(a) Compute the z-score of the fastball with speed 87.1 mph. Round your answer to two decimal places. (b) What percentage of these fastballs would you expect to have speeds between 86.6 mph and 94.8 mph? Express your answer as a decimal. Round to three decimal places. (c) What percentage of these fastballs would you expect to have speeds above 94.8 mph? Express your answer as a decimal. Round to three decimal places.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Finding Probabilities in
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. The data showed that the speeds of the fastballs are normally distributed with mean 92.12 mph and standard deviation 2.43 mph.
(a) Compute the z-score of the fastball with speed 87.1 mph. Round your answer to two decimal places.
(b) What percentage of these fastballs would you expect to have speeds between 86.6 mph and 94.8 mph? Express your answer as a decimal. Round to three decimal places.
(c) What percentage of these fastballs would you expect to have speeds above 94.8 mph? Express your answer as a decimal. Round to three decimal places.
(d) Finding Values: A baseball fan wishes to identify the fastballs among the slowest 22% of all such pitches. Below what speed must a fastball be in order to be included in the slowest 22%? Round your answer to the nearest 0.1 mph.
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