The overall batting average of all major-league baseball hitters in recent years (1970 – 2008) has followed a roughly normal distribution with a mean of .261 and standard deviation .035.
(a) About what percent of hitters bat .226 or lower? (Sketch a normal curve to illustrate and use the 68-95-99.7 rule.)
(b) What percent of hitters bat .331 or higher? (Sketch a normal curve to illustrate and use the 68-95-99.7 rule.)
(c) In 2008, the batting leaders were Chipper Jones of the National League Atlanta Braves with a batting average of .364 and Joe Mauer of the American League Minnesota Twins, whose average was .328. When compared to the entire league, it is clear that Jones’ batting average was higher than Mauer’s, but what about their standing within their respective leagues? Below are the summary statistics for the 2008 top batting averages in each league.
2. According to the National Center for Health Statistics, the distribution of adult male and female heights is reasonably modeled by a normal density curve. In a 2007 study, the NCHS found that all adult males (at least 20 years of age) had a mean height of 69.4 inches with a standard deviation of 2.5 inches. The respective mean and standard deviation for all adult females at least 20 years of age was 64.8 and 2.2 inches. In each case, sketch a standard Normal curve and shade the appropriate area under the curve that is the answer to the question. Show your calculations for z and values from Table A, if appropriate.
Approximately what proportion of adult males are taller than 71.9 inches? (Draw a normal density curve and use the 68-95-99.7 rule).
Find the approximate proportion of adult males who are taller than 68 inches using the standard normal distribution and Table A. Report Table A values to 4 decimal places.
What percent of adult females are between 60 inches and 66 inches tall?
If there were 4000 females in the sample, about how many would we expect to be between 60 and 66 inches tall?
Find the height that corresponds to the 70th percentile of the height distribution for adult males.
To what percentile in the adult female height distribution would your answer in part (e) correspond?
3. The lifetime of a certain brand of tires is approximately normally distributed, with a mean of 40,000 miles and a standard deviation of 2,500 miles under normal driving conditions. Tire wear is greatly affected by road, weather and driver conditions along with proper maintenance of the tires. A driver who aggressively accelerates or makes quick stops, for example, will wear out a tire much more quickly. The brand carries a warranty of 33,000 miles under normal driving conditions, i.e., the company will replace a tire if it wears out before this mileage limit is reached. What percent of the tires will fail before the warranty limit is reached?
If the company sold 250,000 of these tires this year, approximately how many would it expect to have to replace under the warranty conditions?
4. Below is a histogram of the opening day stock price for Apple from January 1, 2000 to July 9, 2009; a total of 496 days.
Draw an appropriate density curve for summarizing the histogram above. How would you describe the shape of this density curve?
Where would the mean and median be located on the density curve you drew in part (a)? Draw in their approximate locations.
Based on the histogram, what is the approximate percentile of the opening price of $35?
Interpret this percentile in the context of this problem.
Based on the histogram, what is the approximate opening stock price which represents the 97th percentile?
The mean and standard deviation of the distribution of the opening day price for Apple stock is $64.94 and $49.50, respectively. What is the z-score for the opening day price of $107.40?
Interpret this z-score in the context of this problem.
5. Syracuse, New York is the snowiest metropolitan area in the United States. Based on 59 years of data from the National Weather Center, the mean annual snowfall is 118.5 inches with a standard deviation of 33.5 inches. The annual snowfall in Syracuse follows a roughly normal distribution.
(a) Sketch a normal curve to illustrate the annul snowfall in Syracuse. Be sure to mark the mean and the points that determine one, two, and three standard deviations away from the mean.
(b) Use the 68-95.99.7 rule to estimate the percent of years where the annual snowfall was between 85 inches and 185.5 inches. Illustrate your method clearly.
(c) Each year the city of Syracuse budgets enough money for snow removal to take care of all but the snowiest 3% of years. It is willing to run some small risk of this happening, especially in a tight budget year. How much would it have to snow in a particular year in order for the city to exceed its snow-removal budget? (Sketch a normal curve and use Table A.)
(d) In 2001, the snowfall totaled 59.4 inches. Was this an unusually low amount of snow for Syracuse? Justify your answer. Include the sketch a normal curve and some numerical calculations to support your answer.