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Chpater 2 Practice

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    5336
  • 2.1 Display Data


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

    1.

    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

    Table2.34

    2.

    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

    Table2.35

    3.

    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

    Table2.36

    4.

    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%

    Table2.37

    5.

    Using the data from Mrs. Ramirez’s math class supplied in Exercise 2.4, construct a bar graph showing the percentages.

    6.

    David County has six high schools. Each school sent students to participate in a county-wide 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%

    Table2.38

    7.

    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.

    8.

    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

    Table2.39

    9.

    What does the frequency column in Table 2.39 sum to? Why?

    10.

    What does the relative frequency column in Table 2.39 sum to? Why?

    11.

    What is the difference between relative frequency and frequency for each data value in Table 2.39?

    12.

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

    13.

    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.

    An empty graph template for use with this question.

    Figure 2.14

    14.

    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

      Table2.40

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

      Table2.41

    3. Tar (mg) in nonfiltered cigarettes Frequency
      10–13 1
      14–17 0
      18–21 15
      22–25 7
      26–29 2

      Table2.42

    15.

    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

    Table2.43

    16.

    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

    Table2.44

    Life expectancy at birth – men Frequency
    49–55 3
    56–62 3
    63–69 1
    70–76 1
    77–83 7
    84–90 5

    Table2.45

    17.

    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

    Table2.46

    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

    Table2.47

    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

    Table2.48

    18.

    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

    Table2.49

    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

    Table2.50

    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.

    2.2 Measures of the Location of the Data

    19.

    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.
    20.

    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.
    21.

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

    22.
    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.
    23.
    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.
    24.

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

    25.

    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.

    26.

    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.

    27.

    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.

    28.

    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% 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"?
    29.

    Suppose that you are buying a house. You and your realtor have determined that the most expensive house you can afford is the 34thpercentile. 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.

    30.

    First quartile = _______

    31.

    Second quartile = median = 50th percentile = _______

    32.

    Third quartile = _______

    33.

    Interquartile range (IQR) = _____ – _____ = _____

    34.

    10th percentile = _______

    35.

    70th percentile = _______

    2.3 Measures of the Center of the Data

    36.

    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

      Table2.51

    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

      Table2.52

    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

      Table2.53

    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

    37.

    Calculate the mean.

    38.

    Identify the median.

    39.

    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:

    40.

    sample mean = x¯x = _______

    41.

    median = _______

    42.

    mode = _______

    2.4 Sigma Notation and Calculating the Arithmetic Mean

    43.

    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

    Table2.54

    44.

    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

    Table2.55

    45.

    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

    Table2.56

    46.

    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

    Table2.57

    47.

    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?

    48.

    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?

    49.

    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

    Table2.58

    50.

    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

    51.

    What is the geometric mean of the data set given? 5, 10, 20

    52.

    What is the geometric mean of the data set given? 9.000, 15.00, 21.00

    53.

    What is the geometric mean of the data set given? 7.0, 10.0, 39.2

    54.

    What is the geometric mean of the data set given? 17.00, 10.00, 19.00

    55.

    What is the average rate of return for the values that follow? 1.0, 2.0, 1.5

    56.

    What is the average rate of return for the values that follow? 0.80, 2.0, 5.0

    57.

    What is the average rate of return for the values that follow? 0.90, 1.1, 1.2

    58.

    What is the average rate of return for the values that follow? 4.2, 4.3, 4.5

    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.

    59.

    1; 1; 1; 2; 2; 2; 2; 3; 3; 3; 3; 3; 3; 3; 3; 4; 4; 4; 5; 5

    60.

    16; 17; 19; 22; 22; 22; 22; 22; 23

    61.

    87; 87; 87; 87; 87; 88; 89; 89; 90; 91

    62.

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

    63.

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

    64.

    What word describes a distribution that has two modes?

    65.

    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 2.15

    66.

    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 2.16

    67.

    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 2.17

    68.

    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 2.18

    69.

    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 2.19

    70.

    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 2.20

    71.

    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 2.21

    72.

    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 2.22

    73.

    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 2.23

    74.

    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?

    75.

    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

    76.

    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

    77.

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

    78.

    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

    79.

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

    80.

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

    81.

    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

    Table2.59

    82.

    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.

    83.

    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

      Table2.60

    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

      Table2.61

    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

      Table2.62