z > 0.94)

= 1 - P(z < 0.94)

Using excel function:

= 1 - NORM.S.DIST(0.94, 1)

= 0.1736

(d) Z score at top 12% using excel = NORM.S.INV(1-0.12) = 1.17

Value of X = µ + z*σ = 92.12 + 1.17*2.43 = 94.96 = 95.0 mph