In: Statistics and Probability
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 88.9 mph. (Round
your answer to two decimal places.)
Approximately what fraction of these four-seam fastballs would you
expect to have speeds between 89.1 mph and 92.5 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 89.1 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 3% of all such pitches. Above what speed must a four-seam
fastball be in order to be included in the fastest 3%? (Round your
answer to the nearest 0.1 mph.)
mph
Given that, mean (μ) = 92.12 mph and
standard deviation = 2.43 mph
a) The z-score of pitch with speed 88.9 mph is,
Z = (88.9 - 92.12) / 2.43 = -3.22 / 2.43 = -1.33
=> Z = -1.33
b) We want to find, P(89.1 < X < 92.5)
Therefore, required probability is 0.4561
c) We want to find, P(X < 89.1)
Therefore, required probability is 0.1075
d) We want to find, the value of x such that, P(X > x) = 0.03
Answer : 96.7 mph