2.174 Explain why we generally prefer the standard deviation to the range as a measure of variation.

2.175 Discuss conditions under which the median is preferred to the mean as a measure of central tendency.

2.176 Give a situation in which we would prefer using a box plot over z-scores to detect an outlier.

2.177 Give a situation in which we would prefer using a stem-and-leaf display over a histogram in describing quantitative data graphically.

2.178 Give a technique that is used to distort information shown on a graph.

2.186 Excavating ancient pottery. Archaeologists excavating the ancient Greek settlement at Phylakopi classified the pottery found in trenches. (Chance, Fall 2000.) The accompanying table describes the collection of 837 pottery pieces uncovered in a particular layer at the excavation site. Construct and interpret a graph that will aid the archaeologists in understanding the distribution of the types of pottery found at the site.

2.187 Japanese reading levels. University of Hawaii language professors C. Hitosugi and R. Day incorporated a 10-week extensive reading program into a second-semester Japanese language course in an effort to improve students’ Japanese reading comprehension. (Reading in a Foreign Language, Apr. 2004.) The professors collected 266 books originally written for Japanese children and required their students to read at least 40 of them as part of the grade in the course. The books were categorized into reading levels (color coded for easy selection) according to length and complexity. The reading levels for the 266 books are summarized in the following table:

1. Calculate the proportion of books at reading level 1 (red).

2. Repeat part a for each of the remaining reading levels.

3. Verify that the proportions in parts a and b sum to 1.

4. Use the previous results to form a bar graph for the reading levels.

5. Construct a Pareto diagram for the data. Use the diagram to identify the reading level that occurs most often.

JREAD 2.188 Reading Japanese books. Refer to Exercise 2.187. Fourteen students participated in a 10-week extensive reading program in a second-semester Japanese language course. The number of books read by each student and the student’s grade in the course are listed in the next table.

1. Construct a stem-and-leaf display for the number of books read by the students.

2. Highlight (or circle) the leaves in the display that correspond to students who earned an A grade in the course. What inference can you make about these students?

   Number of Books	Course Grade
   53	A
   42	A
   40	A
   40	B
   39	A
   34	A
   34	A
   30	A
   28	B
   24	A
   22	C
   21	B
   20	B
   16	B

3. Find the mean, median, and mode of the number of books read. Interpret these values.

4. What do the mean and median indicate about the skewness of the distribution of the data?

5. Find the mean and standard deviation of the number of books read by students who earned an A grade.

6. Find the mean and standard deviation of the number of books read by students who earned either a B or C grade.

7. Refer to parts a and b. Which of the two groups of students has a more variable distribution for number of books read?

8. Find the z-score for an A student who read 40 books. Interpret the result.

9. Find the z-score for a B or C student who read 40 books. Interpret the result.

10. Which of the two groups of students is more likely to have read 40 books? Explain.

2.189 Body length of armadillos. A group of environmentalists reported the results of a study of giant armadillos inhabiting the southeastern region of Venezuela. A sample of 80 armadillos was captured and the body length (not including the tail) of each was measured (in centimeters). A MINITAB graph summarizing the data is shown below.

1. What type of graph is employed?

2. How many armadillos have body lengths between 87 and 91 centimeters?

3. What proportion of the armadillos have body lengths between 87 and 91 centimeters?

4. The dotted vertical lines on the graph show the minimum and maximum sizes of giant armadillos that can be legally captured for commercial purposes. What proportion of the captured armadillos are illegal?

CRASH 2.190 Crash tests on new cars. Each year, the National Highway Traffic Safety Administration (NHTSA) crash tests new car models to determine how well they protect the driver and front-seat passenger in a head-on collision. The NHTSA has developed a “star” scoring system for the frontal crash test, with results ranging from one star (*) to five stars (*****). The more stars in the rating, the better the level of crash protection in a head-on collision. The NHTSA crash test results for 98 cars in a recent model year are stored in the data file named CRASH. The driver-side star ratings for the 98 cars are summarized in the MINITAB printout below. Use the information in the printout to form a pie chart. Interpret the graph.

CRASH 2.191 Crash tests on new cars. Refer to Exercise 2.190 and the NHTSA crash test data. One quantitative variable recorded by the NHTSA is driver’s severity of head injury (measured on a scale from 0 to 1,500). Numerical descriptive statistics for the 98 driver head-injury ratings are displayed in the MINITAB printout at the bottom of the page.

1. Interpret each of the statistics shown on the printout.

2. Find the z-score for a driver head-injury rating of 408. Interpret the result.

TILL 2.192 Estimating the age of glacial drifts. Tills are glacial drifts consisting of a mixture of clay, sand, gravel, and boulders. Engineers from the University of Washington’s Department of Earth and Space Sciences studied the chemical makeup of buried tills in order to estimate the age of the glacial drifts in Wisconsin (American Journal of Science, Jan. 2005). The ratio of the elements aluminum (Al) and beryllium (Be) in sediment is related to the duration of burial. The Al–Be ratios for a sample of 26 buried till specimens are given in the accompanying table.

1. Find and interpret the z-score associated with the largest ratio, the smallest ratio, and the mean ratio.

2. Would you consider the largest ratio to be unusually large? Why or why not?

3. Construct a box plot for the data and identify any outliers.

2.193 Sociology fieldwork methods. University of New Mexico professor Jane Hood investigated the fieldwork methods used by qualitative sociologists. (Teaching Sociology, July 2006.) Searching for all published journal articles, dissertations, and conference proceedings over the previous seven years in the Sociological Abstracts database, she discovered that fieldwork methods could be categorized as follows: Interview, Observation plus Participation, Observation Only, and Grounded Theory. The accompanying table shows the number of papers in each category. Use an appropriate graph to portray the results of the study. Interpret the graph.

MTBE 2.194 Groundwater contamination in wells. Refer to the Environmental Science & Technology (Jan. 2005) study of the factors related to MTBE contamination in 223 New Hampshire wells, presented in Exercise 2.23 (p. 42). The data are saved in the MTBE file. Two of the many quantitative variables measured for each well are the pH level (standard units) and the MTBE level (micrograms per liter).

1. Construct a histogram for the pH levels of the sampled wells. From the histogram, estimate the proportion of wells with pH values less than 7.0.

2. For those wells with detectible levels of MTBE, construct a histogram for the MTBE values. From the histogram, estimate the proportion of contaminated wells with MTBE values that exceed 5 micrograms per liter.

3. Find the mean and standard deviation for the pH levels of the sampled wells, and construct the interval $\overline{x}\pm 2s$. Estimate the percentage of wells with pH levels that fall within the interval. What rule did you apply to obtain the estimate? Explain.

4. Find the mean and standard deviation for the MTBE levels of the sampled wells, and construct the interval $\overline{x}\pm 2s$. Estimate the percentage of wells with MTBE levels that fall within the interval. What rule did you apply to obtain the estimate? Explain.

2.195 Improving SAT scores. The National Education Longitudinal Survey (NELS) tracks a nationally representative sample of U.S. students from eighth grade through high school and college. Research published in Chance (Winter 2001) examined the SAT scores of 265 NELS students who paid a private tutor to help them improve their scores. The table summarizes the changes in both the SAT-Mathematics and SAT-Verbal scores for these students.

1. Suppose one of the 265 students who paid a private tutor is selected at random. Give an interval that is likely to contain the change in this student’s SAT-Math score.

2. Repeat part a for the SAT-Verbal score.

3. Suppose the selected student’s score increased on one of the SAT tests by 140 points. Which test, the SAT-Math or SAT-Verbal, is the one most likely to have had the 140-point increase? Explain.

LICHEN 2.196 Radioactive lichen. Lichen has a high absorbance capacity for radiation fallout from nuclear accidents. Since lichen is a major food source for Alaskan caribou, and caribou are, in turn, a major food source for many Alaskan villagers, it is important to monitor the level of radioactivity in lichen. Researchers at the University of Alaska, Fairbanks, collected data on nine lichen specimens at various locations for this purpose. The amount of the radioactive element cesium-137 was measured (in microcuries per milliliter) for each specimen. The data values, converted to logarithms, are given in the following table (note that the closer the value is to zero, the greater is the amount of cesium in the specimen).

1. Construct a dot plot for the nine measurements.

2. Construct a stem-and-leaf display for the nine measurements.

3. Construct a histogram plot of the nine measurements.

4. Which of the three graphs in parts a–c, respectively, is most informative?

5. What proportion of the measurements has a radioactivity level of $-5.00$ or lower?

PPM 2.197 Ammonia in car exhaust. Three-way catalytic converters have been installed in new vehicles in order to reduce pollutants from motor vehicle exhaust emissions. However, these converters unintentionally increase the level of ammonia in the air. Environmental Science & Technology (Sept. 1, 2000) published a study on the ammonia levels near the exit ramp of a San Francisco highway tunnel. The data in the table represent daily ammonia concentrations (parts per million) on eight randomly selected days during the afternoon drive time in the summer of a recent year. Find an interval that is likely to contain the ammonia level for a randamly selected day during afternoon drive time.

Alternate View

1.53

1.50

1.37

1.51

1.55

1.42

1.41

1.48

2.198 Speed of light from galaxies. Astronomers theorize that cold dark matter caused the formation of galaxies and clusters of galaxies in the universe. The theoretical model for cold dark matter requires an estimate of the velocities of light emitted from galaxy clusters. The Astronomical Journal (July 1995) published a study of observed velocities of galaxies in four different clusters. Galaxy velocity was measured in kilometers per second (km/s), using a spectrograph and high-power telescope.

1. The observed velocities of 103 galaxies located in the cluster named A2142 are summarized in the accompanying histogram. Comment on whether the empirical rule is applicable to describing the velocity distribution for this cluster.

   

2. The mean and standard deviation of the 103 velocities observed in galaxy cluster A2142 were reported as $\overline{x}=27, 117\text{km/s}$ and $s=1, 280\text{km/s,}$ respectively. Use this information to construct an interval that captures approximately 95% of the velocities of the galaxies in the cluster.

3. Recommend a single velocity value to be used in the CDM model for galaxy cluster A2142. Explain your reasoning.

DOLPHIN 2.199 Whistling dolphins. Marine scientists who study dolphin communication have discovered that bottlenose dolphins exhibit an individualized whistle contour known as their signature whistle. A study was conducted to categorize the signature whistles of 10 captive adult bottlenose dolphins in socially interactive contexts (Ethology, July 1995). A total of 185 whistles were recorded during the study period; each whistle contour was analyzed and assigned to a category by a contour similarity (CS) technique. The results are reported in the table on p. 110. Use a graphical method to summarize the results. Do you detect any patterns in the data that might be helpful to marine scientists?

FRECKLE 2.200 Freckling of superalloy ingots. Freckles are defects that sometimes form during the solidification of alloy ingots. A freckle index has been developed to measure the level of freckling on the ingot. A team of engineers conducted several experiments to measure the freckle index of a certain type of superalloy (Journal of Metallurgy, Sept. 2004). The data for $n=18$ alloy tests are shown in the table.

1. Construct a box plot for the data and use it to find any outliers.

2. Find and interpret the z-scores associated with the alloys you identified in part a.

EVOS 2.201 Oil spill impact on seabirds. The Journal of Agricultural, Biological, and Environmental Statistics (Sept. 2000) published a study on the impact of the Exxon Valdez tanker oil spill on the seabird population in Prince William Sound, Alaska. Data were collected on 96 shoreline locations (called transects) of constant width, but variable length. For each transect, the number of seabirds found is recorded, as are the length (in kilometers) of the transect and whether or not the transect was in an oiled area. (The first five and last five observations in the EVOS file are listed in the accompanying table.)

1. a. Identify the variables measured as quantitative or qualitative.

2. b. Identify the experimental unit.

3. c. Use a pie chart to describe the percentage of transects in oiled and unoiled areas.

4. *d. Use a graphical method to examine the relationship between observed number of seabirds and transect length.

5. e. Observed seabird density is defined as observed count divided by length of transect. MINITAB descriptive statistics for seabird densities in unoiled and oiled transects are displayed in the printout at the bottom of the page. Assess whether the distribution of seabird densities differs for transects in oiled and unoiled areas.

6. f. For unoiled transects, give an interval of values that is likely to contain at least 75% of the seabird densities.

7. g. For oiled transects, give an interval of values that is likely to contain at least 75% of the seabird densities.

8. h. Which type of transect, an oiled or unoiled one, is more likely to have a seabird density of 16? Explain.

DIGITS 2.202 Benford’s Law of Numbers. According to Benford’s law, certain digits (1, 2, 3,…, 9) are more likely to occur as the first significant digit in a randomly selected number than are other digits. For example, the law predicts that the number “1” is the most likely to occur (30% of the time) as the first digit. In a study reported in the American Scientist (July–Aug. 1998) to test Benford’s law, 743 first-year college students were asked to write down a six-digit number at random. The first significant digit of each number was recorded and its distribution summarized in the following table. Describe the first digit of the “random guess” data with an appropriate graph. Does the graph support Benford’s law? Explain.

GALAXY 2.203 Speed of light from galaxies. Refer to The Astronomical Journal study of galaxy velocities, presented in Exercise 2.198 (p. 109). A second cluster of galaxies, named A1775, is thought to be a double cluster—that is, two clusters of galaxies in close proximity. Fifty-one velocity observations (in kilometers per second, km/s) from cluster A1775 are listed in the table.

2.204 Standardized test “average.” U.S. News & World Report reported on many factors contributing to the breakdown of public education. One study mentioned in the article found that over 90% of the nation’s school districts reported that their students were scoring “above the national average” on standardized tests. Using your knowledge of measures of central tendency, explain why the schools’ reports are incorrect. Does your analysis change if the term “average” refers to the mean? To the median? Explain what effect this misinformation might have on the perception of the nation’s schools.

2.205 Zinc phosphide in pest control. A chemical company produces a substance composed of 98% cracked corn particles and 2% zinc phosphide for use in controlling rat populations in sugarcane fields. Production must be carefully controlled to maintain the 2% zinc phosphide because too much zinc phosphide will cause damage to the sugarcane and too little will be ineffective in controlling the rat population. Records from past production indicate that the distribution of the actual percentage of zinc phosphide present in the substance is approximately mound shaped, with a mean of 2.0% and a standard deviation of .08%. Suppose one batch chosen randomly actually contains 1.80% zinc phosphide. Does this indicate that there is too little zinc phosphide in this production? Explain your reasoning.

2.206 Grades in statistics. The final grades given by two professors in introductory statistics courses have been carefully examined. The students in the first professor’s class had a grade point average of 3.0 and a standard deviation of .2. Those in the second professor’s class had grade points with an average of 3.0 and a standard deviation of 1.0. If you had a choice, which professor would you take for this course? 

2.207 Salaries of professional athletes. The salaries of superstar professional athletes receive much attention in the media. The multimillion-dollar long-term contract is now commonplace among this elite group. Nevertheless, rarely does a season pass without negotiations between one or more of the players’ associations and team owners for additional salary and fringe benefits for all players in their particular sports.

1. If a players’ association wanted to support its argument for higher “average” salaries, which measure of central tendency do you think it should use? Why?

2. To refute the argument, which measure of central tendency should the owners apply to the players’ salaries? Why?

2.208 The Hite Report. Researcher Shere Hite shocked conservative America with her famous Hite Report on the permissive sexual attitudes of American men and women. In her book Women and Love: A Cultural Revolution in Progress (Knopf Press, 1988), Hite reveals some startling statistics describing how women feel about contemporary relationships:

• Eighty-four percent are not emotionally satisfied with their relationship.

• Ninety-five percent report “emotional and psychological harassment” from their men.

• Seventy percent of those married five years or more are having extramarital affairs.

• Only 13% of those married more than two years are “in love.”

Hite conducted the survey by mailing out 100,000 questionnaires to women across the country over a seven-year period. The questionnaires were mailed to a wide variety of organizations, including church groups, women’s voting and political groups, women’s rights organizations, and counseling and walk-in centers for women. Organizational leaders were asked to circulate the questionnaires to their members. Hite also relied on volunteer respondents who wrote in for copies of the questionnaire. Each questionnaire consisted of 127 open-ended questions, many with numerous subquestions and follow-ups. Hite’s instructions read, “It is not necessary to answer every question! Feel free to skip around and answer those questions you choose.” Approximately 4,500 completed questionnaires were returned, a response rate of 4.5%. These questionnaires form the data set from which the preceding percentages were determined. Hite claims that the 4,500 women respondents are a representative sample of all women in the United States and, therefore, that the survey results imply that vast numbers of women are “suffering a lot of pain in their love relationships with men.” Critically assess the survey results. Do you believe they are reliable?