Organizing Data

Residential Energy Consumption. Construct a relative frequency polygon for the energy-consumption data given in Exercise $2.84$ Use the classes specified in that exercise.

Clocking the Cheetah. Construct a relative-frequency polygon for the speed data given in Exercise 2.89. Use the classes specified in that exercise.

As mentioned, for relative-frequency polygons, we label the horizontal axis with class marks in limit grouping and class midpoints in cut point grouping. How do you think the horizontal axis is labeled in single-value grouping.

Age and Gender. The following bivariate data on age (in years) and gender were obtained from the students in a freshman calculus course. The data show, for example, that the first student on the list is $21$years old and is a male.

a. Group these data in the following contingency table. For the first student, place a tally mark in the box labelled by the " $21-25$ column and the "Male" row, as indicated. Tally the data for the other $49$students.

b. Construct a table like the one in part (a) but with frequencies replacing the tally marks. Add the frequencies in each row and column of your table and record the sums in the proper "Total" boxes.

c. What do the row and column totals in your table in part (b) represent?

d. Add the row totals and add the column totals. Why are those two sums equal, and what does their common value represent?

e. Construct a table that shows the relative frequencies for the data. (Hint: Divide each frequency obtained in part (b) by the total of $50$ students.)

f. Interpret the entries in your table in part (e) as percentages.

The following bivariate data on age (in years) and gender were obtained from students in a freshman calculus course. The data show, for example that the first student on the list is \(21\) years old and is a male.

a. Group these data in the following contingency  table. For the first student, place a tally mark in the box labeled by the "\(21-25\)" column and the "Male" row, as indicated. Tally the data for the other \(49\) students. 

b. Construct a table like the one in part (a) but with frequencies replacing tally marks. Add the frequencies in each row and column of your table and record the sums in the proper "Total" boxes.

c. What do the row and column total in your table in part (b) represent?

d. Add the row totals and add the column totals. Why are those two sums equal, and what does their common value represent?

e. Construct a table that shows the relative frequencies for the data.

f. Interpret the entries in your table in part (e) as percentages. 

Residential Energy Consumption. Refer to the energy consumption data given in Exercise 2.84.

a. Construct a table similar to table $2.14$ for the data, based on the classes specified in Exercise 2.84. Interpret your results.

b. Construct an ogive for the data.

Residential Energy Consumption. Refer to the energy consumption data given in Exercise 2.84.

a. Construct a table similar to Table $2.14$ for the data, based on the classes specified in Exercise 2.84. Interpret your results.

b. Construct an ogive for the data.

Clocking the Cheetah. Refer to the speed data given in Exercise $2.89$.

a. Construct a table similar to Table $2.14$ for the data, based on the classes specified in Exercise 2.89. Interpret your results.

b. Construct an ogive for the data.

For instance, consider the grouped days-to-maturity data given in Table $2.10$ (b) on page $57$.

From that table, we see that the cumulative frequency of investments with a maturity period of less than $50$ day is $4(3+1)$and, therefore, the cumulative relative frequency is$0.1(4 / 40)$.

Table $2.14$shows all cumulative information for the days-to-maturity data.

Cumulative information for days-to-maturity data.

a. Round each observation to the nearest year and then construct a stem-and-leaf diagram of the rounded data.

b. Truncate each observation by dropping the decimal part, and then construct a stem-and-leaf diagram of the truncated data.

c. Compare the stem-and-leaf diagrams that you obtained in parts (a) and (b).

In the special report "Mousetrap: The Most-Visited Shoe and Apparel E-tailers" (Footwear News, Vol. 58 , No. 3, p. 18), we found the following data on the average time, in minutes, spent per user per month from January to June of one year for a sample of $15$ shoe and apparel retail websites.

The following Minitab output shows a stem-and-leaf diagram for these data. The second column gives the stems, and the third column gives the leaves.

Identify and sketch three distribution shapes that are symmetric

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

Give two reasons why the use of smooth curves to describe shapes of distributions is helpful.

We have drawn a smooth curve that  represents a distribution, 

a. Identify the shape of the distirbution with regard to modality.

b. Identify the shape  of the distribution with regard to symmetry (or non symmetry)

c . If the distribution is unimodal and non symmetric, classify it as either right skewed or left skewed.

Suppose that a variable of a population has a bell shaped distribution. If you take a large simple random sample from the population, roughly what shape would you expect the distribution of sample to be? Explain your answer.

Children of U.S presidents. The information please alamanae provides the number of children of each of the U.S presidents. A frequency histogram of number of children  by president, through president Bararck H. Obama , as it follows.

State whether the distribution is symmetric, right skewed, or left skewed.

 Clocking the Cheetah. The cheetah (Acinonyx jubatus) is the fastest land mammal and is highly specialized to run down prey. The cheetah often exceeds speeds of $60 mph$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 $72 mph$. Following is a frequency histogram for the speeds, in miles per hour, for a sample of $35$ cheetahs.

State whether the distribution is symmetric, right skewed, left skewed.

Distribution of a data set.

Population data.

Sample distribution.

Population distribution.

Distribution of a variable.

2.130 Suppose that a variable of a population has a reverse-$J$-shaped distribution and that two simple random samples are taken from the population.

a. Would you expect the distributions of the two samples to have roughly the same shape? If so, what shape?

b. Would you expect some variation in shape for the distributions of the two samples? Explain your answer.

 Suppose that a variable of a population has a reverse-J-shaped distribution and that two simple random samples are taken from the population.

 a) Would you expect the distributions of the two samples to have roughly the same shape? If so, what shape?
 b) Would you expect some variation in shape for the distributions of the two samples? Explain your answers.

Suppose that a variable of a population has a reverse-J-shaped distribution and that two simple random samples are taken from the population.

a. Would you expect the distributions of the two samples to have roughly the same shape? If so, what shape?

b. Would you expect some variation in shape for the distributions of the two samples? Explain your answer.

 Malnutrition and Poverty. R. Reifen et al. studied various nutritional measures of Ethiopian school children and published their findings in the paper "Ethiopian-Born and Native Israeli School Children Have Different Growth Patterns" (Nutrition, Vol. 19 , pp. $427-431$). The study, conducted in Azezo, North West Ethiopia, found that malnutrition is prevalent in primary and secondary school children because of economic poverty. A frequency histogram for the weights, in kilograms (kg), of $60$  randomly selected male Ethiopianborn school children ages $12-15$ years old is as follows.

State whether the distribution is symmetric, right skewed, or left skewed.

The Court's Burrow. The subterranean color (Spalacopus cyan) is a social rodent that lives in large colonies in underground burrows that can reach lengths of up to 600 meters. Zoologists S. Begall and M. Gallardo studied the characteristics of the burrow systems of the subterranean color in central Chile and published their findings in the Journal of Zoology, Landen (Vol. 251, pp. 53-60). A sample of 51 burrows, whose depths were measured in centimeters, yielded the following frequency histogram.

Baltimore Ravens. From Player Roster, the official roster of the 2013 Super Bowl champion Baltimore Ravens, we obtained the heights, in inches, of the players on that team. A dot plot of those heights is as follows.

PCBs and Pelicans. Polychlorinated biphenyls (PCBs), industrial pollutants, are known to be carcinogens and a great danger to natural ecosystems. As a result of several studies, PCB production was banned in the United States in 1979 and by the Stockholm Convention on Persistent Organic Pollutants in 2001. One study, published in 1972 by R. W. Risebrough, is titled "Effects of Environmental Pollutants Upon Animals Other Than Man" (Proceedings of the 6ed Berkeley Symposium on Mathematics and Statistics, VI, University of California Press, pp. 443-463). In that study, 60 Anacapa pelican eggs were collected and measured for their shell thickness, in millimeters (mm), and concentration of PCBs, in parts per million (ppm). Following is a relative-frequency histogram of the PCB concentration data.

Adjusted Gross Incomes. The Internal Revenue Service (IRS) publishes data on adjusted gross incomes in the document Siarisuics of Income, Individual Income Tax Reforms. The following relative-frequency histogram shows one year's individual income tax returns for adjusted gross incomes of less than  50,000.

Snow Goose Nests. In the article "Trophic Interaction Cycles in Tundra Ecosystems and the Impact of Climate Change" (BioScience, Vol. 55, No, 4, pp, 311-321), R. Ims and E. Fuglei provided an overview of animal species in the northern tundra. One threat to the snow goose in arctic Canada is the lemming. Snowy owls act as protection to the snow goose breeding grounds. For two years that are 3 years apart, the following graphs give relative frequency histograms of the distances, in meters, of snow goose nests to the nearest snowy owl nest.

Cholesterol Levels. According to the National Health and Nutrition Examination Survey, published by the Centers for Disease Control and Prevention, the average cholesterol level for children between 4 and 19 years of age is 165mg /dl. A pediatrician who tested the cholesterol levels of several young patients was alarmed to find that many had levels higher than 200mg / dL. The following relative frequency histogram shows the readings for some patients who had high cholesterol levels.

Sickle Cell Disease. A study published by E. Anionwu et al. in the British Medical Journal (Vol, 282, pp .283-286) measured the steady-state hemoglobin levels of patients with three different types of sickle cell disease. Following is a stem-and-leaf diagram of the data.

Stays in Europe and the Mediterranean. The Bureau of Economic Analysis gathers information on the length of stay in Europe and the Mediterranean by U.S. travelers. Data are published in the Survey of Current Bussiness. The following stem-and-leaf diagram portrays the length of stay, in days, of a sample of 36 U.S. residents who traveled to Europe and the Mediterranean last year.

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 given on the WeissStats site.

a. use the technology of your choice to identify the modality and symmetry (or non-symmetry) of the distribution of the data set.

b. if unimodal, classify the distribution as symmetric right-skewed. or left-skewed.

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.

a. use the technology of your choice to identify the modality and symmetry (or non-symmetry) of the distribution of the data set.

b. if unimodal, classify the distribution as symmetric right-skewed. or left-skewed.

Old Faithful. Old Faithful is a geyser in Yellowstone National Park that erupts at relatively predictable intervals. Since 2000, the eruptions of Old Faithfull occur on average every 90 minutes. Information on the daily eruptions of Old Faithful can be found on the website of The Geyser Observation and Study Association (GOSA). The following data provide the times between eruptions, in minutes, for Old Faithful over a two-day period.

a. Use cutpoint grouping with a first class of 60 -under 70 to construct a frequency histogram of the times between eruptions.

b. Identify the shape of the distribution with regard to modality.

c. Identify the shape of the distribution with regard to symmetry (or nonsymmetry).

d. If the distribution is unimodal and nonsymmetric, classify it as either right skewed or left skewed.

Snow Goose Nests. In the article "Trophic Interaction Cycles in Tundra Ecosystems and the Impact of Climate Change" (BioScience, Vol. 55, No. 4, pp. 311-321), R. Ims and E. Fuglei provided an overview of animal species in the northern tundra. One threat to the snow goose in arctic Canada is the lemming. Snowy owls act as protection to the snow goose breeding grounds. For two years that are 3 years apart, the following graphs give relative frequency histograms of the distances, in meters, of snow goose nests to the nearest snowy owl nest.

For each histogram do the following:

(a)  Identify the shape of the distribution with regard to modality.

(b)  Identify the shape of the distribution with regard to symmetry (non-symmetry).

(c)  If the distribution is unimodal and non-symmetric, classify it as either right-skewed or left-skewed.

(d)  Compare the two distributions. 

Old Faithful. Old Faithful is a geyser in Yellowstone National Park that erupts at relatively predictable intervals. Since 2000, the eruptions of Old Faithfull occur on average every 90 minutes. Information on the daily eruptions of Old Faithful can be found on the website of The Geyser Observation and Study Association (GOSA). The following data provide the times between eruptions, in minutes, for Old Faithful over a two-day period.

a. Use cut point grouping with a first class of $60-$under $70$ to construct a frequency histogram of the times between eruptions.

b. Identify the shape of the distribution with regard to modality.

c. Identify the shape of the distribution with regard to symmetry (or non-symmetry).

d. If the distribution is unimodal and nonsymmetric, classify it as either right-skewed or left-skewed.

Class Project: Number of Siblings. This exercise is a class project and works best in relatively large classes.

a. Determine the number of siblings for each student in the class.

b. Obtain a relative-frequency histogram for the number of siblings. Use single-value grouping.

c. Obtain a simple random sample of about one-third of the students in the class.

d. Find the number of siblings for each student in the sample.

e. Obtain a relative-frequency histogram for the number of siblings for the sample. Use single-value grouping.

f. Repeat parts (c)-(c) three more times.

g. Compare the histograms for the samples to each other and to that for the entire population. Relate your observations to Key Fact $2.1$.

Class Project: Random Digits. This exercise can be done individually or, better yet, as a class project. 

a. Use a table of random numbers or a random-number generator to obtain 50 random integers between $0$ and $9$ .

b. Without graphing the distribution of the $50$ numbers you obtained, guess its specific shape. Explain your reasoning.

c. Construct a relative-frequency histogram based on single-value grouping for the $50$ numbers that you obtained in part (a). Is its shape about what you expected?

d. If your answer to part (c) was "no," provide an explanation.

e. What would you do to make getting a "yes" answer to part (c) more plausible?

f. If you are doing this exercise as a class project, repeat parts (a) -(c) for$10000$ random integers.

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.

a. use the technology of your choice to identify the modality and symmetry (or non-symmetry) of the distribution of the data set.

b. if unimodal, classify the distribution as symmetric right-skewed. or left-skewed.

 Bachelor"s Completion. As reported by the U.S. Census bureau in the document Edwrational Arraignment 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.

a. use the technology of your choice to identify the modality and symmetry (or non-symmetry) of the distribution of the data set.

b. if unimodal, classify the distribution as symmetric right-skewed. or left-skewed.

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~F. The results of the study by P. Mackowiak et al. appeared in the article " A Critical Appraisal of 98.6F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl 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 Weists state site.

a. use the technology of your choice to identify the modality and symmetry (or non-symmetry) of the distribution of the data set.

b. if unimodal, classify the distribution as symmetric right-skewed. or left-skewed.