2.1 Display Data
For the next three exercises, use the data to construct a line graph.
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 2.37.
| Number of times in store | Frequency |
|---|---|
| 1 | 4 |
| 2 | 10 |
| 3 | 16 |
| 4 | 6 |
| 5 | 4 |
In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in Table 2.38.
| Years since last purchase | Frequency |
|---|---|
| 0 | 2 |
| 1 | 8 |
| 2 | 13 |
| 3 | 22 |
| 4 | 16 |
| 5 | 9 |
Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.39.
| Number of TV shows | Frequency |
|---|---|
| 0 | 12 |
| 1 | 18 |
| 2 | 36 |
| 3 | 7 |
| 4 | 2 |
The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table 2.40 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% |
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 county-wide science competition. Table 2.41 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% |
Use the data from the David County science competition supplied in Exercise 2.6. Construct a bar graph that shows the county-wide population percentage of students at each school.
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 |
|---|---|---|---|
What is the difference between relative frequency and frequency for each data value in Table 2.42?
What is the difference between cumulative relative frequency and relative frequency for each data value?
To construct the histogram for the data in Table 2.42, 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.
Construct a frequency polygon for the following:
Pulse rates for Females Frequency 60–69 12 70–79 14 80–89 11 90–99 1 100–109 1 110–119 0 120–129 1 Table 2.43Actual speed in a 30 MPH zone Frequency 42–45 25 46–49 14 50–53 7 54–57 3 58–61 1 Table 2.44Tar (mg) in nonfiltered cigarettes Frequency 10–13 1 14–17 0 18–21 15 22–25 7 26–29 2 Table 2.45
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 |
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 |
|---|---|
| 49–55 | 3 |
| 56–62 | 3 |
| 63–69 | 1 |
| 70–76 | 3 |
| 77–83 | 8 |
| 84–90 | 2 |
| Life expectancy at birth – Males | 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.
| 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 |
| 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 |
| 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 |
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 |
| 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 |
- Construct a double time series graph using a common x-axis for both sets of data.
- Which variable increased the fastest? Explain.
- Did the city's increase in police officers have an impact on the murder rate? Explain.
2.2 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 40th percentile.
- Find the 78th 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 37th 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 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.
- 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.
- 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 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.
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 85th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85th 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 78th 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 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.
The University of Wisconsin has two criteria used to set admission standards for students to be admitted to a college in the UW 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 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"?
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.
First quartile = _______
Third quartile = _______
10th 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 2.54Daily 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 2.55Points 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 2.56
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
Identify the median.
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:
sample mean = = _______
mode = _______
2.4 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 2.57 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 |
A group of children are measured to determine the average height of the group. The results are in Table 2.58 below. What is the mean height of the group to the nearest hundredth of an inch?
| Child | Height in inches |
| Adam | 45.21 |
| Betina | 39.45 |
| Chen | 43.78 |
| Donna | 48.76 |
| Edhas | 37.39 |
| Fran | 39.90 |
| George | 45.56 |
| Heather | 46.24 |
A person compares prices for five automobiles. The results are in Table 2.59. What is the mean price of the cars the person has considered?
| Price |
| $20,987 |
| $22,008 |
| $19,998 |
| $23,433 |
| $21,444 |
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 2.60. 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 |
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 2.61.
| 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 |
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?
2.5 Geometric Mean
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.
16; 17; 19; 22; 22; 22; 22; 22; 23
When the data are skewed left, what is the typical relationship between the mean and median?
What word describes a distribution that has two modes?
Describe the relationship between the mode and the median of this distribution.
Describe the shape of this distribution.
Are the mean and the median the exact same in this distribution? Why or why not?
Describe the relationship between the mode and the median of this distribution.
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?
2.7 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 |
Use Table 2.62 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 2.63Daily 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 2.64Points 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 2.65