Which of the three measures would you recommend as the most appropriate repre-

blood pressure readings less than 127 mm Hg were then classified according to their serum cholesterol level. Serum Cholesterol mg100cc Frequency 0.0 –199.9 119 200.0 –219.9 88 220.0 –259.9 127 greater than 259 74 a. Estimate the mode, median, and mean for the serum cholesterol readings if possible. b. Which of the three summary statistics is more informative concerning a typical serum cholesterol level for the group of men? Explain your answer. Env. 3.21 The ratio of DDE related to DDT to PCB concentrations in bird eggs has been shown to have had a number of biological implications. The ratio is used as an indication of the move- ment of contamination through the food chain. The paper “The ratio of DDE to PCB concentra- tions in Great Lakes herring gull eggs and its use in interpreting contaminants data’’ [Journal of Great Lakes Research 1998 241:12 –31] reports the following ratios for eggs collected at 13 study sites from the five Great Lakes. The eggs were collected from both terrestrial- and aquatic-feeding birds. DDE to PCB Ratio Terrestrial Feeders 76.50 6.03 3.51 9.96 4.24 7.74 9.54 41.70 1.84 2.50 1.54 Aquatic Feeders 0.27 0.61 0.54 0.14 0.63 0.23 0.56 0.48 0.16 0.18 a. Compute the mean and median for the 21 ratios, ignoring the type of feeder. b. Compute the mean and median separately for each type of feeder.

c. Using your results from parts a and b, comment on the relative sensitivity of the

mean and median to extreme values in a data set.

d. Which measure, mean or median, would you recommend as the most appropriate

measure of the DDE to PCB level for both types of feeders? Explain your answer. Med. 3.22 A study of the survival times, in days, of skin grafts on burn patients was examined in Woolson and Lachenbruch [Biometrika 1980 67:597– 606]. Two of the patients left the study prior to the failure of their grafts. The survival time for these individuals is some number greater than the reported value. Survival time days: 37, 19, 57, 93, 16, 22, 20, 18, 63, 29, 60 The “’’ indicates that the patient left the study prior to failure of the graft; values given are for the day the patient left the study. a. Calculate the measures of center if possible for the 11 patients. b. If the survival times of the two patients who left the study were obtained, how would these new values change the values of the summary statistics calculated in a? Engin. 3.23 A study of the reliability of diesel engines was conducted on 14 engines. The engines were run in a test laboratory. The time in days until the engine failed is given here. The study was terminated after 300 days. For those engines that did not fail during the study period, an asterisk is placed by the number 300. Thus, for these engines, the time to failure is some value greater than 300. Failure time days: 130, 67, 300, 234, 90, 256, 87, 120, 201, 178, 300, 106, 289, 74 a. Calculate the measures of center for the 14 engines. b. What are the implications of computing the measures of center when some of the exact failure times are not known? Gov. 3.24 Effective tax rates per 100 on residential property for three groups of large cities, ranked by residential property tax rate, are shown in the following table. Group 1 Rate Group 2 Rate Group 3 Rate Detroit, MI 4.10 Burlington, VT 1.76 Little Rock, AR 1.02 Milwaukee, WI 3.69 Manchester, NH 1.71 Albuquerque, NM 1.01 Newark, NJ 3.20 Fargo, ND 1.62 Denver, CO .94 Portland, OR 3.10 Portland ME 1.57 Las Vegas, NV .88 Des Moines, IA 2.97 Indianapolis, IN 1.57 Oklahoma City, OK .81 Baltimore, MD 2.64 Wilmington, DE 1.56 Casper, WY .70 Sioux Falls, IA 2.47 Bridgeport, CT 1.55 Birmingham, AL .70 Providence, RI 2.39 Chicago, IL 1.55 Phoenix, AZ .68 Philadelphia, PA 2.38 Houston, TX 1.53 Los Angeles, CA .64 Omaha, NE 2.29 Atlanta, GA 1.50 Honolulu, HI .59 Source: Government of the District of Columbia, Department of Finance and Revenue, Tax Rates and Tax Burdens in the District of Columbia: A Nationwide Comparison, annual. a. Compute the mean, median, and mode separately for the three groups. b. Compute the mean, median, and mode for the complete set of 30 measurements. c. What measure or measures best summarize the center of these distributions? Explain. 3.25 Refer to Exercise 3.24. Average the three group means, the three group medians, and the three group modes, and compare your results to those of part b. Comment on your findings.

3.5 Describing Data on a Single Variable: Measures of Variability

Engin. 3.26 Pushing economy and wheelchair-propulsion technique were examined for eight wheelchair racers on a motorized treadmill in a paper by Goosey and Campbell [Adapted Physical Activity Quarterly 1998 15:36 –50]. The eight racers had the following years of racing experience: Racing experience years: 6, 3, 10, 4, 4, 2, 4, 7

a. Verify that the mean years’ experience is 5 years. Does this value appear to ade-

quately represent the center of the data set?

b. Verify that c. Calculate the sample variance and standard deviation for the experience data.

How would you interpret the value of the standard deviation relative to the sample mean? 3.27 In the study described in Exercise 3.26, the researchers also recorded the ages of the eight racers. Age years: 39, 38, 31, 26, 18, 36, 20, 31 a. Calculate the sample standard deviation of the eight racers’ ages. b. Why would you expect the standard deviation of the racers’ ages to be larger than the standard deviation of their years of experience? Engin. 3.28 For the data in Exercise 3.26,

a. Calculate the coefficient of variation CV for both the racer’s age and their years of

experience. Are the two CVs relatively the same? Compare their relative sizes to the relative sizes of their standard deviations.

b. Estimate the standard deviations for both the racer’s age and their years of experience

by dividing the ranges by 4. How close are these estimates to the standard deviations calculated in Exercise 3.27? ai y ⫺ y 2 ⫽ ai y ⫺ 5 2 ⫽ 46.