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Chapter 2: Q73E (page 110)
Question: For a set of data with a mound-shaped relative frequency distribution, what can be said about the percentage of the measurements contained in each of the intervals specified in Exercise 2.72?
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Motivation of athletes.A statistician keeps track of every serve that a player hits during the U.S. Open Tennis Championship. The statistician reports that the mean serve speed was 100 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph.
a.Suppose the statistician also observes that the distribution of serve speeds was mound-shaped and symmetric. What percentage of the player鈥檚 serves was between 115 mph and 145 mph?
b.Consider the following serve speeds: 50 mph, 80 mph, and 105 mph. Using the z-score approach for detecting outliers, which of these would represent outliers in the distribution of the player鈥檚 serve speeds?
c.If nothing is known about the shape of the distribution, what percentage of the player鈥檚 serve speeds are less than 70 mph?
Ranking driving performance of professional golfers.A group of researchers developed a new method forranking the total driving performance of golfers on theProfessional Golf Association (PGA) tour (The Sport Journal,Winter 2007). The method requires knowinga golfer鈥檚 average driving distance (yards) and drivingaccuracy (percent of drives that land in the fairway).The values of these two variables are used to compute adriving performance index. Data for 40 PGA golfers (asranked by the new method for a recent tour year) aresaved in the accompanying file. The first five and last fiveobservations are listed in the table below.
a.Find the mean, median, and mode for the 40 driving performance index values.
b.Interpret each of the measures of central tendency, part a.
c.Use the results, part a,to make a statement about the type of skewness in the distribution of driving performance indexes. Support your statement with a graph.
Management system failures.The U.S. Chemical Safety and Hazard Investigation Board (CSB) is responsible for determining the root cause of industrial accidents (Process Safety Progress, December 2004). The accompanying table gives a breakdown of the root causes of 83 incidents caused by management system failures.
Management System Cause Category | Number of Incidents |
Engineering & Design Procedures & Practices Management & Oversight Training & Communication | 27 24 22 10 |
Total | 83 |
a.Find the relative frequency of the number of incidents for each cause category.
b.Construct a Pareto diagram for the data.
c.From the Pareto diagram, identify the cause categories with the highest (and lowest) the relative frequency of incidents.
Compute xbar, s2, and sfor each of the following data sets. If appropriate, specify the units in which your answer is expressed.
a.3, 1, 10, 10, 4
b.8 feet, 10 feet, 32 feet, 5 feet
c.-1, -4, -3, 1, -4, -4
d.1/5 ounce, 1/5 ounce, 1/5 ounce, 2/5 ounce, 1/5 ounce, 4/5 ounce
Corporate sustainability of CPA firms.Refer to the Business and Society(March 2011) study on the sustainability behaviors of CPA corporations, Exercise 2.23 ( p. 83). Recall that level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each of 992 senior managers at CPA firms. Numerical measures of central tendency for level of support are shown in the accompanying Minitab printout.
Descriptive Statistics: Support | |||||||
Variable Support | N | Mean | Minimum | Median | Maximum | Mode | N for mode |
992 | 67.755 | 0.000 | 68.000 | 155.000 | 64 | 20 |
a.Locate the mean on the printout. Comment on the accuracy of the statement: 鈥淥n average, the level of support for corporate sustainability for the 992 senior managers was 67.76 points.鈥�
b.Locate the median on the printout. Comment on the accuracy of the statement: 鈥淗alf of the 992 senior managers reported a level of support for corporate sustainability below 68 points.鈥�
c.Locate the mode on the printout. Comment on the accuracy of the statement: 鈥淢ost of the 992 senior managers reported a level of support for corporate sustainability below 64 points.鈥�
d.Based on the values of the measures of central tendency, make a statement about the type of skewness (if any) that exists in the distribution of 992 support levels. Check your answer by examining the histogram shown in Exercise 2.23.
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