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 94.7 mph. (Round your answer to two decimal places.) Approximately what fraction of these four-seam fastballs would you expect to have speeds between 92.7 mph and 95.5 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 above 95.5 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 22% of all such pitches. Above what speed must a four-seam fastball be in order to be included in the fastest 22%? (Round your answer to the nearest 0.1 mph.) mph

Answer 1

X: Speed of ball follows a normal distribution with a mean and standard deviation

i) z score for speed 94.7

The formula of z score

ii) The fraction of balls expect to have speeds between 92.7 and 95.5 that is P(92.7 < X < 95.5)

The z scores for 92.7 and 95.5

P(92.7 < X < 95.5) becomes P(0.24 < Z < 1.39)

The probabilities using z table for z score 0.24 is 0.595 and the probability for z score 1.39 is 0.918

Between probability = 0.918 - 0.595 = 0.323

The fraction of these four-seam fastballs would you expect to have speeds between 92.7 mph and 95.5 mph is 0.323

iii) fraction of fastballs expect to have speeds above 95.5 mph that is P(X > 95.5)

The z score for x = 95.5 is 1.39 (from previous part)

That is P(X > 95.5) becomes P(Z > 1.39)

The probability using the z table for z score 1.39 is 0.918, but the table provides less than probability. Subtract the probability from 1 to get more than probability.

1 - 0.918 = 0.082

The fraction of these four-seam fastballs would you expect to have speeds above 95.5 mph is 0.082

iv) The four-seam fastballs among the fastest 22% of all such pitches

22% are in the top

First find the z score for the 22%, to find the z score the table needs less than area.

The total area under the normal curve is 100%, the lower area = total area - top area = 100%- 22%= 78% = 0.78

The z score for area 0.78 using z table is 0.77

To find the speed the formula of z score rearrange as,

x = 94.0

The speed must a four-seam fastball be in order to be included in the fastest 22% is 94.0 mph