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 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 = 92.12 mph and a standard deviation of = 2.43 mph.

Approximately what fraction of these four-seam fastballs would you expect to have speeds above 95.3 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 slowest 5% of all such pitches. Below what speed must a four-seam fastball be in order to be included in the slowest 5%? (Round your answer to the nearest 0.1 mph.)

Explanation / Answer

Here Mean = 92.12, Standard deviation = 2.43
To calculate the fraction of four-seam fastballs which have speeds above 95.3 mph, first we need to calculate the z score corresponding to 95.3

Z score can be calculated by the below formula:
Z score = (x - mean)/sd

So Z score = (95.3 - 92.12)/2.43 = 1.308642

Now we can look at the z table and calculate the corresponding p value which is 1 - 0.904672 = 0.095328
So 0.095328 fraction of four-seam fastballs which have speeds above 95.3 mph

For second part we have the p value as 0.05
Z score corresponding to p value of 0.05 is -1.644852

Z score = (x - mean)/sd
-1.644852 = (x - 92.12)/2.43
x - 92.12 = -3.99699
x = 88.12301

So four-seam fastball speed should be less than 88.12301 to be in order to be included in the slowest 5%

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