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 95.9 mph. (Round your answer to two decimal places.

Approximately what fraction of these four-seam fastballs would you expect to have speeds between 91.1 mph and 91.5 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)

Approximately what fraction of these four-seam fastballs would you expect to have speeds below 91.1 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)

A baseball fan wishes to identify the four-seam fastballs among the fastest 8% of all such pitches. Above what speed must a four-seam fastball be in order to be included in the fastest 8%? (Round your answer to the nearest 0.1 mph.)

Answer 1

a)

Here, μ = 92.12, σ = 2.43 and x = 95.9. We need to compute P(X <= 95.9). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (95.9 - 92.12)/2.43 = 1.56

b)

Here, μ = 92.12, σ = 2.43, x1 = 91.1 and x2 = 91.5. We need to compute P(91.1<= X <= 91.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (91.1 - 92.12)/2.43 = -0.42
z2 = (91.5 - 92.12)/2.43 = -0.26

Therefore, we get
P(91.1 <= X <= 91.5) = P((91.5 - 92.12)/2.43) <= z <= (91.5 - 92.12)/2.43)
= P(-0.42 <= z <= -0.26) = P(z <= -0.26) - P(z <= -0.42)
= 0.3974 - 0.3372
= 0.060

c)

z value at 8% = 1.41

z- (x - mean)/s

1.41 = ( x - 92.12)/2.43

x = 2.43 *1.41 + 92.12

x = 95.5