Practice

2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

For each of the following data sets, create a stemplot and identify any outliers.

1.

The miles per gallon rating for 30 cars are shown below (lowest to highest).


 
19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43
2.

The height in feet of 25 trees is shown below (lowest to highest).


 
25, 27, 33, 34, 34, 34, 35, 37, 37, 38, 39, 39, 39, 40, 41, 45, 46, 47, 49, 50, 50, 53, 53, 54, 54
3.

The data are the prices of different laptops at an electronics store. Round each value to the nearest 10.


 
249, 249, 260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610
4.

The data are daily high temperatures in a town for one month.


 
61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95

 


 
For the next three exercises, use the data to construct a line graph.
5.

In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in Table 2.40.

Number of Times in Store Frequency
1 4
2 10
3 16
4 6
5 4
Table 2.40
6.

In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in Table 2.41.

Years Since Last Purchase Frequency
0 2
1 8
2 13
3 22
4 16
5 9
Table 2.41
7.

Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.42.

Number of TV Shows Frequency
0 12
1 18
2 36
3 7
4 2
Table 2.42
8.

The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table 2.43 shows the four seasons, the number of students who have birthdays in each season, and the percentage of students in each group. Construct a bar graph showing the number of students.

Seasons Number of Students Proportion of Population
Spring 8 24%
Summer 9 26%
Autumn 11 32%
Winter 6 18%
Table 2.43
9.

Using the data from Mrs. Ramirez’s math class supplied in Exercise 2.8, construct a bar graph showing the percentages.

10.

David County has six high schools. Each school sent students to participate in a county-wide science competition. Table 2.44 shows the percentage breakdown of competitors from each school and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school.

High School Science Competition Population Overall Student Population
Alabaster 28.9% 8.6%
Concordia 7.6% 23.2%
Genoa 12.1% 15.0%
Mocksville 18.5% 14.3%
Tynneson 24.2% 10.1%
West End 8.7% 28.8%
Table 2.44
11.

Use the data from the David County science competition supplied in Exercise 2.10. Construct a bar graph that shows the county-wide population percentage of students at each school.

2.2 Histograms, Frequency Polygons, and Time Series Graphs

12.

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, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Complete the table.

Data Value (Number of Cars) Frequency Relative Frequency Cumulative Relative Frequency
       
       
       
       
       
 
Table 2.45
13.

What does the frequency column in Table 2.45 sum to? Why?

14.

What does the relative frequency column in Table 2.45 sum to? Why?

15.

What is the difference between relative frequency and frequency for each data value in Table 2.45?

16.

What is the difference between cumulative relative frequency and relative frequency for each data value?

17.

To construct the histogram for the data in Table 2.45, determine appropriate minimum and maximum x- and y-values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling.

An empty graph template for use with this question.
Figure 2.33
18.

Construct a frequency polygon for the following.

  1. 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 2.46
  2. Actual Speed in a 30-MPH Zone Frequency
    42–45 25
    46–49 14
    50–53 7
    54–57 3
    58–61 1
    Table 2.47
  3. Tar (mg) in Nonfiltered Cigarettes Frequency
    10–13 1
    14–17 0
    18–21 15
    22–25 7
    26–29 2
    Table 2.48
19.

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

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
Table 2.49
20.

Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlayed 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?

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 2.50
Life Expectancy at Birth – Men Frequency
49–55 3
56–62 3
63–69 1
70–76 1
77–83 7
84–90 5
Table 2.51
21.

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.

Sex/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 2.52
Sex/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 2.53
Sex/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
Table 2.54
22.

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

Year 1961 1962 1963 1964 1965 1966 1967
Police 260.35 269.8 272.04 272.96 272.51 261.34 268.89
Incidents 8.6 8.9 8.52 8.89 13.07 14.57 21.36
Table 2.55
Year 1968 1969 1970 1971 1972 1973
Police 295.99 319.87 341.43 356.59 376.69 390.19
Incidents 28.03 31.49 37.39 46.26 47.24 52.33
Table 2.56
  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 incident rate? Explain.

2.3 Measures of the Location of the Data

23.

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.

18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

  1. Find the 40th percentile.
  2. Find the 78th percentile.
24.

Listed are 32 ages for Academy Award-winning best actors in order from smallest to largest.

18, 18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

  1. Find the percentile of 37.
  2. Find the percentile of 72.
25.

Jesse was ranked 37th in his graduating class of 180 students. At what percentile is Jesse’s ranking?

26.
  1. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race?
  2. The 20th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20th percentile in the context of the situation.
  3. A bicyclist in the 90th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90th percentile in the context of the situation.
27.
  1. For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race?
  2. The 40th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40th percentile in the context of the situation.
28.

On an exam, would it be more desirable to earn a grade with a high or a low percentile? Explain.

29.

Mina is waiting in line at the Department of Motor Vehicles. Her wait time of 32 minutes is the 85th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85th percentile in the context of this situation.

30.

In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78th percentile. Should Li be pleased or upset by this result? Explain.

31.

In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90th percentile in the context of this problem.

32.

The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:

  1. Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an admissions index score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12 percent of high school students in the state. In this context, what percentile does the top 12 percent represent?
  2. Students whose GPAs are at or above the 96th percentile of all students at their high school are eligible, called eligible in the local context, even if they are not in the top 12 percent of all students in the state. What percentage of students from each high school are eligible in the local context?
33.

Suppose that you are buying a house. You and your real estate agent have determined that the most expensive house you can afford is the 34th percentile. The 34th percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34 percent of the houses or 66 percent of the houses?

Use Exercise 2.25 to calculate the following values.

34.

First quartile = ________

35.

Second quartile = median = 50th percentile = ________

36.

Third quartile = ________

37.

Interquartile range (IQR) = ________ – ________ = ________

38.

10th percentile = ________

39.

70th percentile = ________

2.4 Box Plots

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, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars.

40.

Construct a box plot below. Use a ruler to measure and scale accurately.

41.

Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas but not in others? How can you tell?

2.5 Measures of the Center of the Data

42.

Find the mean for the following frequency tables.

  1. Grade Frequency
    49.5–59.5 2
    59.5–69.5 3
    69.5–79.5 8
    79.5–89.5 12
    89.5–99.5 5
    Table 2.57
  2. Daily Low Temperature Frequency
    49.5–59.5 53
    59.5–69.5 32
    69.5–79.5 15
    79.5–89.5 1
    89.5–99.5 0
    Table 2.58
  3. Points per Game Frequency
    49.5–59.5 14
    59.5–69.5 32
    69.5–79.5 15
    79.5–89.5 23
    89.5–99.5 2
    Table 2.59

Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest. 16, 17, 19, 20, 20, 21, 23, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 29, 30, 32, 33, 33, 34, 35, 37, 39, 40

43.

Calculate the mean.

44.

Identify the median.

45.

Identify the mode.

 


 
Use the following information to answer the next three exercises: 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, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Calculate the following.
46.

sample mean = x¯x = ________

47.

median = ________

48.

mode = ________

2.6 Skewness and the Mean, Median, and Mode

Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right.

49.

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5

50.

16, 17, 19, 22, 22, 22, 22, 22, 23

51.

87, 87, 87, 87, 87, 88, 89, 89, 90, 91

52.

When the data are skewed left, what is the typical relationship between the mean and median?

53.

When the data are symmetrical, what is the typical relationship between the mean and median?

54.

What word describes a distribution that has two modes?

55.

Describe the shape of this distribution.

This is a historgram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak at the first bar and taper lower to the right.
Figure 2.34
56.

Describe the relationship between the mode and the median of this distribution.

This is a histogram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak at the first bar and taper lower to the right. The bar ehighs from left to right are: 8, 4, 2, 2, 1.
Figure 2.35
57.

Describe the relationship between the mean and the median of this distribution.

This is a histogram which  consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak at the first bar and taper lower to the right. The bar heights from left to right are: 8, 4, 2, 2, 1.
Figure 2.36
58.

Describe the shape of this distribution.

This is a histogram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak in the middle and taper down to the right and left.
Figure 2.37
59.

Describe the relationship between the mode and the median of this distribution.

This is a histogram which consists of 5 adjacent bars with the x-axis split intervals of 1 from 3 to 7. The bar heights peak in the middle and taper down to the right and left.
Figure 2.38
60.

Are the mean and the median the exact same in this distribution? Why or why not?

This is a histogram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 2, 4, 8, 5, 2.
Figure 2.39
61.

Describe the shape of this distribution.

This is a histogram which consists of 5 adjacent bars over an x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 1, 1, 2, 4, 7.
Figure 2.40
62.

Describe the relationship between the mode and the median of this distribution.

This is a histogram which consists of 5 adjacent bars over an x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 1, 1, 2, 4, 7.
Figure 2.41
63.

Describe the relationship between the mean and the median of this distribution.

This is a histogram which consists of 5 adjacent bars over an x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 1, 1, 2, 4, 7.
Figure 2.42
64.

The mean and median for the data are the same.

3, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7

Is the data perfectly symmetrical? Why or why not?

65.

Which is the greatest, the mean, the mode, or the median of the data set?

11, 11, 12, 12, 12, 12, 13, 15, 17, 22, 22, 22

66.

Which is the least, the mean, the mode, and the median of the data set?

56, 56, 56, 58, 59, 60, 62, 64, 64, 65, 67

67.

Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why?

68.

In a perfectly symmetrical distribution, when would the mode be different from the mean and median?

2.7 Measures of the Spread of the Data

For each of the examples given below, tell whether the differences in outcomes may be explained by measurement variability, natural variability, induced variability, or sampling variability.

69.

Scientists randomly select five groups of 10 women from a population of 1,000 women to record their body fat percentage. The scientists compute the mean body fat percentage from each group. The differences in outcomes may be attributed to which type of variability?

70.

A pharmaceutical company randomly assigns participants to one of two groups: One is a control group receiving a placebo, and another is a treatment group receiving a new drug to lower blood pressure. The differences in outcomes may be attributed to which type of variability?

71.

Jaiqua and Harold are trying to determine how ramp steepness affects the speed of a ball rolling down the ramp. They measure the time it takes for the ball to roll down ramps of differing slopes. When Jaiqua rolls the ball and Harold works the stopwatch, they get different results than when Harold rolls the ball and Jaiqua works the stopwatch. The differences in outcomes may be attributed to which type of variability?

72.

Twenty people begin the same workout program on the same day and continue for three months. During that time, all participants worked out for the same amount of time and did the same number of exercises and repetitions. Each person was weighed at both the beginning and the end of the program. The differences in outcomes regarding the amount of weight lost may be attributed to which type of variability?

Use the following information to answer the next two exercises: The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.


 
29, 37, 38, 40, 58, 67, 68, 69, 76, 86, 87, 95, 96, 96, 99, 106, 112, 127, 145, 150
73.

Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.

74.

Find the value that is one standard deviation below the mean.

75.

Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team. Which baseball player had the higher batting average when compared to his team?

Baseball Player Batting Average Team Batting Average Team Standard Deviation
Fredo .158 .166 .012
Karl .177 .189 .015
Table 2.60
76.

Use Table 2.60 to find the value that is three standard deviations

  • above the mean, and
  • below the mean.

 


 
77.

Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.

  1. Grade Frequency
    49.5–59.5 2
    59.5–69.5 3
    69.5–79.5 8
    79.5–89.5 12
    89.5–99.5 5
    Table 2.61
  2. Daily Low Temperature Frequency
    49.5–59.5 53
    59.5–69.5 32
    69.5–79.5 15
    79.5–89.5 1
    89.5–99.5 0
    Table 2.62
  3. Points per Game Frequency
    49.5–59.5 14
    59.5–69.5 32
    69.5–79.5 15
    79.5–89.5 23
    89.5–99.5 2
    Table 2.63