1/9/2018 Thinking Mathematically, Sixth Edition12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Random Sampling Blitzer Bonus A Sampling Fiasco In 1936, the Literary

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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs

12.1 Sampling, Frequency Distributions, and Graphs

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Describe the population whose properties are to be analyzed.

2 Select an appropriate sampling technique.

3 Organize and present data.

4 Identify deceptions in visual displays of data.

AT THE END OF THE twentieth century, there were 94 million households in the United States with television sets. The television program viewed by the greatestpercentage of such households in that century was the final episode of Over 50 million American households watched this program.

Numerical information, such as the information about the top three TV shows of the twentieth century, shown in Table 12.1, is called data. The word statistics is oftenused when referring to data. However, statistics has a second meaning: Statistics is also a method for collecting, organizing, analyzing, and interpreting data, as well asdrawing conclusions based on the data. This methodology divides statistics into two main areas. Descriptive statistics is concerned with collecting, organizing,summarizing, and presenting data. Inferential statistics has to do with making generalizations about and drawing conclusions from the data collected.

TABLE 12.1 TV Programs with the Greatest U.S. Audience ViewingPercentage of the Twentieth Century

Program Total Households Viewing Percentage1. Feb. 28, 1983 50,150,000 60.2%2. Dallas Nov. 21, 1980 41,470,000 53.3%3. Roots Part 8 Jan. 30, 1977 36,380,000 51.1%

Source: Nielsen Media Research

Populations and Samples

1 Describe the population whose properties are to be analyzed.

Consider the set of all American TV households. Such a set is called the population. In general, a population is the set containing all the people or objects whoseproperties are to be described and analyzed by the data collector.

The population of American TV households is huge. At the time of the conclusion, there were nearly 84 million such households. Did over 50 millionAmerican TV households really watch the final episode of A friendly phone call to each household (“So, how are you? What's new? Watch any good

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television last night? If so, what?”) is, of course, absurd. A sample, which is a subset or subgroup of the population, is needed. In this case, it would be appropriate tohave a sample of a few thousand TV households to draw conclusions about the population of all TV households.

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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Random Sampling

Blitzer BonusA Sampling Fiasco

In 1936, the Literary Digest mailed out over ten million ballots to voters throughout the country. The results poured in, and the magazine predicted a landslide victoryfor Republican Alf Landon over Democrat Franklin Roosevelt. However, the prediction of the Literary Digest was wrong. Why? The mailing lists the editors usedincluded people from their own subscriber list, directories of automobile owners, and telephone books. As a result, its sample was anything but random. It excludedmost of the poor, who were unlikely to subscribe to the Literary Digest, or to own a car or telephone in the heart of the Depression. Prosperous people in 1936 weremore likely to be Republican than the poor. Thus, although the sample was massive, it included a higher percentage of affluent individuals than the population as awhole did. A victim of both the Depression and the 1936 sampling fiasco, the Literary Digest folded in 1937.

Example 1 Populations and SamplesA group of hotel owners in a large city decide to conduct a survey among citizens of the city to discover their opinions about casino gambling.

a. Describe the population.

b. One of the hotel owners suggests obtaining a sample by surveying all the people at six of the largest nightclubs in the city on a Saturday night. Each person willbe asked to express his or her opinion on casino gambling. Does this seem like a good idea?

SOLUTION

a. The population is the set containing all the citizens of the city.

b. Questioning people at six of the city's largest nightclubs is a terrible idea. The nightclub subset is probably more likely to have a positive attitude toward casinogambling than the population of all the city's citizens.

Check Point 1A city government wants to conduct a survey among the city's homeless to discover their opinions about required residence in city shelters from midnight until 6 A.M.

a. Describe the population.

b. A city commissioner suggests obtaining a sample by surveying all the homeless people at the city's largest shelter on a Sunday night. Does this seem like a goodidea? Explain your answer.

Random SamplingThere is a way to use a small sample to make generalizations about a large population: Guarantee that every member of the population has an equal chance to beselected for the sample. Surveying people at six of the city's largest nightclubs does not provide this guarantee. Unless it can be established that all citizens of the cityfrequent these clubs, which seems unlikely, this sampling scheme does not permit each citizen an equal chance of selection.

Random SamplesA random sample is a sample obtained in such a way that every element in the population has an equal chance of being selected for the sample.

Suppose that you are elated with the quality of one of your courses. Although it's an auditorium section with 120 students, you feel that the professor is lecturing right toyou. During a wonderful lecture, you look around the auditorium to see if any of the other students are sharing your enthusiasm. Based on body language, it's hard totell. You really want to know the opinion of the population of 120 students taking this course. You think about asking students to grade the course on an A to F scale,anticipating a unanimous A. You cannot survey everyone. Eureka! Suddenly you have an idea on how to take a sample. Place cards numbered from 1 through 120, onenumber per card, in a box. Because the course has assigned seating by number, each numbered card corresponds to a student in the class. Reach in and randomlyselect six cards. Each card, and therefore each student, has an equal chance of being selected. Then use the opinions about the course from the six randomly selectedstudents to generalize about the course opinion for the entire 120-student population.










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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Random Sampling

2 Select an appropriate sampling technique.

Your idea is precisely how random samples are obtained. In random sampling, each element in the population must be identified and assigned a number. The numbersare generally assigned in order. The way to sample from the larger numbered population is to generate random numbers using a computer or calculator. Each numberedelement from the population that corresponds to one of the generated random numbers is selected for the sample.

Call-in polls on radio and television are not reliable because those polled do not represent the larger population. A person who calls in is likely to have feelings about anissue that are consistent with the politics of the show's host. For a poll to be accurate, the sample must be chosen randomly from the larger population. The A. C.Nielsen Company uses a random sample of approximately 5000 TV households to measure the percentage of households tuned in to a television program.

Example 2 Selecting an Appropriate Sampling TechniqueWe return to the hotel owners in the large city who are interested in how the city's citizens feel about casino gambling. Which of the following would be the mostappropriate way to select a random sample?

a. Randomly survey people who live in the oceanfront condominiums in the city.

b. Survey the first 200 people whose names appear in the city's telephone directory.

c. Randomly select neighborhoods of the city and then randomly survey people within the selected neighborhoods.

SOLUTION

Keep in mind that the population is the set containing all the city's citizens. A random sample must give each citizen an equal chance of being selected.

a. Randomly selecting people who live in the city's oceanfront condominiums is not a good idea. Many hotels lie along the oceanfront, and the oceanfront propertyowners might object to the traffic and noise as a result of casino gambling. Furthermore, this sample does not give each citizen of the city an equal chance of beingselected.

b. If the hotel owners survey the first 200 names in the city's telephone directory, all citizens do not have an equal chance of selection. For example, individualswhose last name begins with a letter toward the end of the alphabet have no chance of being selected.

c. Randomly selecting neighborhoods of the city and then randomly surveying people within the selected neighborhoods is an appropriate technique. Using thismethod, each citizen has an equal chance of being selected.

In summary, given the three options, the sampling technique in part (c) is the most appropriate.

Surveys and polls involve data from a sample of some population. Regardless of the sampling technique used, the sample should exhibit characteristics typical of thosepossessed by the target population. This type of sample is called a representative sample.
















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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Frequency Distributions

Check Point 2Explain why the sampling technique described in Check Point 1(b) on page 767 is not a random sample. Then describe an appropriate way to select a random sampleof the city's homeless.

Blitzer BonusThe United States Census

A census is a survey that attempts to include the entire population. The U.S. Constitution requires a census of the American population every ten years. When theFounding Fathers invented American democracy, they realized that if you are going to have government by the people, you need to know who and where they are.Nowadays about $400 billion per year in federal aid is distributed based on the Census numbers, for everything from jobs to bridges to schools. For every 100 peoplenot counted, states and communities could lose as much as $130,000 annually, or $1300 per person each year, so this really matters.

Although the Census generates volumes of statistics, its main purpose is to give the government block-by-block population figures. The U.S. Census is not foolproof.The 1990 Census missed 1.6% of the American population, including an estimated 4.4% of the African-American population, largely in inner cities. Only 67% ofhouseholds responded to the 2000 Census, even after door-to-door canvassing. About 6.4 million people were missed and 3.1 million were counted twice. Althoughthe 2010 Census was one of the shortest forms in history, counting each person was not an easy task, particularly with concerns about immigration status and privacyof data.

Of course, there would be more than $400 billion to spread around if it didn't cost so much to count us in the first place: about $15 billion for the 2010 Census. Thatincluded $338 million for ads in 28 languages, a Census-sponsored NASCAR entry, and $2.5 million for a Super Bowl ad. The ads were meant to boost the responserate, since any household that did not mail back its form got visited by a Census worker, another pricey item. In all, the cost of the 2010 Census worked out toappoximately $49 per person.

Frequency Distributions

3 Organize and present data.

After data have been collected from a sample of the population, the next task facing the statistician is to present the data in a condensed and manageable form. In thisway, the data can be more easily interpreted.

Suppose, for example, that researchers are interested in determining the age at which adolescent males show the greatest rate of physical growth. A random sample of35 ten-year-old boys is measured for height and then remeasured each year until they reach 18. The age of maximum yearly growth for each subject is as follows:

12, 14, 13, 14, 16, 14, 14, 17, 13, 10, 13, 18, 12, 15, 14, 15, 15, 14, 14, 13, 15, 16, 15, 12, 13, 16, 11, 15, 12, 13, 12, 11, 13, 14, 14.

A piece of data is called a data item. This list of data has 35 data items. Some of the data items are identical. Two of the data items are 11 and 11. Thus, we can saythat the data value 11 occurs twice. Similarly, because five of the data items are 12, 12, 12, 12, and 12, the data value 12 occurs five times.

Collected data can be presented using a frequency distribution. Such a distribution consists of two columns. The data values are listed in one column. Numerical dataare generally listed from smallest to largest. The adjacent column is labeled frequency and indicates the number of times each value occurs.

















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Example 3 Constructing a Frequency DistributionConstruct a frequency distribution for the data of the age of maximum yearly growth for 35 boys:

12, 14, 13, 14, 16, 14, 14, 17, 13, 10, 13, 18, 12, 15, 14, 15, 15, 14, 14, 13, 15, 16, 15, 12, 13, 16, 11, 15, 12, 13, 12, 11, 13, 14, 14.

SOLUTION

It is difficult to determine trends in the data above in their current format. Perhaps we can make sense of the data by organizing them into a frequency distribution. Letus create two columns. One lists all possible data values, from smallest (10) to largest (18). The other column indicates the number of times the value occurs in thesample. The frequency distribution is shown in Table 12.2.

TABLE 12.2 A Frequency Distribution for a Boy's Age ofMaximum Yearly Growth

Age of Maximum Growth Number of Boys (Frequency)10 111 212 513 714 915 616 317 118 1

Total:

The frequency distribution indicates that one subject had maximum growth at age 10, two at age 11, five at age 12, seven at age 13, and so on. The maximum growthfor most of the subjects occurred between the ages of 12 and 15. Nine boys experienced maximum growth at age 14, more than at any other age within the sample.The sum of the frequencies, 35, is equal to the original number of data items.

The trend shown by the frequency distribution in Table 12.2 indicates that the number of boys who attain their maximum yearly growth at a given age increases untilage 14 and decreases after that. This trend is not evident in the data in their original format.

Check Point 3Construct a frequency distribution for the data showing final course grades for students in a precalculus course, listed alphabetically by student name in a grade book:

F, A, B, B, C, C, B, C, A, A, C, C, D, C, B, D, C, C, B, C.

A frequency distribution that lists all possible data items can be quite cumbersome when there are many such items. For example, consider the following data items.These are statistics test scores for a class of 40 students.

It's difficult to determine how well the group did when the grades are displayed like this. Because there are so many data items, one way to organize these data so thatthe results are more meaningful is to arrange the grades into groups, or classes, based on something that interests us. Many grading systems assign an A to grades inthe 90–100 class, B to grades in the 80–89 class, C to grades in the 70–79 class, and so on. These classes provide one way to organize the data.

Looking at the 40 statistics test scores, we see that they range from a low of 43 to a high of 94. We can use classes that run from 40 through 49, 50 through 59, 60through 69, and so on up to 90 through 99, to organize the scores. In Example 4, we go through the data and tally each item into the appropriate class. This method fororganizing data is called a grouped frequency distribution.




n = 35

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35 is the sum of

the frequencies.

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Example 4 Constructing a Grouped Frequency DistributionUse the classes 40–49, 50–59, 60–69, 70–79, 80–89, and 90–99 to construct a grouped frequency distribution for the 40 test scores on the previous page.

SOLUTION

We use the 40 given scores and tally the number of scores in each class.

Tallying Statistics Test ScoresTest Scores (Class) Tally Number of Students (Frequency)

40–49 3

50–59 6

60–69 6

70–79 11

80–89 9

90–99 5

Omitting the tally column results in the grouped frequency distribution in Table 12.3. The distribution shows that the greatest frequency of students scored in the 70–79 class. The number of students decreases in classes that contain successively lower and higher scores. The sum of the frequencies, 40, is equal to the originalnumber of data items.

TABLE 12.3 A Grouped FrequencyDistribution for Statistics Test ScoresClass Frequency40–49 350–59 660–69 670–79 1180–89 990–99 5

Total:

The leftmost number in each class of a grouped frequency distribution is called the lower class limit. For example, in Table 12.3, the lower limit of the first class is 40and the lower limit of the third class is 60. The rightmost number in each class is called the upper class limit. In Table 12.3, 49 and 69 are the upper limits for the firstand third classes, respectively. Notice that if we take the difference between any two consecutive lower class limits, we get the same number:

The number 10 is called the class width.

When setting up class limits, each class, with the possible exception of the first or last, should have the same width. Because each data item must fall into exactly oneclass, it is sometimes helpful to vary the width of the first or last class to allow for items that fall far above or below most of the data.

Check Point 4Use the classes in Table 12.3 to construct a grouped frequency distribution for the following 37 exam scores:



||| ◀

The second score in the list,

47, is shown as the first

tally in this row.

|||||

|||||

||||| ||||

|||||||| ◀

The first score in the list,

82, is shown as the first

tally in this row.

||||

n = 40

▲

40, the sum of the

frequencies, is the

number of data items.

50 − 40 = 10, 60 − 50 = 10, 70 − 60 = 10, 80 − 70 = 10, 90 − 80 = 10.

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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Histograms and Frequency Polygons

Histograms and Frequency PolygonsTake a second look at the frequency distribution for the age of a boy's maximum yearly growth in Table 12.2. A bar graph with bars that touch can be used to visuallydisplay the data. Such a graph is called a histogram. Figure 12.1 illustrates a histogram that was constructed using the frequency distribution in Table 12.2. A series ofrectangles whose heights represent the frequencies are placed next to each other. For example, the height of the bar for the data value 10, shown in Figure 12.1, is 1.This corresponds to the frequency for 10 given in Table 12.2. The higher the bar, the more frequent the age. The break along the horizontal axis, symbolized by ,eliminates listing the ages 1 through 9.

TABLE 12.2 A Frequency Distribution for a Boy's Age ofMaximum Yearly Growth

Age of Maximum Growth Number of Boys (Frequency)10 111 212 513 714 915 616 317 118 1

Total:

dFIGURE 12.1 A histogram for a boy's age of maximum yearly growth

A line graph called a frequency polygon can also be used to visually convey the information shown in Figure 12.1. The axes are labeled just like those in a histogram.Thus, the horizontal axis shows data values and the vertical axis shows frequencies. Once a histogram has been constructed, it's fairly easy to draw a frequencypolygon. Figure 12.2 shows a histogram with a dot at the top of each rectangle at its midpoint. Connect each of these midpoints with a straight line. To complete thefrequency polygon at both ends, the lines should be drawn down to touch the horizontal axis. The completed frequency polygon is shown in Figure 12.3.

dFIGURE 12.2 A histogram with a superimposed frequency polygon



n = 35

▲

35 is the sum of

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dFIGURE 12.3 A frequency polygon

Stem-and-Leaf PlotsA unique way of displaying data uses a tool called a stem-and-leaf plot. Example 5 illustrates how we sort the data, revealing the same visual impression created by ahistogram.

Example 5 Constructing a Stem-and-Leaf PlotUse the data showing statistics test scores for 40 students to construct a stem-and-leaf plot:


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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Histograms and Frequency Polygons

SOLUTION

The plot is constructed by separating each data item into two parts. The first part is the stem. The stem consists of the tens digit. For example, the stem for the scoreof 82 is 8. The second part is the leaf. The leaf consists of the units digit for a given value. For the score of 82, the leaf is 2. The possible stems for the 40 scores are4, 5, 6, 7, 8, and 9, entered in the left column of the plot.

Begin by entering each data item in the first row:

We continue in this manner and enter all the data items. Figure 12.4 shows the completed stem-and-leaf plot. If you turn the page so that the left margin is on thebottom and facing you, the visual impression created by the enclosed leaves is the same as that created by a histogram. An advantage over the histogram is that thestem-and-leaf plot preserves exact data items. The enclosed leaves extend farthest to the right when the stem is 7. This shows that the greatest frequency of studentsscored in the 70s.

dFIGURE 12.4 A stem-and-leaf plot displaying 40 test scores

Check Point 5Construct a stem-and-leaf plot for the data in Check Point 4 on page 771.



82 47 75 64 57 82 63 93.

Entering

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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Deceptions in Visual Displays of Data

Deceptions in Visual Displays of Data

4 Identify deceptions in visual displays of data.

Benjamin Disraeli, Queen Victoria's prime minister, stated that there are “lies, damned lies, and statistics.” The problem is not that statistics lie, but rather that liars usestatistics. Graphs can be used to distort the underlying data, making it difficult for the viewer to learn the truth. One potential source of misunderstanding is the scale onthe vertical axis used to draw the graph. This scale is important because it lets a researcher “inflate” or “deflate” a trend. For example, both graphs in Figure 12.5present identical data for the percentage of people in the United States living below the poverty level from 2001 through 2005. The graph on the left stretches the scaleon the vertical axis to create an overall impression of a poverty rate increasing rapidly over time. The graph on the right compresses the scale on the vertical axis tocreate an impression of a poverty rate that is slowly increasing, and beginning to level off, over time.

dFIGURE 12.5 Source: U.S. Census Bureau

There is another problem with the data in Figure 12.5. Look at Table 12.4 that shows the poverty rate from 2001 through 2011. Depending on the time frame chosen,the data can be interpreted in various ways. Carefully choosing a time frame can help represent data trends in the most positive or negative light.

TABLE 12.4 U.S.Poverty Rate from

2001 to 2011Year Poverty Rate2001 11.7%2002 12.1%2003 12.5%2004 12.7%2005 12.6%2006 12.3%2007 12.5%2008 13.2%2009 14.3%2010 15.1%2011 15.0%

Things to Watch for in Visual Displays of Data1. Is there a title that explains what is being displayed?

2. Are numbers lined up with tick marks on the vertical axis that clearly indicate the scale? Has the scale been varied to create a more or less dramatic impressionthan shown by the actual data?

3. Do too many design and cosmetic effects draw attention from or distort the data?

4. Has the wrong impression been created about how the data are changing because equally spaced time intervals are not used on the horizontal axis?Furthermore, has a time interval been chosen that allows the data to be interpreted in various ways?

5. Are bar sizes scaled proportionately in terms of the data they represent?

6. Is there a source that indicates where the data in the display came from? Do the data come from an entire population or a sample? Was a random sample usedand, if so, are there possible differences between what is displayed in the graph and what is occurring in the entire population? (We'll discuss these margins of errorin Section 12.4.) Who is presenting the visual display, and does that person have a special case to make for or against the trend shown by the graph?


















12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Concept and Vocabulary Check

Table 12.5 contains two examples of misleading visual displays.

TABLE 12.5 Examples of Misleading Visual DisplaysGraphic Display Presentation Problems

dSource: Bureau of Labor Statistics

Although the length of each dollar bill is proportional to its spending power, the visualdisplay varies both the length and width of the bills to show the diminishing power ofthe dollar over time. Because our eyes focus on the areas of the dollar-shaped bars,this creates the impression that the purchasing power of the dollar diminished evenmore than it really did. If the area of the dollar were drawn to reflect its purchasingpower, the 2005 dollar would be approximately twice as large as the one shown inthe graphic display.

dSource: U.S. Census Bureau

Cosmetic effects of homes with equal heights, but different frontal additions andshadow lengths, make it impossible to tell if they proportionately depict the givenareas. Time intervals on the horizontal axis are not uniform in size, making it appearthat dwelling swelling has been linear from 1980 through 2010. The data indicatethat this is not the case. There was a greater increase in area from 1980 through1990, averaging 34 square feet per year, than from 1990 through 2010, averaging15.6 square feet per year.

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. A sample obtained in such a way that every member of the population has an equal chance of being selected is called a/an ____________________ sample.

2. If data values are listed in one column and the adjacent column indicates the number of times each value occurs, the data presentation is called a/an____________________________.

3. If the data presentation in Exercise 2 is varied by organizing the data into classes, the data presentation is called a/an _________________________________. Ifone class in such a distribution is 80–89, the lower class limit is ____________________ and the upper class limit is ____________________.

4. Data can be displayed using a bar graph with bars that touch each other. This visual presentation of the data is called a/an ____________________. The heights ofthe bars represent the ____________________ of the data values.

5. If the midpoints of the tops of the bars for the data presentation in Exercise 4 are connected with straight lines, the resulting line graph is a data presentation calleda/an ________________________. To complete such a graph at both ends, the lines are drawn down to touch the ______________________.

6. A data presentation that separates each data item into two parts is called a/an ________________________.










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12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Exercise Set 12.1

In Exercises 7–10, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

7. A sample is the set of all the people or objects whose properties are to be described and analyzed by the data collector. ________________

8. A call-in poll on radio or television is not reliable because the sample is not chosen randomly from a larger population. ________________

9. One disadvantage of a stem-and-leaf plot is that it does not display the data items. ________________

10. A deception in the visual display of data can result by stretching or compressing the scale on a graph's vertical axis. ________________

Exercise Set 12.1Practice and Application Exercises

1. The government of a large city needs to determine whether the city's residents will support the construction of a new jail. The government decides to conduct asurvey of a sample of the city's residents. Which one of the following procedures would be most appropriate for obtaining a sample of the city's residents?

a. Survey a random sample of the employees and inmates at the old jail.

b. Survey every fifth person who walks into City Hall on a given day.

c. Survey a random sample of persons within each geographic region of the city.

d. Survey the first 200 people listed in the city's telephone directory.

2. The city council of a large city needs to know whether its residents will support the building of three new schools. The council decides to conduct a survey of asample of the city's residents. Which procedure would be most appropriate for obtaining a sample of the city's residents?

a. Survey a random sample of teachers who live in the city.

b. Survey 100 individuals who are randomly selected from a list of all people living in the state in which the city in question is located.

c. Survey a random sample of persons within each neighborhood of the city.

d. Survey every tenth person who enters City Hall on a randomly selected day.

A questionnaire was given to students in an introductory statistics class during the first week of the course. One question asked, “How stressed have you been in the last weeks, on a scale of 0 to 10, with 0 being not at all stressed and 10 being as stressed as possible?” The students' responses are shown in the frequency

distribution. Use this frequency distribution to solve Exercises 3–6.

Stress Rating Frequency0 21 12 33 124 165 186 137 318 269 1510 14

Source: Journal of Personality and Social Psychology, 69, 1102–1112

3. Which stress rating describes the greatest number of students? How many students responded with this rating?

4. Which stress rating describes the least number of students? How many responded with this rating?

5. How many students were involved in this study?

6. How many students had a stress rating of 8 or more?

7. A random sample of 30 college students is selected. Each student is asked how much time he or she spent on homework during the previous week. The followingtimes (in hours) are obtained:

16, 24, 18, 21, 18, 16, 18, 17, 15, 21, 19, 17, 17, 16, 19, 18, 15, 15, 20, 17, 15, 17, 24, 19, 16, 20, 16, 19, 18, 17.

Construct a frequency distribution for the data.

8. A random sample of 30 male college students is selected. Each student is asked his height (to the nearest inch). The heights are as follows:

72, 70, 68, 72, 71, 71, 71, 69, 73, 71, 73, 75, 66, 67, 75, 74, 73, 71, 72, 67, 72, 68, 67, 71, 73, 71, 72, 70, 73, 70.

Construct a frequency distribution for the data.

A college professor had students keep a diary of their social interactions for a week. Excluding family and work situations, the number of social interactions of tenminutes or longer over the week is shown in the following grouped frequency distribution. Use this information to solve Exercises 9–16.



21

2








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Number of Social Interactions FrequencyNumber of Social Interactions Frequency0–4 125–9 1610–14 1615–19 1620–24 1025–29 1130–34 435–39 340–44 345–49 3

Source: Society for Personality and Social Psychology

9. Identify the lower class limit for each class.

10. Identify the upper class limit for each class.










11. What is the class width?

12. How many students were involved in this study?

13. How many students had at least 30 social interactions for the week?

14. How many students had at most 14 social interactions for the week?

15. Among the classes with the greatest frequency, which class has the least number of social interactions?

16. Among the classes with the smallest frequency, which class has the least number of social interactions?

17. As of 2011, the following are the ages, in chronological order, at which U.S. presidents were inaugurated:

57, 61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46, 54, 49, 50, 47, 55, 55, 54, 42, 51, 56, 55, 51, 54, 51, 60, 62, 43, 55, 56, 61, 52, 69, 64, 46, 54, 47.

Source: Time Almanac

Construct a grouped frequency distribution for the data. Use 41–45 for the first class and use the same width for each subsequent class.

18. The IQ scores of 70 students enrolled in a liberal arts course at a college are as follows:

102, 100, 103, 86, 120, 117, 111, 101, 93, 97, 99, 95, 95, 104, 104, 105, 106, 109, 109, 89, 94, 95, 99, 99, 103, 104, 105, 109, 110, 114, 124, 123, 118, 117, 116, 110,114, 114, 96, 99, 103, 103, 104, 107, 107, 110, 111, 112, 113, 117, 115, 116, 100, 104, 102, 94, 93, 93, 96, 96, 111, 116, 107, 109, 105, 106, 97, 106, 107, 108.

Construct a grouped frequency distribution for the data. Use 85–89 for the first class and use the same width for each subsequent class.

19. Construct a histogram and a frequency polygon for the data involving stress ratings in Exercises 3–6.

20. Construct a histogram and a frequency polygon for the data in Exercise 7.

21. Construct a histogram and a frequency polygon for the data in Exercise 8.

The histogram shows the distribution of starting salaries (rounded to the nearest thousand dollars) for college graduates based on a random sample of recent graduates.

d

In Exercises 22–25, determine whether each statement is true or false according to the graph at the bottom of the previous column.

22. The graph is based on a sample of approximately 500 recent college graduates.

23. More college graduates had starting salaries in the $51,000–$55,000 range than in the $36,000–$40,000 range.

24. If the sample is truly representative, then for a group of 400 college graduates, we can expect about 28 of them to have starting salaries in the $31,000–$35,000range.

25. The percentage of starting salaries falling above those shown by any rectangular bar is equal to the percentage of starting salaries falling below that bar.

The frequency polygon shows a distribution of IQ scores.










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d

In Exercises 26–29, determine whether each statement is true or false according to the graph.

26. The graph is based on a sample of approximately 50 people.

27. More people had an IQ score of 100 than any other IQ score, and as the deviation from 100 increases or decreases, the scores fall off in a symmetrical manner.

28. More people had an IQ score of 110 than a score of 90.

29. The percentage of scores above any IQ score is equal to the percentage of scores below that score.

30. Construct a stem-and-leaf plot for the data in Exercise 17 showing the ages at which U.S. presidents were inaugurated.

31. A random sample of 40 college professors is selected from all professors at a university. The following list gives their ages:

63, 48, 42, 42, 38, 59, 41, 44, 45, 28, 54, 62, 51, 44, 63, 66, 59, 46, 51, 28, 37, 66, 42, 40, 30, 31, 48, 32, 29, 42, 63, 37, 36, 47, 25, 34, 49, 30, 35, 50.

Construct a stem-and-leaf plot for the data. What does the shape of the display reveal about the ages of the professors?










32. In “Ages of Oscar-Winning Best Actors and Actresses” (Mathematics Teacher magazine) by Richard Brown and Gretchen Davis, the stem-and-leaf plots shownbelow compare the ages of 30 actors and 30 actresses at the time they won the award.

a. What is the age of the youngest actor to win an Oscar?

b. What is the age difference between the oldest and the youngest actress to win an Oscar?

c. What is the oldest age shared by two actors to win an Oscar?

d. What differences do you observe between the two stem-and-leaf plots? What explanations can you offer for these differences?

Actors Stems Actresses 2 14666798753221 3 0011334445577888776543322100 4 111296651 5 210 6 0116 7 4 8 0

In Exercises 33–37, describe what is misleading in each visual display of data.

33.

dSource: U.S. Census Bureau

34.










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dSource: R. R. Bowker

35.

dSource: Computer Industry Almanac

36.

dSource: Nielsen Media Research












37.

dSource: Entertainment Weekly

Writing in Mathematics38. What is a population? What is a sample?

39. Describe what is meant by a random sample.

40. Suppose you are interested in whether or not the students at your college would favor a grading system in which students may receive final grades of and so on. Describe how you might obtain a random sample of 100 students from the entire student population.

41. For Exercise 40, would questioning every fifth student as he or she is leaving the campus library until 100 students are interviewed be a good way to obtain arandom sample? Explain your answer.

42. What is a frequency distribution?

43. What is a histogram?

44. What is a frequency polygon?

45. Describe how to construct a frequency polygon from a histogram.

46. Describe how to construct a stem-and-leaf plot from a set of data.

47. Describe two ways that graphs can be misleading.

Critical Thinking ExercisesMake Sense? In Exercises 48–51, determine whether each statement makes sense or does not make sense, and explain your reasoning.

48. The death rate from this new strain of flu is catastrophic because 25% of the people hospitalized with the disease have died.

49. The following graph indicates that for the period from 2000 through 2010, the percentage of female college freshmen describing their health as “above average”has rapidly decreased.

dSource: John Macionis, Sociology, Fourteenth Edition, Pearson, 2012.



A+, A, A−, B+, B, B−, C+, C, C − ,








Page 779

50. A public radio station needs to survey its contributors to determine their programming interests, so they should select a random sample of 100 of their largestcontributors.

51. Improperly worded questions can steer respondents toward answers that are not their own.

52. Construct a grouped frequency distribution for the following data, showing the length, in miles, of the 25 longest rivers in the United States. Use five classes thathave the same width.

Source: U.S. Department of the Interior

Group Exercises53. The classic book on distortion using statistics is How to Lie with Statistics by Darrell Huff. This activity is designed for five people. Each person should select twochapters from Huff's book and then present to the class the common methods of statistical manipulation and distortion that Huff discusses.

54. Each group member should find one example of a graph that presents data with integrity and one example of a graph that is misleading. Use newspapers,magazines, the Internet, books, and so forth. Once graphs have been collected, each member should share his or her graphs with the entire group. Be sure to explainwhy one graph depicts data in a forthright manner and how the other graph misleads the viewer.


2540

1460

1280

906

743

2340

1450

1240

886

724

1980

1420

1040

862

692

1900

1310

990

800

659

1900

1290

926

774

649






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1/9/2018 Thinking Mathematically, Sixth Edition12 Statistics > 12.1 Sampling, Frequency Distributions, and Graphs > Random Sampling Blitzer Bonus A Sampling Fiasco In 1936, the Literary