3.7: Practice (Chapter 3)
- Page ID
- 59116
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)3.1: Bar Charts and Pie Charts (Categorical Data)
- What type of data is best represented with a bar chart? What about a pie chart?
- Draw a bar chart to represent the following class survey: Cats (10), Dogs (8), Fish (4), Other (3)
- Explain a limitation of pie charts when many categories are included.
- Can a bar chart be used with relative frequencies instead of counts? Explain.
- True or False: “There should be no space between bars in a pie chart.”
- Choose a categorical variable from your dataset. Would a pie chart or bar chart be more appropriate? Why?
- The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. The following table 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 chart 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 Ms. Ramirez’s math class supplied in the table above, construct a pie chart showing the percentages.
- The bar plot and the pie chart below show the distribution of pre-existing medical conditions of children involved in a study on the optimal duration of antibiotic use in treatment of tracheitis, which is an upper respiratory infection.
- What features are apparent in the bar plot but not in the pie chart?
- What features are apparent in the pie chart but not in the bar plot?
- Which graph would you prefer to use for displaying these categorical data?
3.2: Histograms, Dot Plots, and Stem Plots (Quantitative Data)
- Draw a histogram for the values: {2, 2, 3, 5, 5, 5, 6, 7, 8, 10}
- What is the key difference between a bar chart and a histogram?
- What makes dot plots difficult to use for large sets of numerical data?
- Construct a stem-and-leaf plot for: {34, 36, 37, 41, 42, 42, 43, 46}
- What does the height of a bar in a histogram represent?
- Choose a quantitative variable from your dataset. Would you rather display it with a histogram, dot plot, or stem plot? Why?
- Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed 100 shoppers. The samples yielded the following information.
Data table of distance shoppers live from mall. Javier Ercilia \(\bar{x}\) 6.0 miles 6.0 miles s 4.0 miles 7.0 miles - How can you determine which survey was correct ?
- Explain what the difference in the results of the surveys implies about the data.
- If the two histograms depict the distribution of values for each supervisor, which one depicts Ercilia's sample? How do you know?
- Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X = the number of pairs of sneakers owned. Use the data in the table to create a stem plot and find the sample mean \(\bar{x}\)
Data table of number of shoes students own. X Frequency 1 2 2 5 3 8 4 12 5 12 6 0 7 1 - Describe one advantage of a stem plot over a frequency histogram.
3.3: Frequency Tables and Relative Frequency
- Explain the difference between frequency and relative frequency.
- Fill in the missing relative frequencies for the following table (total = 40):
A = 8, B = 12, C = ___, D = 4 - Why are frequency tables useful before creating graphs like histograms?
- Construct a frequency table for: {2, 2, 3, 3, 3, 4, 5, 5}
- Does a frequency table preserve original data values? Why does that matter?
- Create a relative frequency table from your dataset for a chosen variable.
- 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 does the frequency column in a table sum to? Why?
- What does the relative frequency column in table sum to? Why?
- What is the difference between relative frequency and frequency for each data value in a table?
- What is the difference between cumulative relative frequency and relative frequency for each data value?
Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each.
- What is the percentage of people who own at most three T-shirts costing more than $19 each?
- If the data were collected by asking the first 111 people who entered the store, then what type of sampling was used?
- Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in the table below.
| Salary ($) | Relative Frequency |
|---|---|
| < 20,000 | 0.02 |
| 20,000–25,000 | 0.09 |
| 25,000–30,000 | 0.19 |
| 30,000–40,000 | 0.26 |
| 40,000–50,000 | 0.18 |
| 50,000–75,000 | 0.17 |
| 75,000–99,999 | 0.02 |
| 100,000+ | 0.01 |
- What percentage of the survey answered "not sure"?
- What percentage think that middle-class is from $25,000 to $50,000?
- Construct a histogram of the data.
- Should all bars have the same width, based on the data? Why or why not?
- How should the <20,000 and the 100,000+ intervals be handled? Why?
- Find the 40th and 80th percentiles
- List all the measurements for the data set represented by the following data frequency table.
\[\begin{array}{c|ccccc}x & 31 & 32 & 33 & 34 & 35 \\ \hline f & 1 & 5 & 6 & 4 & 2\end{array}\]
- Construct the data frequency table for the following data set.
\[\begin{array}22 & 25 & 22 & 27 & 24 & 23 \\ 26 & 24 & 22 & 24 & 26 &\end{array}\]
- The IQ scores of ten students randomly selected from an elementary school are given. \[\begin{array}108 & 100 & 99 & 125 & 87 \\ 105 & 107 & 105 & 119 & 118\end{array}\]Grouping the measures in the \(80s\), the \(90s\), and so on, construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram.
- The IQ scores of ten students randomly selected from an elementary school for academically gifted students are given. \[\begin{array}133 & 140 & 152 & 142 & 137 \\ 145 & 160 & 138 & 139 & 138\end{array}\]Grouping the measures by their common hundreds and tens digits, construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram.
- During a one-day blood drive \(300\) people donated blood at a mobile donation center. The blood types of these \(300\) donors are summarized in the table. \[\begin{array}{c|ccc}Blood\: Type\hspace{0.167em} & O & A & B & AB \\ \hline Frequency & 136 & 120 & 32 & 12\end{array}\]Construct a relative frequency histogram for the data set.
- In a particular kitchen appliance store an electric automatic rice cooker is a popular item. The weekly sales for the last \(20\)weeks are shown. \[\begin{array}20 & 15 & 14 & 14 & 18 \\ 15 & 17 & 16 & 16 & 18 \\ 15 & 19 & 12 & 13 & 9 \\ 19 & 15 & 15 & 16 & 15\end{array}\]Construct a relative frequency histogram with classes \(6-10\), \(11-15\), and \(16-20\).
- Consider the data set represented by the ordered stem and leaf diagram \[\begin{array}{c|c c c c c c c c c c c c c c c c c c} 10 &0 &0 \\ 9 &1 &1 &1 &1 &2 &3\\ 8 &0 &1 &1 &2 &2 &3 &4 &5 &7 &8 &8 &9\\ 7 &0 &0 &0 &1 &1 &2 &4 &4 &5 &6 &6 &6 &7 &7 &7 &8 &8 &9\\ 6 &0 &1 &2 &2 &2 &3 &4 &4 &5 &7 &7 &7 &7 &8 &8\\ 5 &0 &2 &3 &3 &4 &4 &6 &7 &7 &8 &9\\ 4 &2 &5 &6 &8 &8\\ 3 &9 &9 \end{array}\]
- Find the three quartiles.
- Give the five-number summary of the data.
- Find the range and the IQR.
- For the following stem and leaf diagram the units on the stems are thousands and the units on the leaves are hundreds, so that for example the largest observation is \(3,800\). \[\begin{array}{c|c c c c c c c c c c c} 3 &5 &6 &8 \\ 3 &0 &0 &1 &1 &2 &4\\ 2 &5 &6 &6 &7 &7 &8 &8 &9 &9 \\ 2 &0 &0 &0 &0 &1 &2 &2 &4 \\ 1 &5 &5 &5 &6 &6 &7 &7 &7 &8 &8 &9 \\ 1 &0 &0 &1 &3 &4 &4 &4 \\ 0 &5 &6 &8 &8\\ 0 &4 \end{array}\]
- Find the percentile rank of \(800\).
- Find the percentile rank of \(3,200\).
- Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:
Part-time Student Course Loads # of Courses Frequency Relative Frequency Cumulative Relative Frequency 1 30 0.6 2 15 3 - Fill in the blanks in the table.
- What percent of students take exactly two courses?
- What percent of students take one or two courses?
- The infant mortality rate is defined as the number of infant deaths per 1,000 live births. This rate is often used as an indicator of the level of health in a country. The relative frequency histogram below shows the distribution of estimated infant death rates in 2012 for 222 countries.68
- Estimate Q1, the median, and Q3 from the histogram.
- Would you expect the mean of this data set to be smaller or larger than the median? Explain your reasoning.
3.4: Shapes of Distributions
- Describe how you can identify whether a histogram is right-skewed, left-skewed, or symmetric.
- Which measure of center is better to use with skewed data: mean or median? Why?
- Draw a rough histogram with a left-skewed distribution. Label axes.
- Provide a real-world example of a variable that likely has a right-skewed distribution.
- How would a boxplot help verify the shape of a distribution?
- What does a unimodal vs. bimodal distribution tell you?
- 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?
- True/False: A bimodal distribution has two modes and two medians.
- If the mean time to respond to a stimulus is much higher than the median time to respond, what can you say about the shape of the distribution of response times?
- True/False: The best way to describe a skewed distribution is to report the mean.
- For each of the following graphs, describe the shape of the distribution.
- Describe the distribution in the histograms below and match them to the box plots.
- For each of the following, describe whether you expect the distribution to be symmetric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR.
- Housing prices in a country where 25% of the houses cost below $350,000, 50% of the houses cost below $450,000, 75% of the houses cost below $1,000,000 and there are a meaningful number of houses that cost more than $6,000,000.
- Housing prices in a country where 25% of the houses cost below $300,000, 50% of the houses cost below $600,000, 75% of the houses cost below $900,000 and very few houses that cost more than $1,200,000.
- Number of alcoholic drinks consumed by college students in a given week.
- Annual salaries of the employees at a Fortune 500 company.
3.5: Time Plots and Trends Over Time
- Identify two types of data (outside this class) that would make good candidates for a time plot.
- Create a time plot for this data:
Months: Jan–Jun; Visitors: 204, 228, 305, 311, 298, 287 - What does it mean for a variable to "trend upward" over time?
- Can a time plot use bar-style charts instead of points and lines? Why or why not?
- Look at your project data. If it includes any times tamped data or price listed dates, could you use a time plot? What would it show?
- How would a time plot be different than a histogram for the same data?
- Construct a times series graph for the total number of births in Scotland.
Table of the number of births in Scotland Year 1855 1856 1857 1858 1859 1860 1861 Total Births 93,349 101,821 103,415 104,018 106,543 105,629 107,009 Number of births in Scotland (continued) Year 1862 1863 1864 1865 1866 1867 1868 1869 Total Births 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354 Number of births in Scotland (continued) Year 1871 1870 1872 1871 1872 1827 1874 1875 Total Births 116,128 115,390 118,765 116,128 118,765 119,700 123,711 123,578 - The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for the city of Detroit, Michigan during the period from 1961 to 1973.
Data table of Police and Homicides over time 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 Data table of Police and Homicides (continued) 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 Detroit’s increase in police officers have an impact on the murder rate? Explain.