2.13: Practice
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 45662
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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 2.34.
Number of times in store  Frequency 

1  4 
2  10 
3  16 
4  6 
5  4 
Table 2.34
In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in Table 2.35.
Years since last purchase  Frequency 

0  2 
1  8 
2  13 
3  22 
4  16 
5  9 
Table 2.35
Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.36.
Number of TV shows  Frequency 

0  12 
1  18 
2  36 
3  7 
4  2 
Table 2.36
The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table 2.37 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 2.37
Using the data from Mrs. Ramirez’s math class supplied in Exercise 2.4, construct a bar graph showing the percentages.
David County has six high schools. Each school sent students to participate in a countywide science competition. Table 2.38 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 2.38
Use the data from the David County science competition supplied in Exercise 2.6. Construct a bar graph that shows the countywide population percentage of students at each school.
Sixtyfive 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 2.39
What does the frequency column in Table 2.39 sum to? Why?
What does the relative frequency column in Table 2.39 sum to? Why?
What is the difference between relative frequency and frequency for each data value in Table 2.39?
What is the difference between cumulative relative frequency and relative frequency for each data value?
To construct the histogram for the data in Table 2.39, 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 2.14
Construct a frequency polygon for the following:

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 2.40

Actual speed in a 30 MPH zone Frequency 42–45 25 46–49 14 50–53 7 54–57 3 58–61 1 Table 2.41

Tar (mg) in nonfiltered cigarettes Frequency 10–13 1 14–17 0 18–21 15 22–25 7 26–29 2 Table 2.42
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 2.43
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 2.44
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 2.45
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 2.46
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 2.47
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 2.48
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.
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 2.49
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 2.50
 Construct a double time series graph using a common xaxis for both sets of data.
 Which variable increased the fastest? Explain.
 Did Detroit’s increase in police officers have an impact on the murder rate? Explain.
2.3 Measures of the Location of the Data
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
 Find the 40^{th} percentile.
 Find the 78^{th} percentile.
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
 Find the percentile of 37.
 Find the percentile of 72.
Jesse was ranked 37^{th} in his graduating class of 180 students. At what percentile is Jesse’s ranking?
 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?
 The 20^{th} percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20^{th} percentile in the context of the situation.
 A bicyclist in the 90^{th} 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 90^{th} percentile in the context of the situation.
 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?
 The 40^{th} percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40^{th} percentile in the context of the situation.
On an exam, would it be more desirable to earn a grade with a high or low percentile? Explain.
Mina is waiting in line at the Department of Motor Vehicles (DMV). Her wait time of 32 minutes is the 85^{th} percentile of wait times. Is that good or bad? Write a sentence interpreting the 85^{th} percentile in the context of this situation.
In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78^{th} percentile. Should Li be pleased or upset by this result? Explain.
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 90^{th} percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90^{th} percentile in the context of this problem.
The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:
 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?
 Students whose GPAs are at or above the 96^{th} 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"?
Suppose that you are buying a house. You and your realtor have determined that the most expensive house you can afford is the 34^{th} percentile. The 34^{th} 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. Sixtyfive 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.
First quartile = _______
Second quartile = median = 50^{th} percentile = _______
Third quartile = _______
Interquartile range (IQR) = _____ – _____ = _____
10^{th} percentile = _______
70^{th} percentile = _______
2.3 Measures of the Center of the Data
Find the mean for the following frequency tables.

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 2.51

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 2.52

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 2.53
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
Calculate the mean.
Identify the median.
Identify the mode.
Use the following information to answer the next three exercises: Sixtyfive 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:
sample mean = x¯𝑥¯ = _______
median = _______
mode = _______
2.5 Sigma Notation and Calculating the Arithmetic Mean
A group of 10 children are on a scavenger hunt to find different color rocks. The results are shown in the Table 2.54 below. The column on the right shows the number of colors of rocks each child has. What is the mean number of rocks?
Child  Rock colors 
1  5 
2  5 
3  6 
4  2 
5  4 
6  3 
7  7 
8  2 
9  1 
10  10 
Table 2.54
A group of children are measured to determine the average height of the group. The results are in Table 2.55 below. What is the mean height of the group to the nearest hundredth of an inch?
Child  Height in inches 
Adam  45.21 
Betty  39.45 
Charlie  43.78 
Donna  48.76 
Earl  37.39 
Fran  39.90 
George  45.56 
Heather  46.24 
Table 2.55
A person compares prices for five automobiles. The results are in Table 2.56. What is the mean price of the cars the person has considered?
Price 
$20,987 
$22,008 
$19,998 
$23,433 
$21,444 
Table 2.56
A customer protection service has obtained 8 bags of candy that are supposed to contain 16 ounces of candy each. The candy is weighed to determine if the average weight is at least the claimed 16 ounces. The results are in given in Table 2.57. What is the mean weight of a bag of candy in the sample?
Weight in ounces 
15.65 
16.09 
16.01 
15.99 
16.02 
16.00 
15.98 
16.08 
Table 2.57
A teacher records grades for a class of 70, 72, 79, 81, 82, 82, 83, 90, and 95. What is the mean of these grades?
A family is polled to see the mean of the number of hours per day the television set is on. The results, starting with Sunday, are 6, 3, 2, 3, 1, 3, and 7 hours. What is the average number of hours the family had the television set on to the nearest whole number?
A city received the following rainfall for a recent year. What is the mean number of inches of rainfall the city received monthly, to the nearest hundredth of an inch? Use Table 2.58.
Month  Rainfall in inches 
January  2.21 
February  3.12 
March  4.11 
April  2.09 
May  0.99 
June  1.08 
July  2.99 
August  0.08 
September  0.52 
October  1.89 
November  2.00 
December  3.06 
Table 2.58
A football team scored the following points in its first 8 games of the new season. Starting at game 1 and in order the scores are 14, 14, 24, 21, 7, 0, 38, and 28. What is the mean number of points the team scored in these eight games?
2.6 Geometric Mean
What is the geometric mean of the data set given? 5, 10, 20
What is the geometric mean of the data set given? 9.000, 15.00, 21.00
What is the geometric mean of the data set given? 7.0, 10.0, 39.2
What is the geometric mean of the data set given? 17.00, 10.00, 19.00
What is the average rate of return for the values that follow? 1.0, 2.0, 1.5
What is the average rate of return for the values that follow? 0.80, 2.0, 5.0
What is the average rate of return for the values that follow? 0.90, 1.1, 1.2
What is the average rate of return for the values that follow? 4.2, 4.3, 4.5
2.7 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.
1; 1; 1; 2; 2; 2; 2; 3; 3; 3; 3; 3; 3; 3; 3; 4; 4; 4; 5; 5
16; 17; 19; 22; 22; 22; 22; 22; 23
87; 87; 87; 87; 87; 88; 89; 89; 90; 91
When the data are skewed left, what is the typical relationship between the mean and median?
When the data are symmetrical, what is the typical relationship between the mean and median?
What word describes a distribution that has two modes?
Describe the shape of this distribution.
Figure 2.15
Describe the relationship between the mode and the median of this distribution.
Figure 2.16
Describe the relationship between the mean and the median of this distribution.
Figure 2.17
Describe the shape of this distribution.
Figure 2.18
Describe the relationship between the mode and the median of this distribution.
Figure 2.19
Are the mean and the median the exact same in this distribution? Why or why not?
Figure 2.20
Describe the shape of this distribution.
Figure 2.21
Describe the relationship between the mode and the median of this distribution.
Figure 2.22
Describe the relationship between the mean and the median of this distribution.
Figure 2.23
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?
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
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
Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why?
In a perfectly symmetrical distribution, when would the mode be different from the mean and median?
2.8 Measures 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
Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.
Find the value that is one standard deviation below the mean.
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 2.59
Use Table 2.59 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.
Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.

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 2.60

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 2.61

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 2.62