For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:

• [latex]f[/latex] = frequency
• [latex]n[/latex] = total number of data values (or the sum of the individual frequencies), and
• [latex]RF[/latex] = relative frequency,

then: [latex]\text{RF}=\frac{f}{n}[/latex]

For example, if three students in Mr. Ahab's English class of 40 students received from 90% to 100%, then, [latex]f = 3[/latex], [latex]n = 40[/latex], and [latex]RF = \frac{f}{n}[/latex] = [latex]\frac{3}{40} = 0.075[/latex]. 7.5% of the students received 90–100%. 90–100% are quantitative measures.

To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995). If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

Example

The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured.

60; 60.5; 61; 61; 61.5

63.5; 63.5; 63.5

64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5

66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5

68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5

70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71

72; 72; 72; 72.5; 72.5; 73; 73.5

74

The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.

60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.

The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.

[latex]\frac{74.05-59.95}{8}=1.76[/latex]

Note: We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are:

• 59.95
• 59.95 + 2 = 61.95
• 61.95 + 2 = 63.95
• 63.95 + 2 = 65.95
• 65.95 + 2 = 67.95
• 67.95 + 2 = 69.95
• 69.95 + 2 = 71.95
• 71.95 + 2 = 73.95
• 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the x-axis and relative frequency on the y-axis.

Your Turn!

The following data are the shoe sizes of 50 male students. The sizes are continuous data since shoe size is measured. If you were to construct a histogram, how might you calculate the width of each bar or class interval. Suppose you choose six bars.

9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5

11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5

12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14

Solution

• Smallest value: 9
• Largest value: 14
• Convenient starting value: 9 – 0.05 = 8.95
• Convenient ending value: 14 + 0.05 = 14.05
• [latex]\frac{14.05-8.95}{6}=0.85[/latex]
• The calculations suggest using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one.

Example

Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data, since books are counted.

1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1

2; 2; 2; 2; 2; 2; 2; 2; 2; 2

3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3

4; 4; 4; 4; 4; 4

5; 5; 5; 5; 5

6; 6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.

Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex], the 5 in the middle of the interval from [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex], and the [latex]\underline{\hspace{2cm}}[/latex] in the middle of the interval from [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex].

• 3.5 to 4.5
• 4.5 to 5.5
• 6
• 5.5 to 6.5

Calculate the number of bars as follows:

[latex]\frac{6.5-0.5}{\mathrm{number of bars}}=1[/latex] where 1 is the width of a bar. Therefore, bars = 6.

The following histogram displays the number of books on the x-axis and the frequency on the y-axis.

Your Turn!

The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.

1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1

2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2

3; 3; 3; 3; 3; 3; 3; 3

20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.

Fill in the blanks for the following sentence.

Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to [latex]\underline{\hspace{2cm}}[/latex], the 2 in the middle of the interval from [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex], and the 3 in the middle of the interval from [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex].

Solution

Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, and the 3 in the middle of the interval from 2.5 to 3.5. [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex].

Example

Using this data set for the number of hours classmate's spent playing video games on the weekend, construct a histogram.

9.95; 10; 2.25; 16.75; 0; 19.5; 22.5; 7.5; 15; 12.75; 5.5; 11; 10; 20.75; 17.5; 23; 21.9; 24; 23.75; 18; 20; 15; 22.9; 18.8; 20.5

Solution

 

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x-axis and y-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

Example

A frequency polygon was constructed from the frequency table below.

Table 1: Frequency Distribution for Calculus Final Test Scores

Lower Bound	Upper Bound	Frequency	Cumulative Frequency
49.5	59.5	5	5
59.5	69.5	10	15
69.5	79.5	30	45
79.5	89.5	40	85
89.5	99.5	15	100

 

The first label on the x-axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x-axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Your Turn!

Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in the table below.

Table 2: Frequency Table of Ages at Inauguration

Age at Inauguration	Frequency
41.5–46.5	4
46.5–51.5	11
51.5–56.5	14
56.5–61.5	9
61.5–66.5	4
66.5–71.5	2

Solution

The first label on the x-axis is 39. This represents an interval extending from 36.5 to 41.5. Since there are no ages less than 41.5, this interval is used only to allow the graph to touch the x-axis. The point labeled 44 represents the next interval, or the first “real” interval from the table, and contains four scores. This reasoning is followed for each of the remaining intervals with the point 74 representing the interval from 71.5 to 76.5. Again, this interval contains no data and is only used so that the graph will touch the x-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

Example

We will construct an overlay frequency polygon comparing the scores from the data below with the students’ final numeric grade.

Table 3: Frequency Distribution for Calculus Final Test Scores

Lower Bound	Upper Bound	Frequency	Cumulative Frequency
49.5	59.5	5	5
59.5	69.5	10	15
69.5	79.5	30	45
79.5	89.5	40	85
89.5	99.5	15	100

Table 4: Frequency Distribution for Calculus Final Grades

Lower Bound	Upper Bound	Frequency	Cumulative Frequency
49.5	59.5	10	10
59.5	69.5	10	20
69.5	79.5	30	50
79.5	89.5	45	95
89.5	99.5	5	100

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Example

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

Table 5: Consumer Price Index, 2003-2012 for January - July

Year	Jan	Feb	Mar	Apr	May	Jun	Jul
2003	181.7	183.1	184.2	183.8	183.5	183.7	183.9
2004	185.2	186.2	187.4	188.0	189.1	189.7	189.4
2005	190.7	191.8	193.3	194.6	194.4	194.5	195.4
2006	198.3	198.7	199.8	201.5	202.5	202.9	203.5
2007	202.416	203.499	205.352	206.686	207.949	208.352	208.299
2008	211.080	211.693	213.528	214.823	216.632	218.815	219.964
2009	211.143	212.193	212.709	213.240	213.856	215.693	215.351
2010	216.687	216.741	217.631	218.009	218.178	217.965	218.011
2011	220.223	221.309	223.467	224.906	225.964	225.722	225.922
2012	226.665	227.663	229.392	230.085	229.815	229.478	229.104

Table 6: Consumer Price Index, 2003-2012 for August - December and Annual

Year	Aug	Sep	Oct	Nov	Dec	Annual
2003	184.6	185.2	185.0	184.5	184.3	184.0
2004	189.5	189.9	190.9	191.0	190.3	188.9
2005	196.4	198.8	199.2	197.6	196.8	195.3
2006	203.9	202.9	201.8	201.5	201.8	201.6
2007	207.917	208.490	208.936	210.177	210.036	207.342
2008	219.086	218.783	216.573	212.425	210.228	215.303
2009	215.834	215.969	216.177	216.330	215.949	214.537
2010	218.312	218.439	218.711	218.803	219.179	218.056
2011	226.545	226.889	226.421	226.230	225.672	224.939
2012	230.379	231.407	231.317	230.221	229.601	229.594

Solution

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

Your Turn!

The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO₂ emissions for the United States.

Table 7: CO₂ Emissions for the US

Year	Ukraine	United Kingdom	United States
2003	352,259	540,640	5,681,664
2004	343,121	540,409	5,790,761
2005	339,029	541,990	5,826,394
2006	327,797	542,045	5,737,615
2007	328,357	528,631	5,828,697
2008	323,657	522,247	5,656,839
2009	272,176	474,579	5,299,563

Solution

Videos

Below are helpful videos for the content covered in this Section. Videos are provided from YouTube.

• Dot Plots and Frequency Tables
• How to Make a Histogram Using a Frequency Distribution Table
• How to Make a Frequency Polygon
• Describing Quantitative Data Using a Histogram
• Determine the Possible Minimum and Maximum from a Histogram
• Determine the Sample Size From a Histogram

Section 2.2 Review

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on the y-axis with the frequency being graphed on the x-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

Section 2.2 Practice

1. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table below
2. What does the frequency column in the table sum to? Why?
3. What does the relative frequency column in the table sum to? Why?
4. What is the difference between relative frequency and frequency for each data value in the table?
5. What is the difference between cumulative relative frequency and relative frequency for each data value?
6. To construct the histogram for the data in the table, determine appropriate minimum and maximum [latex]x[/latex] and [latex]y[/latex] values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling.

Table 8: Blank Frequency Table for Question 1

Data Value (# cars)	Frequency	Relative Frequency	Cumulative Relative Frequency

Solution

The completed table is:

Table 9: Completed Frequency Table for Question 1

Data Value (# cars)	Frequency	Relative Frequency	Cumulative Relative Frequency
3	14	0.22	0.22
4	19	0.29	0.51
5	12	0.18	0.69
6	9	0.14	0.83
7	11	0.17	1

The frequency tells the number of data points that have each value. The frequency column sums to 65, because that is the total number of data values (the randomly selected car salespersons). The relative frequency shows the proportion of data points that have each value. The total of the relative frequency column should be 1. The cumulative relative frequency shows the sum of the relative frequency in that row and all previous relative frequencies.

Answers will vary for the histogram. One possible histogram is shown:

Construct a frequency polygon for the following:

1. Pulse rates for 40 randomly selected women. Table below.
2. Actual speed of 50 cars travelling in a 30 MPH speed zone. Table below.
3. Amount of tar in mg of 25 nonfiltered cigarettes. Table below.
   

   Table 10: Frequency Table of Pulse Rates for 40 Women

   Pulse Rates for Women	Frequency
   60–69	12
   70–79	14
   80–89	11
   90–99	1
   100–109	1
   110–119	0
   120–129	1

   Table 11: Frequency Table of Actual Speed in 30 MPH Zone

   Actual Speed in a 30 MPH Zone	Frequency
   42–45	25
   46–49	14
   50–53	7
   54–57	3
   58–61	1

   Table 12: Frequency Table of Tar in Nonfiltered Cigarettes

   Tar (mg) in Nonfiltered Cigarettes	Frequency
   10–13	1
   14–17	0
   18–21	15
   22–25	7
   26–29	2

Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger.

Tabel 13: Frequency Table of Depth of Hunger for Countries

Depth of Hunger	Frequency
230–259	21
260–289	13
290–319	5
320–349	7
350–379	1
380–409	1
410–439	1

Solution

Find the midpoint for each class. These will be graphed on the x-axis. The frequency values will be graphed on the y-axis values.

Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlaid frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers.

What can we conclude about the life expectancy of women compared to men?

Table 14: Frequency Table of Life Expectancy at Birth – Women

Life Expectancy at Birth – Women	Frequency
49–55	3
56–62	3
63–69	1
70–76	3
77–83	8
84–90	2

Table 15: Frequency Table of Life Expectancy at Birth – Men

Life Expectancy at Birth – Men	Frequency
49–55	3
56–62	3
63–69	1
70–76	1
77–83	7
84–90	5

Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births.

Table 16: Years 1855-1861, Female, Male, and Total Number of Births

Year	1855	1856	1857	1858	1859	1860	1861
Female	45,545	49,582	50,257	50,324	51,915	51,220	52,403
Male	47,804	52,239	53,158	53,694	54,628	54,409	54,606
Total	93,349	101,821	103,415	104,018	106,543	105,629	107,009

Table 17: Years 1862-1869, Female, Male, and Total Number of Births

Year	1862	1863	1864	1865	1866	1867	1868	1869
Female	51,812	53,115	54,959	54,850	55,307	55,527	56,292	55,033
Male	55,257	56,226	57,374	58,220	58,360	58,517	59,222	58,321
Total	107,069	109,341	112,333	113,070	113,667	114,044	115,514	113,354

Table 18: Years 1871-1875, Female, Male, and Total Number of Births

Year	1871	1870	1872	1871	1872	1827	1874	1875
Female	56,099	56,431	57,472	56,099	57,472	58,233	60,109	60,146
Male	60,029	58,959	61,293	60,029	61,293	61,467	63,602	63,432
Total	116,128	115,390	118,765	116,128	118,765	119,700	123,711	123,578

 

Solution

The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for the city of Detroit, Michigan during the period from 1961 to 1973.

Table 19: Years 1961-1967, Police and Homicides Count

Year	1961	1962	1963	1964	1965	1966	1967
Police	260.35	269.8	272.04	272.96	272.51	261.34	268.89
Homicides	8.6	8.9	8.52	8.89	13.07	14.57	21.36

Table 20: Years 1968-1973, Police and Homicides Count

Year	1968	1969	1970	1971	1972	1973
Police	295.99	319.87	341.43	356.59	376.69	390.19
Homicides	28.03	31.49	37.39	46.26	47.24	52.33

1. Construct a double time series graph using a common x-axis for both sets of data.
2. Which variable increased the fastest? Explain.
3. Did Detroit’s increase in police officers have an impact on the murder rate? Explain.

Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.

Table 21: Single Travelers Incomplete Frequency Table

Amount($)	Frequency	Rel. Frequency
51–100	5
101–150	10
151–200	15
201–250	15
251–300	10
301–350	5

Table 22: Couple Travelers Incomplete Frequency Table

Amount($)	Frequency	Rel. Frequency
100–150	5
201–250	5
251–300	5
301–350	5
351–400	10
401–450	10
451–500	10
501–550	10
551–600	5
601–650	5

1. Fill in the relative frequency for each group.
2. Construct a histogram for the singles group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.
3. Construct a histogram for the couples group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.
4. Compare the two graphs:
   • List two similarities between the graphs.
   • List two differences between the graphs.
   • Overall, are the graphs more similar or different?
5. Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x-axis by $50, scale it by $100. Use relative frequency on the y-axis.
6. Compare the graph for the singles with the new graph for the couples:
   • List two similarities between the graphs.
   • Overall, are the graphs more similar or different?
7. How did scaling the couples graph differently change the way you compared it to the singles graph?
8. Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences.

Solution

Table 23: Singles Travelers Complete Frequency Table

Amount($)	Frequency	Relative Frequency
51–100	5	0.08
101–150	10	0.17
151–200	15	0.25
201–250	15	0.25
251–300	10	0.17
301–350	5	0.08

Table 24: Couples Travelers Complete Frequency Table

Amount($)	Frequency	Relative Frequency
100–150	5	0.07
201–250	5	0.07
251–300	5	0.07
301–350	5	0.07
351–400	10	0.14
401–450	10	0.14
451–500	10	0.14
501–550	10	0.14
551–600	5	0.07
601–650	5	0.07

1. See the Singles and Couples tables above.
2. The histogram is shown below. Note, data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where both boundary values are included).
3. The histogram is shown below. Note, the data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where values on both boundaries are included).
4. Compare the two graphs:
   • List two similarities between the graphs.
     • Answers may vary. Possible answers include:
       • Both graphs have a single peak.
       • Both graphs use class intervals with width equal to $50.
   • List two differences between the graphs.
     • Answers may vary. Possible answers include:
       • The couples graph has a class interval with no values.
       • It takes almost twice as many class intervals to display the data for couples.
   • Overall, are the graphs more similar or different?
     • Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same.
5. Check student's solution.
6. Compare the graph for the Singles with the new graph for the Couples:
   • List two similarities between the graphs.
     • Answers may vary. Possible answers include:
       • Both graphs have a single peak.
       • Both graphs display 6 class intervals.
       • Both graphs show the same general pattern.
   • Overall, are the graphs more similar or different?
     • Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
7. Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
8. Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

 

*Note: The title of the image is incorrect. The image cannot be fixed.

Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows.

Table 25: Incomplete Frequency Table of Number of Movies Watched

# of movies	Frequency	Relative Frequency	Cumulative Relative Frequency
0	5
1	9
2	6
3	4
4	1

1. Construct a histogram of the data.
2. Complete the columns of the chart.

Use the following information to answer the next two exercises.

Suppose one hundred eleven people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each.

1. The percentage of people who own at most three T-shirts costing more than $19 each is approximately:

1. • 21
   • 59
   • 41
   • Cannot be determined
2. If the data were collected by asking the first 111 people who entered the store, then the type of sampling is:
   • cluster
   • simple random
   • stratified
   • convenience

Solution

1. 41
2. cluster

Following are the 2010 obesity rates by U.S. states and Washington, DC.

Table 26: US States and Percent of Obesity Rates

State	Percent (%)	State	Percent (%)	State	Percent (%)
Alabama	32.2	Kentucky	31.3	North Dakota	27.2
Alaska	24.5	Louisiana	31.0	Ohio	29.2
Arizona	24.3	Maine	26.8	Oklahoma	30.4
Arkansas	30.1	Maryland	27.1	Oregon	26.8
California	24.0	Massachusetts	23.0	Pennsylvania	28.6
Colorado	21.0	Michigan	30.9	Rhode Island	25.5
Connecticut	22.5	Minnesota	24.8	South Carolina	31.5
Delaware	28.0	Mississippi	34.0	South Dakota	27.3
Washington, DC	22.2	Missouri	30.5	Tennessee	30.8
Florida	26.6	Montana	23.0	Texas	31.0
Georgia	29.6	Nebraska	26.9	Utah	22.5
Hawaii	22.7	Nevada	22.4	Vermont	23.2
Idaho	26.5	New Hampshire	25.0	Virginia	26.0
Illinois	28.2	New Jersey	23.8	Washington	25.5
Indiana	29.6	New Mexico	25.1	West Virginia	32.5
Iowa	28.4	New York	23.9	Wisconsin	26.3
Kansas	29.4	North Carolina	27.8	Wyoming	25.1

Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint: Label the x-axis with the states.

Solution

Answers will vary.

References

Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker

“Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013).

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“Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).

“Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).

“CO₂ emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).

“Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html (accessed April 3, 2013).

“Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).

Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach. CRC Press: 1980.

“Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).