Major League Baseball now records information about every pitch thrown in every

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 ML.B 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 flow a Normal model with mean ? 9212 mph and a standard deviation of?-2.43 mph. Compute the 2-score of pitch with speed 85 mph. (Round your answer to 2 decimal places) Approximately what fraction of these four-seam fastballs would you expect to have speeds between 90.2 mph and 96.8 mph? Express your answer as a decimal, not a percent, and round to 4 decimal places) Approximately what fraction of these four-seam fastballs would you expect to have speeds below 90.2 mph? (Express your answer as a decimal, not a percent, and round to 4 decimal places) A baseball fan wishes to identify the four-seam fastballs among the fastest 4% of all such patches. Above what speed must a four-seam fastball be in order to be included in the fastest 4967 (Round your answer to the nearest 0.1 mph) mph

Explanation / Answer

mean = 92.12 ,s = 2.43

a)
z = (x -mean)/s
= ( 85 - 92.12)/2.43
= -2.93

b)
P(90.2 < x < 96.8)
= P((90.2 - 92.12)/2.43 < z < (96.8 - 92.12)/2.43)
= P(-0.7901 < z < 1.9259)

P(90.2 < x < 96.8) = P(-0.7901 < z < 1.9259) = 0.7582

c)
P(x < 90.2)
z = (x -mean)/s
= (90.2 - 92.12)/2.43
= -0.7901

P(x < 90.2) = P(z < -0.7901) = 0.2147

d)
z value at 4% = 1.7507

z = ( x -mean)/s
1.7507 = ( x - 92.12)/2.43
x = 87.8658

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