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Chapter 2: 140SE (page 142)
Discuss the conditions under which the median is preferred to the mean as a measure of central tendency.
When there is skewness in the data set.
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Surface roughness of oil field pipe.Oil field pipes are internally coated to prevent corrosion. Researchers at the University of Louisiana, Lafayette, investigated the influence that coating may have on the surface roughness of oil field pipes (Anti-corrosion Methods and Materials, Vol. 50, 2003). A scanning probe instrument was used to measure the surface roughness of each in a sample of 20 sections of coated interior pipe. The data (in micrometers) are provided in the table.
(1.72, 2.50, 2.16, 2.13, 1.06, 2.24, 2.3,1 2.03, 1.09, 1.40, 2.57, 2.64, 1.26, 2.05, 1.19, 2.13, 1.27, 1.51, 2.41, 1.95)
a.Find and interpret the mean of the sample.
b.Find and interpret the median of the sample.
c.Which measure of central tendency鈥攖he mean or the median鈥攂est describes the surface roughness of the sampled pipe sections? Explain.
Using z-scores for grades. At one university, the students are given z-scores at the end of each semester rather than the traditional GPAs. The mean and standard deviation of all students鈥� cumulative GPAs, on which the z-scores are based, are 2.7 and .5, respectively.
a. Translate each of the following z-scores to the corresponding GPA: z = 2.0, z = -1.0, z = .5, z = -2.5.
b. Students with z-scores below 鈭�1.6 are put on probation. What is the corresponding probationary GPA?
c. The president of the university wishes to graduate the top 16% of the students with cum laude honors and the top 2.5% with summa cum laude honors. Where (approximately) should the limits be set in terms of z-scores? In terms of GPAs? What assumption, if any, did you make about the distribution of the GPAs at the university?
Do social robots walk or roll? A social (or service) robot is designed to entertain, educate, and care for human users. In a paper published by the International Conference on Social Robotics (Vol. 6414, 2010), design engineers investigated the trend in the design of social robots. Using a random sample of 106 social robots obtained through a Web search, the engineers found that 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. This information is portrayed in the accompanying graph.
a. What type of graph is used to describe the data?
b. Identify the variable measured for each of the 106 robot designs.
c. Use the graph to identify the social robot design that is currently used the most.
d. Compute class relative frequencies for the different categories shown in the graph.
e. Use the results from part d to construct a Pareto diagram for the data.
Explain why we generally prefer the standard deviation to the range as a measure of variability for quantitative data.
Explain how the relationship between the mean and median provides information about the symmetry or skewness of the data鈥檚 distribution.
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