Introductory Statistics 2e

# Practice

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

For each of the following data sets, create a stem plot 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 ten.
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 2.38.

Number of times in storeFrequency
14
210
316
46
54
Table 2.38
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 2.39.

Years since last purchase Frequency
0 2
1 8
2 13
3 22
4 16
5 9
Table 2.39
7.

Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.40.

Number of TV Shows Frequency
0 12
1 18
2 36
3 7
4 2
Table 2.40
8.

The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table 2.41 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 2.41
9.

Using the data from Ms. Ramirez’s math class supplied in Table 2.41, 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 2.42 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
Alabaster28.9%8.6%
Concordia7.6%23.2%
Genoa12.1%15.0%
Mocksville18.5%14.3%
Tynneson24.2%10.1%
West End8.7%28.8%
Table 2.42
11.

Use the data from the David County science competition supplied in Table 2.42. Construct a bar graph that shows the county-wide population percentage of students at each school.

## 2.2Histograms, 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; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table.

Data Value (# cars) Frequency Relative Frequency Cumulative Relative Frequency
Table 2.43
13.

What does the frequency column in Table 2.43 sum to? Why?

14.

What does the relative frequency column in Table 2.43 sum to? Why?

15.

What is the difference between relative frequency and frequency for each data value in Table 2.43?

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 2.43, 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.

Figure 2.31
18.

Construct a frequency polygon for the following:

1. Pulse Rates for Females Frequency
60–6912
70–7914
80–8911
90–991
100–1091
110–1190
120–1291
Table 2.44
2. Actual Speed in a 30 MPH Zone Frequency
42–4525
46–4914
50–537
54–573
58–611
Table 2.45
3. Tar (mg) in Nonfiltered Cigarettes Frequency
10–131
14–170
18–2115
22–257
26–292
Table 2.46
19.

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

Depth of Hunger Frequency
230–25921
260–28913
290–3195
320–3497
350–3791
380–4091
410–4391
Table 2.47
20.

Use the two frequency tables to compare the life expectancy of males and females 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 females compared to males?

Life Expectancy at Birth – Females Frequency
49–553
56–623
63–691
70–763
77–838
84–902
Table 2.48
Life Expectancy at Birth – Males Frequency
49–553
56–623
63–691
70–761
77–837
84–905
Table 2.49
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 2.50
 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 2.51
 Sex/Year 1870 1871 1872 1873 1874 1875 Female 56,431 56,099 57,472 58,233 60,109 60,146 Male 58,959 60,029 61,293 61,467 63,602 63,432 Total 115,390 116,128 118,765 119,700 123,711 123,578
Table 2.52
22.

The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for a city 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 Homicides 8.6 8.9 8.52 8.89 13.07 14.57 21.36
Table 2.53
 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
Table 2.54
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 the city's increase in police officers have an impact on the murder rate? Explain.

## 2.3Measures 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 low percentile? Explain.

29.

Mina is waiting in line at the Department of Motor Vehicles (DMV). 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 Wisconsin has two criteria used to set admission standards for students to be admitted to a college in the UW 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% of high school students in the state. In this context, what percentile does the top 12% 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% 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 realtor 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% of the houses or 66% of the houses?

Use the following information to answer the next six 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; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.

34.

First quartile = _______

35.

Second quartile = median = 50th percentile = _______

36.

Third quartile = _______

37.

Interquartile range (IQR) = _____ – _____ = _____

38.

10th percentile = _______

39.

70th percentile = _______

## 2.4Box Plots

Use the following information to answer the next two 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; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven 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?

## 2.5Measures of the Center of the Data

42.

Find the mean for the following frequency tables.

49.5–59.52
59.5–69.53
69.5–79.58
79.5–89.512
89.5–99.55
Table 2.55
2. Daily Low Temperature Frequency
49.5–59.553
59.5–69.532
69.5–79.515
79.5–89.51
89.5–99.50
Table 2.56
3. Points per Game Frequency
49.5–59.514
59.5–69.532
69.5–79.515
79.5–89.523
89.5–99.52
Table 2.57

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; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Calculate the following:

46.

sample mean = $x¯x¯$ = _______

47.

median = _______

48.

mode = _______

## 2.6Skewness 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.

Figure 2.32
56.

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

Figure 2.33
57.

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

Figure 2.34
58.

Describe the shape of this distribution.

Figure 2.35
59.

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

Figure 2.36
60.

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

Figure 2.37
61.

Describe the shape of this distribution.

Figure 2.38
62.

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

Figure 2.39
63.

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

Figure 2.40
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?

## 2.7Measures of the Spread of the Data

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

69.

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

70.

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

71.

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 0.158 0.166 0.012
Karl 0.177 0.189 0.015
Table 2.58
72.

Use Table 2.58 to find the value that is three standard deviations:

• above the mean
• below the mean

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

73.

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

49.5–59.52
59.5–69.53
69.5–79.58
79.5–89.512
89.5–99.55
Table 2.59
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 2.60
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 2.61