3.7: Practice (Chapter 3)

3.1: Graphical Displays for Categorical Data

1. For each of the following, describe as categorical or numerical
   1. Hair color
   2. College GPA
   3. Humidity
   4. Character's level in a video game
   5. Subscribers to a content creator
   6. Score in a tennis match
   7. B-vitamins
   8. Round in the NCAA division tournament
   9. Musical notes in a melody
   10. Type of surround sound
   11. Speed limits
2. Give an example of a data set that would be best represented with a bar chart.
3. Give an example of a data set that would be best represented with a pie chart
4. Draw a bar chart to represent the following class survey data about animal companions: Cats (10), Dogs (8), Fish (4), Other (3)
5. Explain a limitation of pie charts when many categories are included.
6. Can a bar chart be used with relative frequencies instead of counts? Explain.
7. True or False: “There should be no space between bars in a bar chart.”
8. True or False: “There should be no space between slices in a pie chart.”
9. How do we decide if a pie chart or bar chart would be more appropriate?
10. 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.

Birthday data for Math Class

Seasons	Number of students	Proportion of population
Spring	8	24%
Summer	9	26%
Autumn	11	32%
Winter	6	18%

 

11. Using the data from Ms. Ramirez’s math class supplied in the table above, construct a pie chart showing the percentages.
12. 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.

    

    1. What features are apparent in the bar plot but not in the pie chart?
    2. What features are apparent in the pie chart but not in the bar plot?
    3. Which graph is better to use for this data and why?

3.2: Graphical Displays for Quantitative Data

1. What makes dot plots difficult to use for large sets of numerical data?

2. The following dot plot represents the average amount of sleep per night that students in a class report.

   

    

   1. What are the values in the original data set?
   2. If you created a similar plot with students in your class, do you think the results would be the same? Why or why not?
   3. Where do your data appear to cluster? How might you interpret the clustering?

3. Create dot plots for the following data sets and then use the dot plots to state how the means and standard deviations compare.

   1. set 1 {3, 5, 5, 5, 8, 11, 11, 11, 13},  set 2 {3, 5, 5, 5, 8, 11, 11, 11, 20}

   2. set 1 {-20, 0, 0, 0, 15, 25, 30, 30}, set 2 {-40, 0, 0, 0, 15, 25, 30, 30}

   3. set 1 {0, 2, 4, 6, 8, 10}, set 2 {20, 22, 24, 26, 28, 30}

   4. set 1 {100, 200, 300, 400, 500}, set 2 {0, 50, 300, 550, 600}

4. For Susan Dean's spring pre-calculus class, scores for the first exam were as follows (smallest to largest):

   33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100

   Stem-and-Leaf Graph

   Stem	Leaf
   3	3
   4	2 9 9
   5	3 5 5
   6	1 3 7 8 8 9 9
   7	2 3 4 8
   8	0 3 8 8 8
   9	0 2 4 4 4 4 6
   10	0

   What does the Stemplot tell you about the class's performance on the exam?

5. Consider a data set and Stemplot of the distances (in kilometers) from a home to local supermarkets. 

   1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3

   Stem	Leaf
   1	1 5
   2	3 5 7
   3	2 3 3 5 8
   4	0 2 5 5 7 8
   5	5 6
   6	5 7
   7
   8
   9
   10
   11
   12	3

   1. Do the data seem to have any concentration of values?

   2. What does the bottom row of the Stemplot tell you?

6. Consider the data set represented by the ordered stem and leaf diagram: 

   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 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

    
   1. Find the three quartiles.
   2. Give the five-number summary of the data.
   3. Find the range and the IQR.
7. 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. 

   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

    
   1. Find the percentile rank of 800.
   2. Find the percentile rank of 3,200.
8. Construct a stem-and-leaf plot for: {34, 36, 37, 41, 42, 42, 43, 46}

9. Construct a Stemplot using the data in the table below

   Table : Presidential Ages at Inauguration

   President	Ageat Inauguration	President	Age	President	Age
   Pierce	48	Harding	55	Obama	47
   Polk	49	T. Roosevelt	42	G.H.W. Bush	64
   Fillmore	50	Wilson	56	G. W. Bush	54
   Tyler	51	McKinley	54	Reagan	69
   Van Buren	54	B. Harrison	55	Ford	61
   Washington	57	Lincoln	52	Hoover	54
   Jefferson	57	Grant	46	Truman	60
   Madison	57	Hayes	54	Eisenhower	62
   J. Q. Adams	57	Arthur	51	L. Johnson	55
   Monroe	58	Garfield	49	Kennedy	43
   J. Adams	61	A. Johnson	56	F. Roosevelt	51
   Jackson	61	Cleveland	47	Nixon	56
   Taylor	64	Taft	51	Clinton	47
   Buchanan	65	Coolidge	51	Trump	70
   W. H. Harrison	68	Cleveland	55	Carter	52

10. 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.

    Data table of number of shoes students own.

    X	Frequency
    1	2
    2	5
    3	8
    4	12
    5	12
    6	0
    7	1

3.3: Frequency Tables and Relative Frequency

1. Fill in the missing frequency for a data set with sample size 40: A = 8, B = 12, C = ___, D = 4
2. Why is it useful to create a frequency table before creating a histogram?
3. Does a frequency table preserve original data values? Why does that matter?
4. What does the frequency column in a table sum to? Why?
5. What does the relative frequency column in table sum to? Why?
6. Explain the difference between frequency and relative frequency.
7. What is the difference between cumulative relative frequency and relative frequency for each data value?
8. What is the key difference between a bar chart and a histogram?
9. What does the height of a bar in a histogram represent?
10. Describe one advantage of a stem plot over a frequency histogram.
11. Choose a quantitative variable from your dataset. Would you rather display it with a histogram, dot plot, or stem plot? Why?
12. Construct a frequency table for: {2, 2, 3, 3, 3, 4, 5, 5}
13. Draw a histogram for the values: {2, 2, 3, 5, 5, 5, 6, 7, 8, 10}
14. 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.​

Empty data table to complete

Data Value (# cars)	Frequency	Relative Frequency	Cumulative Relative Frequency

15. 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.

1. What is the percentage of people who own at most three T-shirts costing more than $19 each?
2. If the data were collected by asking the first 111 people who entered the store, then what type of sampling was used?

16. 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.

Table of income Americans consider "middle-class"

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

1. What percentage of the survey answered "not sure"?
2. What percentage think that middle-class is from $25,000 to $50,000?
3. Construct a histogram of the data.
   1. Should all bars have the same width, based on the data? Why or why not?
   2. How should the <20,000 and the 100,000+ intervals be handled? Why?
4. Find the 40^th and 80^th percentiles

17. List all the measurements for the data set represented by the following data frequency table. 

    X	31	32	33	34	35
    frequency	1	5	6	4	2

18. Construct the data frequency table for the following data set. 

    22

    25

    22

    27

    24

    23

    26

    24

    22

    24

    26

19. The IQ scores of ten students randomly selected from an elementary school are given. {108, 100, 99, 125, 87, 105, 107, 105, 119, 118}. Grouping the measures in the 80's, the 90's, and so on, construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram.
20. The IQ scores of ten students randomly selected from an elementary school for academically gifted students are given. {133, 140, 152, 142, 137, 145, 160, 138, 139, 138}. Grouping the measures by their common hundreds and tens digits, construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram.
21. 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. Construct a relative frequency histogram for the data set. 

    Blood Type	O	A	B	AB
    Frequency	136	120	32	12

22. 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. Construct a frequency histogram with a "nice" class width. 

    20	15	14	14	18	17	16	16	18
    15	16	19	9	15	13	12	15	15
23. 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

    1. Fill in the blanks in the table.
    2. What percent of students take exactly two courses?
    3. What percent of students take one or two courses?
24. 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
    mean	6.0 miles	6.0 miles
    s	4.0 miles	7.0 miles

    1. How can you determine which survey was correct ?
    2. Explain what the difference in the results of the surveys implies about the data.
    3. If the two histograms depict the distribution of values for each supervisor, which one depicts Ercilia's sample? How do you know?

       

 

3.4: Shapes of Distributions

1. Describe how you can identify whether a histogram is right-skewed, left-skewed, or symmetric.
2. Which measure of center is better to use with skewed data: mean or median? Why?
3. Draw a rough histogram with a left-skewed distribution. Label axes.
4. How would a boxplot help verify the shape of a distribution?
5. What are the similarities and differences between unimodal and bimodal?
6. When the data are skewed left, what is the typical relationship between the mean and median?
7. When the data are symmetrical, what is the typical relationship between the mean and median?
8. True/False: A bimodal distribution has two modes and two medians.
9. True/False: The best way to describe a skewed distribution is to report the mean.
10. 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?
11. Provide a real-world example of a variable that likely has a right-skewed distribution.
12. For each of the following graphs, describe the shape of the distribution.

13. Describe the shape of the distributions in the histograms below and match them to the box plots.

    

14. 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.
    1. 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.
    2. 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.
    3. Number of alcoholic drinks consumed by college students in a given week.
    4. Annual salaries of the employees at a Fortune 500 company.

 

3.5: Time Series Plots and Trends Over Time

1. Identify two types of data (outside this class) that would make good candidates for a time plot.
2. What does it mean for a variable to "trend upward" over time?
3. Can a time plot use bars instead of points and lines? Why or why not?
4. Look at your project data. If it includes any time-stamped data or dates, could you use a time plot? What would it show?
5. How would a time plot be different than a histogram for the same data?
6. The time series plot below shows the finishing times for male and female winners of the New York Marathon between 1980 and 1999.
   1. What points in time are significant for this data?
   2. What happened at those points in time?
7. Create a time plot for this data: Months: Jan–Jun; Visitors: 204, 228, 305, 311, 298, 287
8. 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

9. 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

   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.

10. The following data shows the Annual Consumer Price Index for ten years. Construct a time series graph for the Annual Consumer Price Index.

    Year	Annual
    2003	184.0
    2004	188.9
    2005	195.3
    2006	201.6
    2007	207.342
    2008	215.303
    2009	214.537
    2010	218.056
    2011	224.939
    2012	229.594

11. The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO₂ emissions.

    CO₂ Emissions

    	Ukraine	United Kingdom	United States
    2003	352,259	540,640	5,681,664
    2004	343,121	540,409	5,790,761
    2005	339,029	541,990	5,826,394
    2006	327,797	542,045	5,737,615
    2007	328,357	528,631	5,828,697
    2008	323,657	522,247	5,656,839
    2009	272,176	474,579	5,299,563

12. The Gross Domestic Product Purchasing Power Parity (GDP PPP) is an indication of a country’s currency value compared to another country. The table below shows the GDP PPP of Cuba as compared to US dollars. Construct a time series plot of the data.

    Year	Cuba’s PPP	Year	Cuba’s PPP
    1999	1,700	2006	4,000
    2000	1,700	2007	11,000
    2002	2,300	2008	9,500
    2003	2,900	2009	9,700
    2004	3,000	2010	9,900
    2005	3,500

13. For the following situations, determine which types of graphs are appropriate ways of displaying the data. Then choose which graph type would be best and explain why.
    1. A study on whether IQ level is related to birth order (oldest, middle, youngest)
    2. A study on the relationship between the annual income of a family and the amount of money the family spends on entertainment
    3. A store asked 250 of its customers whether they were satisfied with the service or not
    4. The median US household cost for housing each year since 1989
    5. Peoples' favorite television show
    6. The average length of time spent commuting to work
    7. The average cost of bread each day over the course of a month

Practice problems include parts of:

Introductory Statistics 2e by OpenStax is licensed under CC BY 4.0

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

and

OpenIntro Statistics by  David Diez, Christopher Barr, & Mine Çetinkaya-Rundel is licensed under CC BY 3.0