Organizing Data

Construct a stem-and-leaf diagram for the data, using one line per stem.

Construct a stem-and-leaf diagram for the data, using five lines per stem.

Construct a stem-and-leaf diagram for the data, using two lines per stem.

We have presented some quantitative data sets and specified a grouping method for practicing the concepts.

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative-frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Use cutpoint grouping with a first class midpoint of 0.5 and a class width of 1.

We have presented some quantitative data sets and specified a grouping method for practicing the concepts.

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative-frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Use cutpoint grouping with a first cutpoint of 25 and a class width of 3.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Professor Weiss asked his introductory statistics students to state how many siblings they have. The responses are shown in the following table. Use single-value grouping.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

The U.S. Census Bureau conducts nationwide surveys on characteristics of U.S. households and publishes the results in Current Population Reports. Following are the data on the number of people per household for a sample of 40 households. Use single-value grouping.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

In the paper "The Eastern Cottonmouth at the Northern Edge of Its Range", C. Blem and L. Blem examined the reproductive characteristics of the eastern cottonmouth, a once widely distributed snake whose numbers have decreased recently due to encroachment by humans. A simple random sample of 24 female cottonmouths in Florida yielded the following data on number of young per litter. Use single-value grouping.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Every year since 1998, Arbitron and Edison Research have conducted a nationally representative survey focusing on trends in digital platforms. According to their publication The Infinite Dial, in 2002, there was an average of 1.2 working computers in the U.S. home. A random sample of 45 households taken this year yielded the following data on number of working computers. Use single-value grouping.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

The U.S. Energy Information Administration collects data on residential energy consumption ad expenditures. Results are published in the document Residential Energy Consumption Survey: Consumption and Expenditures. The following table gives one year's energy consumption for a sample of 50 households in the South. Data are in millions of BTUs. Use limit grouping with a first class of 40-49 and a class width of 10.

 Consumer Reports provides information on new automobile models, including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of 35 new models yielded the following data on fuel tank capacity, in gallons. Use cutpoint grouping with 12 as the first cutpoint and classes of equal width 2 .

a. determine a frequency distribution.

b. obtain a relative-frequency distribution.

c. construct a frequency histogram based on your result from part (a).

d. construct a relative-frequency histogram based on your result from part (b).

Age of Passenger Cars. According to R. L. Polk & Co., the average age of passenger cars in the United States was $9.1$ years in $2000$. A sample of  passenger cars taken this year provided the ages, in years, displayed in the following table. Construct a dotplot for the ages.

Exam Scores. Construct a dot plot for the following exam scores of the students in an introductory statistics class.

Dementia is a person's loss of intellectual and social abilities that is severe enough to interfere with judgment, behavior, and daily functioning. Alzheimer's disease is the most common type of dementia. In the article "Living with Early Onset Dementia: Exploring the Experience and Developing EvidenceBased Guidelines for Practice" (Alzheimer's Care Quarterly, Vol. 5 , Issue 2, pp. 111-122), P. Harris and J. Keady explored the experience and struggles of people diagnosed with dementia and their families. A simple random sample of 21 people with early-onset dementia gave the following data on age, in years, at diagnosis. Use limit grouping with a first class of $40-44$ and a class width of 5 .

a. determine a frequency distribution.

b. obtain a relative-frequency distribution.

c. construct a frequency histogram based on your result from part (a).

d. construct a relative-frequency histogram based on your result from part (b).

The U.S. Department of Agriculture reports in Food Consumption, Prices, and Expenditures that the average American consumed about $33.5\mathrm{lb}$ of cheese in 2011. Cheese consumption has increased steadily since 1960 , when the average American ate only $8.3\mathrm{lb}$ of cheese annually. The following table provides last year's cheese consumption, in pounds, for 35 randomly selected Americans. Use limit grouping with a first class of $20-22$ and a class width of 3 .

a. determine a frequency distribution.

b. obtain a relative-frequency distribution.

c. construct a frequency histogram based on your result from part (a).

d. construct a relative-frequency histogram based on your result from part (b).

 Patients who undergo chronic hemodialysis often experience severe anxiety. Videotapes of progressive relaxation exercises were shown to one group of patients and neutral videotapes to another group. Then both groups took

the State-Trait Anxiety Inventory, a psychiatric questionnaire used to measure anxiety, on which higher scores correspond to higher anxiety. In the paper "The Effectiveness of Progressive Relaxation in Chronic Hemodialysis Patients" (Journal of Chronic Diseases, Vol. 35, No. 10), R. Alarcon et al. presented the results of the study. The following data give score results for the group that viewed relaxation-exercises videotapes. Use limit grouping with a first class of 12-17 and a class width of 6 .

a. determine a frequency distribution.

b. obtain a relative-frequency distribution.

c. construct a frequency histogram based on your result from part (a).

d. construct a relative-frequency histogram based on your result from part (b).

The cheetah (Acinonyx jubatus) is the fastest land mammal and is highly specialized to run down prey. The cheetah often exceeds speeds of $60\mathrm{mph}$ and, according to the online document "Cheetah Conservation in Southern Africa" (Trade \& Environment Database (TED) Case Studies, Vol. 8, No. 2) by J. Urbaniak, the cheetah is capable of speeds up to$72\mathrm{mph}$. The following table gives the speeds, in miles per hour, over $1 / 4$ mile for 35 chectahs. Use cutpoint grouping with 52 as the first cutpoint and classes of equal width 2 .

a. determine a frequency distribution.

b. obtain a relative-frequency distribution.

c. construct a frequency histogram based on your result from part (a).

d. construct a relative-frequency histogram based on your result from part (b).

In the article "Distribution of Oxygen in Surface Sediments from Central Sagami Bay, Japan: In Situ Measurements by Microelectrodes and Planar Optodes" (Deep Sea Research Part I: Oceanographic Research Papers, Vol. 52, Issue 10, pp. 1974-1987), R. Glud et al. explored the distributions of oxygen in surface sediments from central Sagami Bay. The oxygen distribution gives important information on the general biogeochemistry of marine sediments. Measurements were performed at 16 sites. A sample of 22 depths yielded the following data, in millimoles per square meter per day, on diffusive oxygen uptake. Use cutpoint grouping with a first class of 0 -under 1 .

a. determine a frequency distribution.

b. obtain a relative-frequency distribution.

c. construct a frequency histogram based on your result from part (a).

d. construct a relative-frequency histogram based on your result from part (b).

Construct a stem-and-leaf diagram for the data, using the specified number of lines per stem.

In the article "Distribution of Oxygen in Surface Sediments from Central Sagami Bay, Japan: In Situ Measurements by Microelectrodes and Planar Optodes", R. Glud et al. explored the distributions of oxygen distribution gives important information on the general biogeochemistry of marine sediments. Measurements were performed at $16$ sites. A sample of $22$ depths yielded the following data, in millimoles per square meter per day, on diffusive oxygen uptake. Use cut-point grouping with a first class of $0$-under $1$.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Dementia is a person's loss of intellectual and social abilities that is severe enough to interfere with judgment, behavior and daily functioning. Alzheimer's disease is the most common type of dementia. In the article "Living with Early Onset Dementia: Exploring the Experience and Developing Evidence Based Guidelines for Practice", P. Harris and J. Keady explored the experience and struggles of people diagnosed with dementia and their families. A simple random sample of 21 people diagnosed with dementia and their families. A simple random sample of 21 people with early-onset dementia gave the following data on age, in years, at diagnosis. Use limit grouping with a fast class of 40-44 and a class width of 5.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

The U.S. Department of Agriculture reports in Food Consumption, Prices and Expenditures that the avrage American consumed about 33.5lb of cheese in 2011. Cheese consumption has increased steadily since 1960, when the average American ate only 8.3lb of cheese annually. The following table provides last year's cheese consumption, in pounds, for 35 randomly selected Americans. Use limit grouping with a first class of 20-22 and a class width of 3.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Patients who undergo chronic hemodialysis often experience severe anxiety. Videotapes of progressive relaxation exercises were shown to one group of patients and neutral videotapes to another group. Then both groups took the State-Trait Anxiety Inventory, a psychiatric questionnaire used to measure anxiety, on which higher scores correspond to higher anxiety. In the paper "The Effectiveness of Progressive Relaxation in Chronic Hemodialysis Patients", R. Alarcon et al. presented the results of the study. The following data give score results for the group that viewed relaxation-exercises videotapes. Use limit grouping with a first class of 12-17 and a class width of 6.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

From the TVbytheNumbers website, we obtain the viewing audiences, in millions, for the top 20 primetime broadcast TV shows for the week ending August 18,2013. Use cutpoint grouping with a first class of 4-under 5.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

The cheetah is the fastest land mammal and is highly specialized to run down prey. The cheetah often exceed speeds of 60mph and according to the online document "Cheetah Conservation in Southern Africa" by J. Urbaniak, the cheetah is capable of speeds up to 72mph. The following table gives the speeds, in miles per hour, over $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ mile for 35 cheetahs. Use cutpoint grouping with 52 as the first cutpoint and classes of equal width 2.

Use the specified grouping method to

Part (a): Determine a frequency distribution.

Part (b): Obtain a relative frequency distribution.

Part (c): Construct a frequency histogram based on your result from part (a).

Part (d): Construct a relative frequency histogram based on your result from part (b).

Consumer Reports provides information on new automobile models, including price, mileage ratings, engine size, body size and indicators of features. A simple random sample of 35 new models yielded the following data on fuel tank capacity, in gallons. Use cutpoint grouping with 12 as the first cutpoint and classes of equal width 2.

Stressed-Out Bus Drivers. Frustrated passengers, congested streets, time schedules, and air and noise pollution are just some of the physical and social pressures that lead many urban bus drivers to retire prematurely with disabilities such as coronary heart disease and stomach disorders. An intervention program designed by the Stockholm Transit District was implemented to improve the work conditions of the city's bus drivers. Improvements were evaluated by G. Evans et al., who collected physiological and psychological data for bus drivers who drove on the improved routes (intervention) and for drivers who were assigned the normal routes (control). Their findings were published in the article "Hassles on the Job: A Study of a Job Intervention With Urban Bus Drivers" (Journal of Organizational Behavior, $Vol. 20, pp. 199-208$). Following are data, based on the results of the study, for the heart rates, in beats per minute, of the intervention and control drivers.

Acute Postoperative Days. Several neurosurgeons wanted to determine whether a dynamic system (Z-plate) reduced the number of acute postoperative days in the hospital relative to a static system (ALPS plate). R. Jacobowitz, Ph.D., an Arizona State University professor, along with G. Vishteh, M.D., and other neurosurgeons obtained the following data on the number of acute postoperative days in the hospital using the dynamic and static systems.

M&Ms. In the article "Sweetening Statistics-What M&M's Can Teach Us" (Minitab Inc., August 2008), M. Paret and E. Martz discussed several statistical analyses that they performed on bags of M&Ms. The authors took a random sample of $30$ small bags of peanut M&Ms and obtained the following weights, in grams (g). Construct a stem-and-leaf diagram for these weights.

Women in the Workforce. In an issue of Science (Vol. 308, No. 5721 , p. 483), D. Normile reported on a study from the Japan Statistics Bureau of the$30$ industrialized countries in the Organization for Economic Co-operation and Development (OECD) titled "Japan Mulls Workforce Goals for Women." Following are the percentages of women in scientific workforces for a sample of $17$ countries. Construct a stem-and-leaf diagram for these percentages.

Process Capability. R. Morris and E. Watson studied various aspects of process capability in the paper "Determining Process Capability in a Chemical Batch Process" (Quality Engineering, Vol. $10\left(2\right)$, pp. 389-396). In one part of the study, the researchers compared the variability in product of a particular piece of equipment to a known analytic capability to decide whether product consistency could be improved. The following data were obtained for 10 batches of product.

Construct a stem-and-leaf diagram for these data with

a. one line per stem.

b. two lines per stem.

c. Which stem-and-leaf diagram do you find more useful? Why?

University Patents. The number of patents a university receives is an indicator of the research level of the university. From a study titled Science and Engineering Indicators issued by the National Science Foundation, we found the number of U.S. patents awarded to a sample of $36$ private and public universities to be as follows.

Construct a stem-and-leaf diagram for these data with

a. one line per stem.

b. two lines per stem.

c. Which stem-and-leaf diagram do you find more useful? Why?

San Francisco Giants. From the Baseball Almanac website, we found the heights, in inches, of the players on the 2012 World Series-winning San Francisco Giants baseball team.

a. Construct a stem-and-leaf diagram of these data with five lines per stem.

b. Why is it better to use five lines per stem here instead of one or two lines per stem?

Detroit Tigers. From the Baseball Almanac website, we found the heights, in inches, of the players on the 2012 World Series runner-up Detroit Tigers baseball team.

Use the graph to answer the following questions. Note that cholesterol levels are always expressed as whole numbers.

a. Construct a stem-and-leaf diagram of these data with five lines per stem.

b. Why is it better to use five lines per stem here instead of one or two lines per stem?

Adjusted Gross Incomes. The Internal Revenue Service (IRS) publishes data on adjusted gross incomes in Statistics of Income, and Individual Income Tax Returns. The following relative frequency histogram shows one year's individual income tax returns for adjusted gross incomes of less than $\backslash (50,000$.

Use the histogram and the fact that adjusted gross incomes are expressed to the nearest whole dollar to answer each of the following questions.

a. Approximately what percentage of the individual income tax returns had an adjusted gross income between$\backslash )10,000$ and $\backslash (19,999$, inclusive?

b. Approximately what percentage had an adjusted gross income of less than $\backslash )30,000$ ?

c. The IRS reported that $\backslash (89,928,000$ individual income tax returns had an adjusted gross income of less than $\backslash )50,000$. Approximately how many had an adjusted gross income between $\backslash (30,000$ and $\backslash )49,999$, inclusive?

Cholesterol Levels. According to the National Health and

Control and Prevention, the average cholesterol level for children between $4$ and $19$ years of age is $165\mathrm{mg}/\mathrm{dL}$. A pediatrician who tested the cholesterol levels of several young patients was alarmed to find that many had levels higher than $200\mathrm{mg}/\mathrm{dL}$. The following relativefrequency histogram shows the readings for some patients who had high cholesterol levels.

Use the graph to answer the following questions. Note that cholesterol levels are always expressed as whole numbers.

a. What percentage of the patients have cholesterol levels between $205$ and $209$ , inclusive?

b. What percentage of the patients have levels of $215$ or higher?

c. If the number of patients is $20$ , how many have levels between $210$ and $214$ , inclusive?

Hospital Beds. The number of hospital beds available in community hospitals is decreasing. Through advancements in care and technology, hospitals are getting more efficient. However, the aging and increasingly obese and diabetic population intensify the need. The American Hospital Association Annual Survey, published by the American Hospital Association (AHA), gave the number of hospital beds available per $1000$ people for each of the $50$ states plus the District of Columbia for $2010$. Following is a stem-and-leaf diagram of the data. The stems represent units and the leaves tenths (e.g., the first entry is $1.7$).

How many states (including the District of Columbia) had

a. at least three but less than four hospital beds per $1000$ people available?

b. at least four and a half hospital beds per $1000$ people available?

Parkinson's Disease. Parkinson's disease affects internally generated movements such as movements recalled from memory. L-Dopa is a drug that is used in clinical treatment of Parkinson's disease. In the article, "L-Dopa Induces Under-Damped Visually Guided Motor Responses in Parkinson's Disease" (Experimental Brain Research, Vol. 202, Issue 3, pp. 553-559), Wing-Lok Au et al. explore the effects of L-Dopa on patients with Parkinson's disease. Following is a stem-and-leaf diagram of the ages, in years, at which a sample of $14$ patients experienced the onset of symptoms of Parkinson's disease.

How many of these $14$ patients were

a. under $45$ years old at the onset of symptoms?

b. at least $65$ years old at the onset of symptoms?

c. between $50$ years old and $64$years old, inclusive, at the onset of symptoms?

The Beatles. In the article, "Length of The Beatles' Songs" (Chance, Vol. 25, No. 1, pp. 30-33), T. Koyama discusses aspects and interpretations of the lengths of songs by The Beatles. Data on the length, in seconds, of 229 Beatles' songs are presented on the WeissStats site. Use the technology of your choice to

a. obtain frequency and relative-frequency distributions.

b. get and interpret a frequency histogram or a relative-frequency histogram.

c. construct a dotplot.

d. Compare your graphs from parts (b) and (c).

Days to Maturity for Short-Term Investments. Refer to the days-to-maturity data in Table 2.6 on page 53 . Note that there are $40$ observations, the smallest and largest of which are $36$ and $99$ , respectively. Apply the preceding procedure to choose classes for limit grouping. Use approximately seven classes. Note: If in Step $2$ you decide on $10$ for the class width and in Step $3$ you choose $30$ for the lower limit of the first class, then you will get the same classes as used in Example 2.13; otherwise, you will get different classes (which is fine).

Exam Scores. The exam scores for the students in an introductory statistics class are as follows.

a. Group these exam scores, using the classes $30-39,40-49,50-59, 60-69,70-79,80-89$, and $90-100$.

b. What are the widths of the classes?

c. If you wanted all the classes to have the same width, what classes would you use?

The Great White Shark. In an article titled "Great White, Deep Trouble" (National Geographic, Vol. 197(4), pp. 2-29), Peter Benchley-the author of $JAWS$-discussed various aspects of the Great White Shark (Carcharodon carcharias). Data on the number of pups borne in a lifetime by each of 80 Great White Shark females are provided on the WeissStats site. Use the technology of your choice to

a. obtain frequency and relative-frequency distributions, using single-value grouping.

b. construct and interpret either a frequency histogram or a relative frequency histogram.

Weights of 18- to 24-Year-Old Males. Refer to the weight data in Table $2.8$ on page 54 . Note that there are 37 observations, the smallest and largest of which are $129.2$ and$278.8$, respectively. Apply the preceding procedure to choose classes for cut point grouping. Use approximately eight classes. Note: If in Step 2 you decide on 20 for the class width and in Step 3 you choose 120 for the lower cut point of the first class, then you will get the same classes as used in Example 2.14; otherwise, you will get different classes (which is fine).

The Beatles. In the article, "Length of The Beatles' Songs" (Chance, Vol. 25, No. 1, pp. 30-33), T. Koyama discusses aspects and interpretations of the lengths of songs by The Beatles. Data on the length, in seconds, of 229 Beatles' songs are presented on the WeissStats site. Use the technology of your choice to

 a. obtain frequency and relative frequency distributions.
 b. get and interpret a frequency histogram or a relative-frequency histogram.
 c. construct a dotplot.
 d. Compare your graphs from parts (b) and (c).

High School Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who has completed high school is provided on the WeissStats site. Apply the technology of your choice to construct a stem-and-leaf diagram of the percentages with
 a. one line per stem.
 b. two lines per stem.
 c. five lines per stem.
 d. Which stem-and-leaf diagram do you consider most useful? Explain your answer.

Bachelor's Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who has completed a bachelor's degree is provided on the WeissStats site. Apply the technology of your choice to construct a stem-and-leaf diagram of the percentages with
 a. one line per stem.
 b. two lines per stem.
 c. five lines per stem.
 d. Which stem-and-leaf diagram do you consider most useful? Explain your answer.

 Exam Scores. The exam scores for the students in an introductory statistics class are as follows.

88    82    89    70    85

63    100    86    67    39

90    96    76    34    81

64    75    84    89    96

a. Group these exam scores, using the classes 30-39,40-49,50-59, 60-69,70-79,80-89, and 90-100.

b. What are the widths of the classes?

c. If you wanted all the classes to have the same width, what classes would you use?

High School Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who has completed high school is provided on the WeissStats site. Apply the technology of your choice to construct a stem-and-leaf diagram of the percentages with

a. one line per stem.

b. two lines per stem.

c. five lines per stem.

d. Which stem-and-leaf diagram do you consider most useful? Explain your answer.

High School Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who has completed high school is provided on the WeissStats site. Apply the technology of your choice to construct a stem-and-leaf diagram of the percentages with

a. one line per stem.

b. two lines per stem.

c. five lines per stem.

d. Which stem-and-leaf diagram do you consider most useful? Explain your answer.

Body Temperature. A study by researchers at the University of Maryland addressed the question of whether the mean body temperature of humans is $98.6^{\circ }F$. The results of the study by P. Mackowiak et al. appeared in the article "A Critical Appraisal of $98.6^{\circ }F$" the Upper Limit of the Normal Body Temperature, and Other Legacies of Cart Reinhold August Wunderlich" (Journal of the American Medical Association, Vol. 268, pp. 1578-1580). Among other data, the researchers obtained the body temperatures of 93 healthy humans, as provided on the WeissStats site. Use the technology of your choice to obtain and interpret

a. a frequency histogram or a relative-frequency histogram of the temperatures.

b. a dot-plot of the temperatures.

c. a stem-and-leaf diagram of the temperatures.

d. Compare your graphs from parts (a)-(c). Which do you find most useful?

In the article "Length of the Beatles" songs, T. Koyama discusses aspects and interpretations of lengths of songs by the Beatles. Data on the length, in seconds, of \(229\) Beatles' songs are presented on the WeissStats site. Use the technology of your choice to

a. obtain frequency and relative-frequency distributions

b. get and interpret a frequency histogram or a relative-frequency histogram

c. construct a dotpot.

d. compare your graphs from part (b) and (c)