1. Using all 1991 birth records in the computerized national birth certificate registry compiled by the National Center for Health Statistics (NCHS), statisticians Traci Clemons and Marcello Pagano found that the birth weights of babies in the United States are not symmetric ("Are babies normal?" The American Statistician, Nov 1999, 53:4). However, they also found that when infants born outside of the "typical" 37-43 weeks and infants born to mothers with a history of diabetes are excluded, the birth weights of the remaining infants do follow a Normal model with mean μ = 3432 g and standard deviation σ = 482 g. The following questions refer to infants born from 37 to 43 weeks whose mothers did not have a history of diabetes. Compute the z-score of an infant who weighs 2002 g. (Round your answer to two decimal places.)? Approximately what fraction of infants would you expect to have birth weights between 2880 g and 4040 g? (Express your answer as a decimal, not a percent, and round to three decimal places.)? Approximately what fraction of infants would you expect to have birth weights below 2880 g? (Express your answer as a decimal, not a percent, and round to three decimal places.)? A medical researcher wishes to study infants with low birth weights and seeks infants with birth weights among the lowest 2%. Below what weight must an infant's birth weight be in order for the infant be included in the study? (Round your answer to the nearest gram.)?
1. Using all 1991 birth records in the computerized national birth certificate registry compiled by the National Center for Health Statistics (NCHS), statisticians Traci Clemons and Marcello Pagano found that the birth weights of babies in the United States are not symmetric ("Are babies normal?" The American Statistician, Nov 1999, 53:4). However, they also found that when infants born outside of the "typical" 37-43 weeks and infants born to mothers with a history of diabetes are excluded, the birth weights of the remaining infants do follow a Normal model with mean μ = 3432 g and standard deviation σ = 482 g. The following questions refer to infants born from 37 to 43 weeks whose mothers did not have a history of diabetes.
Compute the z-score of an infant who weighs 2002 g. (Round your answer to two decimal places.)?
Approximately what fraction of infants would you expect to have birth weights between 2880 g and 4040 g? (Express your answer as a decimal, not a percent, and round to three decimal places.)?
Approximately what fraction of infants would you expect to have birth weights below 2880 g? (Express your answer as a decimal, not a percent, and round to three decimal places.)?
A medical researcher wishes to study infants with low birth weights and seeks infants with birth weights among the lowest 2%. Below what weight must an infant's birth weight be in order for the infant be included in the study? (Round your answer to the nearest gram.)?
2. 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.
A. Compute the z-score of pitch with speed 98.4 mph. (Round your answer to two decimal places.)?
B. Approximately what fraction of these four-seam fastballs would you expect to have speeds between 89.1 mph and 93.7 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)?
C. 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.)?
D. A baseball fan wishes to identify the four-seam fastballs among the fastest 14% of all such pitches. Above what speed must a four-seam fastball be in order to be included in the fastest 14%? (Round your answer to the nearest 0.1 mph.)?
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