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4.8 3.7 4.0 3.5 3.6 4.3 3.6 3.9 3.3 4.4 3.5 3.5 4.3 3.4 3.9 3.9 4.2 4.2 4.0 a. Construct a stem and leaf plot by using the leading digit as the stem. b. Construct a stem and leaf plot by using each leading digit twice. Does this technique improve the presentation of the data? Explain. 1.21 A discrete variable can take on only the values 0, 1, or 2. A set of 20 measurements on this variable is shown here. Construct a relative frequency histogram for the data. 1.22 Refer to Exercise 1.21. a. Draw a dotplot to describe the data. b. How could you define the stem and the leaf for this data set? c. Draw the stem and leaf plot using your decision from part b. d. Compare the dotplot, the stem and leaf plot, and the relative frequency histogram (Exercise 1.21). Do they all convey roughly the same information? 1.23 Navigating a Maze An experimental psychologist measured the length of time it took for a rat to successfully navigate a maze on each of five days. The results are shown in the table. Create a line chart to describe the data. Do you think that any learning is taking place? Day Time (sec.) 1 45 2 43 3 46 4 32 5 25 1.24 Measuring over Time The value of a quantitative variable is measured once a year EX0124 for a 10-year period. Here are the data: Year Measurement Year Measurement 1 2 3 4 5 61.5 62.3 60.7 59.8 58.0 6 7 8 9 10 58.2 57.5 57.5 56.1 56.0 a. Create a line chart to describe the variable as it changes over time. b. Describe the measurements using the chart constructed in part a. 1.25 Test Scores The test scores on a 100-point test were recorded for 20 students: EX0125 61 94 93 89 91 67 86 62 55 72 63 87 86 68 82 65 76 75 57 84 a. Use an appropriate graph to describe the data. b. Describe the shape and location of the scores. c. Is the shape of the distribution unusual? Can you think of any reason the distribution of the scores would have such a shape? |
APPLICATIONS 1.26 A Recurring Illness The length of time (in months) between the onset of a EX0126 particular illness and its recurrence was recorded for n 50 patients: 2.1 14.7 4.1 14.1 1.6 4.4 9.6 18.4 1.0 3.5 2.7 16.7.2 2.4 11.4 32.3 7.4 6.1 2.4 18.0 9.9 8.2 13.5 18.0 26.7 9.0 19.2 7.4 8.7 3.7 2.0 6.9.2 24.0 12.6 6.6 4.3 8.3 1.4 23.1 3.9 3.3.3 8.2 5.6 1.6 1.2 1.3 5.8.4 a. Construct a relative frequency histogram for the data. b. Would you describe the shape as roughly symmetric, skewed right, or skewed left? c. Give the fraction of recurrence times less than or equal to 10 months. 1.27 Education Pays Off! Education pays off, according to a snapshot provided in a report to the city of Riverside by the Riverside County Office of Education.7 The average annual incomes for six different levels of education are shown in the table: Educational Level Average Annual Income High school graduate Some college, no degree Bachelor’s degree Master’s degree Doctorate Professional (Doctor, Lawyer) Source: U.S. Census Bureau $26,795 29,095 50,623 63,592 85,675 101,375 a. What graphical methods could you use to describe the data? b. Select the method from part a that you think best describes the data. c. How would you summarize the information that you see in the graph regarding educational levels and salary? 1.28 Preschool The ages (in months) at which 50 children were first enrolled in a EX0128 preschool are listed below. 38 47 32 55 42 40 35 34 39 50 30 34 41 33 37 35 43 30 32 39 39 41 46 32 33 40 36 35 45 45 48 41 40 42 38 36 43 30 41 46 31 48 46 36 36 36 40 37 50 31 1.5 RELATIVE FREQUENCY HISTOGRAMS ❍ 31 a. Construct a stem and leaf display for the data. b. Construct a relative frequency hist |
ogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. c. Compare the graphs in parts a and b. Are there any significant differences that would cause you to choose one as the better method for displaying the data? d. What proportion of the children were 35 months (2 years, 11 months) or older, but less than 45 months (3 years, 9 months) of age when first enrolled in preschool? e. If one child were selected at random from this group of children, what is the probability that the child was less than 50 months old (4 years, 2 months) when first enrolled in preschool? Text not available due to copyright restrictions EX0130 1.30 How Long Is the Line? To decide on the number of service counters needed for stores to be built in the future, a supermarket chain wanted to obtain information on the length of time (in minutes) required to service customers. To find the distribution of customer service times, a sample of 1000 customers’ service times was recorded. Sixty of these are shown here: 3.6 1.1 1.4.6 1.1 1.6 2.1.3 1.3 2.5.8 5.2 1.4 1.1 4.5 1.3 1.7.6 1.9 1.8.2 2.8 1.2 1.9 1.8.8.9.8 1.1.7.3 1.1 3.1 1.1 1.0.5 1.6 1.7.7 1.3 2.2.6.2 1.2 2.3 1.2.7.3.8.5.4.4.9 1.8 1.0.6 1.8.4 3.1 1.1 32 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS a. Construct a stem and leaf plot for the data. b. What fraction of the service times are less than or equal to 1 minute? c. What is the smallest of the 60 measurements? 1.31 Service Times, continued Refer to Exercise 1.30. Construct a relative frequency histogram for the supermarket service times. a. Describe the shape of the distribution. Do you see any outliers? b. Assuming that the outliers in this data |
set are valid observations, how would you explain them to the management of the supermarket chain? c. Compare the relative frequency histogram with the stem and leaf plot in Exercise 1.30. Do the two graphs convey the same information? 1.32 Calcium Content The calcium (Ca) content of a powdered mineral substance was EX0132 analyzed ten times with the following percent compositions recorded:.0271.0271.0282.0281.0279.0269.0281.0275.0268.0276 a. Draw a dotplot to describe the data. (HINT: The scale of the horizontal axis should range from.0260 to.0290.) b. Draw a stem and leaf plot for the data. Use the numbers in the hundredths and thousandths places as the stem. c. Are any of the measurements inconsistent with the other measurements, indicating that the technician may have made an error in the analysis? 1.33 American Presidents Listed below are the ages at the time of death for the 38 EX0133 deceased American presidents from George Washington to Ronald Reagan:5 Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren W. H. Harrison Tyler Polk Taylor Fillmore Pierce Buchanan Lincoln A. Johnson Grant Hayes 67 90 83 85 73 80 78 79 68 71 53 65 74 64 77 56 66 63 70 Garfield Arthur Cleveland B. Harrison Cleveland McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. D. Roosevelt Truman Eisenhower Kennedy L. Johnson Nixon Reagan 49 56 71 67 71 58 60 72 67 57 60 90 63 88 78 46 64 81 93 a. Before you graph the data, try to visualize the distribution of the ages at death for the presidents. What shape do you think it will have? b. Construct a stem and leaf plot for the data. Describe the shape. Does it surprise you? c. The five youngest presidents at the time of death appear in the lower “tail” of the distribution. Three of the five youngest have one common trait. Identify the five youngest presidents at death. What common trait explains these measurements? EX0134 1.34 RBC Counts The red blood cell count of a healthy person was measured on each of 15 days. The number recorded is measured in 106 cells per microliter (L). 5.4 5.3 5.3 5.2 5.4 4.9 5.0 5. |
2 5.4 5.2 5.1 5.2 5.5 5.3 5.2 a. Use an appropriate graph to describe the data. b. Describe the shape and location of the red blood cell counts. c. If the person’s red blood cell count is measured today as 5.7 106/L, would you consider this unusual? What conclusions might you draw? 1.35 Batting Champions The officials of major league baseball have crowned a batting EX0135 champion in the National League each year since 1876. A sample of winning batting averages is listed in the table:5 Year 2005 2000 1915 1917 1934 1911 1898 1924 1963 1992 1954 1975 1958 1942 1948 1971 1996 1961 1968 1885 Name Average Derreck Lee Todd Helton Larry Doyle Edd Roush Paul Waner Honus Wagner Willie Keeler Roger Hornsby Tommy Davis Gary Sheffield Willie Mays Bill Madlock Richie Ashburn Ernie Lombardi Stan Musial Joe Torre Tony Gwynn Roberto Clemente Pete Rose Roger Connor.335.372.320.341.362.334.379.424.326.330.345.354.350.330.376.363.353.351.335.371 a. Construct a relative frequency histogram to describe the batting averages for these 20 champions. b. If you were to randomly choose one of the 20 sites nearby? The table shows the number of hazardous waste sites in each of the 50 states and the District of Columbia in the year 2006:5 1.5 RELATIVE FREQUENCY HISTOGRAMS ❍ 33 names, what is the chance that you would choose a player whose average was above.400 for his championship year? 1.36 Top 20 Movies The table that follows shows the weekend gross ticket sales for the EX0136 top 20 movies during the week of August 4, 2006:9 Movie 1.Talladega Nights: The Ballad of Ricky Bobby 2. Barnyard 3. Pirates of the Caribbean: Dead Man’s Chest 4. Miami Vice 5. The Descent 6. John Tucker Must Die 7. Monster House 8. The Ant Bully 9. You, Me and Dupree 10. The Night Listener 11. The Devil Wears Prada 12. Lady in the Water 13. Little Man 14. Superman Returns 15. Scoop 16. Little Miss Sunshine 17. Clerks II 18. My Super Ex-Girlfriend 19. Cars 20. Click Source: www.radiofree |
.com/mov-tops.shtml Weekend Gross ($ millions) $47.0 15.8 11.0 10.2 8.9 6.2 6.1 3.9 3.6 3.6 3.0 2.7 2.5 2.2 1.8 1.5 1.3 1.2 1.1 0.8 a. Draw a stem and leaf plot for the data. Describe the shape of the distribution. Are there any outliers? b. Construct a dotplot for the data. Which of the two graphs is more informative? Explain. 1.37 Hazardous Waste How safe is your neighborhood? Are there any hazardous waste EX0137 AL AK AZ AR CA CO CT DE DC FL GA 15 6 9 10 95 19 16 15 1 50 17 HI ID IL IN IA KS KY LA ME MD 3 9 48 30 12 11 14 14 12 18 MA MI MN MS MO MT NE NV NH NJ 33 68 24 5 26 15 14 1 21 117 NM 13 87 NY 31 NC 0 ND 37 OH 11 OK 11 OR 96 PA 12 RI 26 SC SD TN TX UT VT VA WA WV WI WY 2 14 44 18 11 28 47 9 38 2 a. What variable is being measured? Is the variable discrete or continuous? b. A stem and leaf plot generated by MINITAB is shown here. Describe the shape of the data distribution. Identify the unusually large measurements marked “HI” by state. Stem-and-Leaf Display: Hazardous Waste Stem-and-leaf of Sites N = 51 Leaf Unit = 1.0 0 011223 0 56999 1 0111122234444 6 11 24 (8) 1 55567889 2 14 19 2 668 17 3 013 14 3 78 11 4 4 9 4 78 8 5 0 6 HI 68, 87, 95, 96, 117 c. Can you think of any reason these five states would have a large number of hazardous waste sites? What other variable might you measure to help explain why the data behave as they do? As you continue to work through the exercises in this chapter, you will become more experienced in recognizing different types of data and in determining the most appropriate graphical method to use. Remember that the type of graphic you use is not as important as the interpretation that accompanies the picture. Look for these important characteristics: • Location of the center of the data • Shape of the distribution of data • Unusual observations in the data |
set 34 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS Using these characteristics as a guide, you can interpret and compare sets of data using graphical methods, which are only the first of many statistical tools that you will soon have at your disposal. CHAPTER REVIEW Key Concepts I. How Data Are Generated 1. Experimental units, variables, measurements 2. Samples and populations 3. Univariate, bivariate, and multivariate data II. Types of Variables 1. Qualitative or categorical 2. Quantitative a. Discrete b. Continuous III. Graphs for Univariate Data Distributions 1. Qualitative or categorical data a. Pie charts b. Bar charts 2. Quantitative data a. Pie and bar charts b. Line charts c. Dotplots d. Stem and leaf plots e. Relative frequency histograms 3. Describing data distributions a. Shapes—symmetric, skewed left, skewed right, unimodal, bimodal b. Proportion of measurements in certain intervals c. Outliers Easy access to the web has made it possible for you to understand statistical concepts using an interactive web tool called an applet. These applets provide visual reinforcement for the concepts that have been presented in the chapter. Sometimes you will be able to perform statistical experiments, sometimes you will be able to interact with a statistical graph to change its form, and sometimes you will be able to use the applet as an interactive “statistical table.” At the end of each chapter, you will find exercises designed specifically for use with a particular applet. The applets have been customized specifically to match the presentation and notation used in your text. They can be found on the Premium Website. If necessary, follow the instructions to download the latest web browser and/or Java plug-in, or just click the appropriate link to load the applets. Your web browser will open the index of applets, organized by chapter and name. When you click a particular applet title, the applet will appear in your browser. To return to the index of applets, simply click the link at the bottom of the page. CHAPTER REVIEW ❍ 35 Dotplots Click the Chapter 1 applet called Building a Dotplot. If you move your cursor over the applet marked Dotplot Demo you will see a green line with a value that changes as |
you move along the horizontal axis. When you left-click your mouse, a dot will appear at that point on the dotplot. If two measurements are identical, the dots will pile up on top of each other (Figure 1.17). Follow the directions in the Dotplot Demo, using the sample data given there. If you make a mistake, the applet will tell you. The second applet will not correct your mistakes; you can add as many dots as you want! FI GUR E 1.1 7 Building a Dotplot applet ● Histograms Click the Chapter 1 applet called Building a Histogram. If you scroll down to the applet marked Histogram Demo, you will see the interval boundaries (or interval midpoints) for the histogram along the horizontal axis. As you move the mouse across the graph, a light gray box will show you where the measurement will be added at your next mouse click. When you release the mouse, the box turns dark blue (dark blue in Figure 1.18). The partially completed histogram in Figure 1.18 contains one 3, one 4, one 5, three 6s, and one 7. Follow the directions in the Histogram Demo using the sample data given there. Click the link to compare your results to the correct histogram. The second applet will be used for some of the MyApplet Exercises. Click the applet called Flipping Fair Coins, and scroll down to the applet marked sample size 3. The computer will collect some data by “virtually” tossing 3 coins and recording the quantitative discrete variable x number of heads observed Click on “New Coin Flip.” You will see the result of your three tosses in the upperleft-hand corner, along with the value of x. For the experiment in Figure 1.19 we observed x 2. The applet begins to build a relative frequency histogram to describe the data set, which at this point contains only one observation. Click “New Coin Flip” a few more times. Watch the coins appear, along with the value of x, and watch the 36 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS F IG URE 1. 18 Building a Histogram applet ● F IG URE 1. 19 Flipping Fair Coins applet ● F IG URE 1. 20 Flipping Fair Coins applet ● MY MINITAB ❍ 37 relative frequency histogram grow. The red area (light blue |
in Figures 1.19 and 1.20) represents the current data added to the histogram, and the dark blue area in Figure 1.20 is contributed from the previous coin flips. You can flip the three coins 10 at a time or 100 at a time to generate data more quickly. Figure 1.20 shows the relative frequency histogram for 500 observations in our data set. Your data set will look a little different. However, it should have the same approximate shape—it should be relatively symmetric. For our histogram, we can say that the values x 0 and x 3 occurred about 12–13% of the time, while the values x 1 and x 2 occurred between 38% and 40% of the time. Does your histogram produce similar results? Introduction to MINITABTM MINITAB is a computer software package that is available in many forms for different computer environments. The current version of MINITAB at the time of this printing is MINITAB 15, which is used in the Windows environment. We will assume that you are familiar with Windows. If not, perhaps a lab or teaching assistant can help you to master the basics. Once you have started Windows, there are two ways to start MINITAB: If there is a MINITAB shortcut icon on the desktop, double-click on the icon. • • Click the Start button on the taskbar. Follow the menus, highlighting All Programs MINITAB Solutions MINITAB 15 Statistical Software English. Click on MINITAB 15 Statistical Software English to start the program. When MINITAB is opened, the main MINITAB screen will be displayed (see Figure 1.21). It contains two windows: the Data window and the Session window. Clicking FI GUR E 1.2 1 ● 38 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS anywhere on the window will make that window active so that you can either enter data or type commands. Although it is possible to manually type MINITAB commands in the Session window, we choose to use the Windows approach, which will be familiar to most of you. If you prefer to use the typed commands, consult the MINITAB manual for detailed instructions. At the top of the Session window, you will see a Menu bar. Highlighting and clicking on any command on the Menu bar will cause a menu to drop down, from which you may then select the necessary command. We will use the standard notation to indicate a sequence of |
commands from the drop-down menus. For example, File Open Worksheet will allow you to retrieve a “worksheet”—a set of data from the Data window—which you have previously saved. To close the program, the command sequence is File Exit. MINITAB 15 allows multiple worksheets to be saved as “projects.” When you are working on a project, you can add new worksheets or open worksheets from other projects to add to your current project. As you become more familiar with MINITAB, you will be able to organize your information into either “worksheets” or “projects,” depending on the complexity of your task. Graphing with MINITAB The first data set to be graphed consists of qualitative data whose frequencies have already been recorded. The class status of 105 students in an introductory statistics class are listed in Table 1.13. Before you enter the data into the Minitab Data window, start a project called “Chapter 1” by highlighting File New. A Dialog box called “New” will appear. Highlight Minitab Project and click OK. Before you continue, let’s save this project as “Chapter 1” using the series of commands File Save Project. Type Chapter 1 in the File Name box, and select a location using the white box marked “Save in:” at the top of the Dialog box. Click Save. In the Data window at the top of the screen, you will see your new project name, “Chapter 1.MPJ.” TABLE 1.13 ● Status of Students in Statistics Class Status Freshman Sophomore Junior Senior Grad Student Frequency 5 23 32 35 10 To enter the data into the worksheet, click on the gray cell just below the name C1 in the Data window. You can enter your own descriptive name for the categories— possibly “Status.” Now use the down arrow or your mouse to continue down column C1, entering the five status descriptions. Notice that the name C1 has changed to C1-T because you are entering text rather than numbers. Continue by naming column 2 (C2) “Frequency,” and enter the five numerical frequencies into C2. The Data window will appear as in Figure 1.22. To construct a pie chart for these data, click on Graph Pie Chart, and |
a Dialog box will appear (see Figure 1.23). In this box, you must specify how you want to create the chart. Click the radio button marked Chart values from a table. Then place your cursor in the box marked “Categorical variable.” Either (1) highlight C1 in the list at the left and choose Select, (2) double-click on C1 in the list at the left, or (3) type C1 in the “Categorical variable” box. Similarly, place the cursor in the box marked MY MINITAB ❍ 39 FI GUR E 1.2 2 ● FI GUR E 1.2 3 ● “Summary variables” and select C2. Click Labels and select the tab marked Slice Labels. Check the boxes marked “Category names” and “Percent.” When you click OK, MINITAB will create the pie chart in Figure 1.24. We have removed the legend by selecting and deleting it. As you become more proficient at using the pie chart command, you may want to take advantage of some of the options available. Once the chart is created, right-click on the pie chart and select Edit Pie. You can change the colors and format of the chart, “explode” important sectors of the pie, and change the order of the categories. If you right-click on the pie chart and select Update Graph Automatically, the pie chart will automatically update when you change the data in columns C1 and C2 of the MINITAB worksheet. If you would rather construct a bar chart, use the command Graph Bar Chart. In the Dialog box that appears, choose Simple. Choose an option in the “Bars represent” drop-down list, depending on the way that the data has been entered into the 40 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS F IG URE 1. 24 ● worksheet. For the data in Table 1.13, we choose “Values from a table” and click OK. When the Dialog box appears, place your cursor in the “Graph variables” box and select C2. Place your cursor in the “Categorical variable” box, and select C1. Click OK to finish the bar chart, shown in Figure 1.25. Once the chart is created, right-click |
on various parts of the bar chart and choose Edit to change the look of the chart. MINITAB can create dotplots, stem and leaf plots, and histograms for quantitative data. The top 40 stocks on the over-the-counter (OTC) market, ranked by percentage of outstanding shares traded on a particular day, are listed in Table 1.14. Although we could simply enter these data into the third column (C3) of Worksheet 1 in the “Chapter 1” project, let’s start a new worksheet within “Chapter 1” using File New, highlighting Minitab Worksheet, and clicking OK. Worksheet 2 will appear on the screen. Enter the data into column C1 and name them “Stocks” in the gray cell just below the C1. TABLE 1.14 ● Percentage of OTC Stocks Traded 11.88 7.99 7.15 7.13 6.27 6.07 5.98 5.91 5.49 5.26 5.07 4.94 4.81 4.79 4.55 4.43 4.40 4.05 3.94 3.93 3.78 3.69 3.62 3.48 3.44 3.36 3.26 3.20 3.11 3.03 2.99 2.89 2.88 2.74 2.74 2.69 2.68 2.63 2.62 2.61 To create a dotplot, use Graph Dotplot. In the Dialog box that appears, choose One Y Simple and click OK. To create a stem and leaf plot, use Graph Stemand-Leaf. For either graph, place your cursor in the “Graph variables” box, and select “Stocks” from the list to the left (see Figure 1.26). FI GUR E 1.2 5 ● MY MINITAB ❍ 41 FI GUR E 1.2 6 ● You can choose from a variety of formatting options before clicking OK. The dotplot appears as a graph, while the stem and leaf plot appears in the Session window. To print either a Graph window or the Session window, click on the window to make it active and use File Print Graph (or Print Session Window). To create a histogram, use Graph Histogram. In the Dialog box that appears, choose Simple and click OK, selecting “Stocks” for the “ |
Graph variables” box. 42 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS Select Scale Y-Scale Type and click the radio button marked “Frequency.” (You can edit the histogram later to show relative frequencies.) Click OK twice. Once the histogram has been created, right-click on the Y-axis and choose Edit Y Scale. Under the tab marked “Scale,” you can click the radio button marked “Position of ticks” and type in 0 5 10 15. Then click the tab marked “Labels,” the radio button marked “Specified” and type 0 5/40 10/40 15/40. Click OK. This will reduce the number of ticks on the y-axis and change them to relative frequencies. Finally, double-click on the word “Frequency” along the y-axis. Change the box marked “Text” to read “Relative frequency” and click OK. To adjust the type of boundaries for the histogram, right-click on the bars of the histogram and choose Edit Bars. Use the tab marked “Binning” to choose either “Cutpoints” or “Midpoints” for the histogram; you can specify the cutpoint or midpoint positions if you want. In this same Edit box, you can change the colors, fill type, and font style of the histogram. If you right-click on the bars and select Update Graph Automatically, the histogram will automatically update when you change the data in the “Stocks” column. As you become more familiar with MINITAB for Windows, you can explore the various options available for each type of graph. It is possible to plot more than one variable at a time, to change the axes, to choose the colors, and to modify graphs in many ways. However, even with the basic default commands, it is clear that the distribution of OTC stocks is highly skewed to the right. Make sure to save your work using the File Save Project command before you exit MINITAB! F IG URE 1. 27 ● SUPPLEMENTARY EXERCISES ❍ 43 Supplementary Exercises 1.38 Quantitative or Qualitative? Identify each variable as quantitative or qualitative: a. Ethnic origin of a candidate for public office b. Score (0–100) on a placement |
examination c. Fast-food establishment preferred by a student (McDonald’s, Burger King, or Carl’s Jr.) d. Mercury concentration in a sample of tuna a. Number of people in line at a supermarket checkout counter b. Depth of a snowfall c. Length of time for a driver to respond when faced with an impending collision d. Number of aircraft arriving at the Atlanta airport in a given hour 1.39 Symmetric or Skewed? Do you expect the distributions of the following variables to be symmetric or skewed? Explain. a. Size in dollars of nonsecured loans b. Size in dollars of secured loans c. Price of an 8-ounce can of peas d. Height in inches of freshman women at your university EX0143 1.43 Aqua Running Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who want a low-impact aerobics program. A study reported in the Journal of Sports Medicine investigated the relationship between exercise cadence and heart rate by measuring the heart rates of 20 healthy volunteers at a cadence of 48 cycles per minute (a cycle consisted of two steps).10 The data are listed here: e. Number of broken taco shells in a package of 100 shells f. Number of ticks found on each of 50 trapped cottontail rabbits 1.40 Continuous or Discrete? Identify each variable as continuous or discrete: a. Number of homicides in Detroit during a one-month period b. Length of time between arrivals at an outpatient clinic c. Number of typing errors on a page of manuscript d. Number of defective lightbulbs in a package containing four bulbs e. Time required to finish an examination 1.41 Continuous or Discrete, again Identify each variable as continuous or discrete: a. Weight of two dozen shrimp b. A person’s body temperature c. Number of people waiting for treatment at a hospital emergency room d. Number of properties for sale by a real estate agency e. Number of claims received by an insurance company during one day 1.42 Continuous or Discrete, again Identify each variable as continuous or discrete: 87 101 109 79 91 78 80 112 96 94 95 98 90 94 92 96 81 107 98 96 Construct a stem and leaf plot to describe the data. Discuss the characteristics of the data distribution. 1.44 Major World Lakes A lake is a body of water surrounded by land. Hence, some EX0144 bodies of water named “seas,” |
like the Caspian Sea, are actual salt lakes. In the table that follows, the length in miles is listed for the major natural lakes of the world, excluding the Caspian Sea, which has an area of 143,244 square miles, a length of 760 miles, and a maximum depth of 3,363 feet.5 Name Length (mi) Name Length (mi) Superior Victoria Huron Michigan Aral Sea Tanganyika Baykal Great Bear Nyasa Great Slave Erie Winnipeg Ontario Balkhash Ladoga Maracaibo Onega Source: The World Almanac and Book of Facts 2007 350 250 206 307 260 420 395 192 360 298 241 266 193 376 124 133 145 Eyre Titicaca Nicaragua Athabasca Reindeer Turkana Issyk Kul Torrens Vänern Nettilling Winnipegosis Albert Nipigon Gairdner Urmia Manitoba Chad 90 122 102 208 143 154 115 130 91 67 141 100 72 90 90 140 175 44 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS a. Use a stem and leaf plot to describe the lengths of the world’s major lakes. b. Use a histogram to display these same data. How does this compare to the stem and leaf plot in part a? c. Are these data symmetric or skewed? If skewed, what is the direction of the skewing? 1.45 Ages of Pennies We collected 50 pennies and recorded their ages, by calculaEX0145 ting AGE CURRENT YEAR YEAR ON PENNY. 5 1 5 0 19 17 28 0 10 0 0 3 0 20 6 25 3 2 16 0 20 25 5 0 17 9 4 19 19 23 0 8 1 22 5 1 4 21 1 36. Before drawing any graphs, try to visualize what the distribution of penny ages will look like. Will it be mound-shaped, symmetric, skewed right, or skewed left? b. Draw a relative frequency histogram to describe the distribution of penny ages. How would you describe the shape of the distribution? 1.46 Ages of Pennies, continued The data below represent the ages of a different set of EX0146 50 pennies, again calculated using AGE CURRENT YEAR YEAR ON PENNY. 3 4 41 25 0 2 4 7 3 8 5 14 28 5 0 3 14 2 3 23 8 12 5 17 17 21 24 1 16 9 3 19 20 0 2 9 10 1 9 7 0 4 14 3 3 0 0 4 0 7 a. |
Draw a relative frequency histogram to describe the distribution of penny ages. Is the shape similar to the shape of the relative frequency histogram in Exercise 1.41? b. Draw a stem and leaf plot to describe the penny ages. Are there any unusually large or small measurements in the set? 1.47 Presidential Vetoes Here is a list of the 43 presidents of the United States along EX0147 with the number of regular vetoes used by each:5 Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren W. H. Harrison Tyler Polk Taylor Fillmore. Harrison Cleveland McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. D. Roosevelt Truman Eisenhower Kennedy 19 42 6 42 30 33 5 20 21 372 180 73 12 9 Pierce 4 Buchanan 2 Lincoln 21 A. Johnson 45 Grant 12 Hayes 0 Garfield 4 Arthur Cleveland 304 Source: The World Almanac and Book of Facts 2007 L. Johnson Nixon Ford Carter Reagan G. H. W. Bush Clinton G. W. Bush 16 26 48 13 39 29 36 1 Use an appropriate graph to describe the number of vetoes cast by the 43 presidents. Write a summary paragraph describing this set of data. EX0148 1.48 Windy Cities Are some cities more windy than others? Does Chicago deserve to be nicknamed “The Windy City”? These data are the average wind speeds (in miles per hour) for 55 selected cities in the United States:5 7.8 12.9 8.9 9.2 10.3 7.1 7.8 10.5 9.1 5.5 8.7 8.8 8.3 10.7 10.2 9.2 10.2 8.7 8.2 9.3 12.2 10.5 9.5 6.2 8.4 7.7 11.3 7.6 9.6 6.9 11.5 10.5 8.8 35.1 8.0 10.2 8.8 8.7 8.8 11.0 9.4 12.2 9.0 8.7 7.9 10.4 7.7 9.6 12.4 11.8 9.0 10.8 8.6 5.8 9.0 Source: The World Almanac and Book of Facts 2007 a. Construct a relative frequency histogram for the data. (HINT: Choose the class boundaries without including the value x 35.1 in the range of values |
.) b. The value x 35.1 was recorded at Mt. Washington, New Hampshire. Does the geography of that city explain the observation? c. The average wind speed in Chicago is recorded as 10.3 miles per hour. Do you consider this unusually windy? 1.49 Kentucky Derby The following data set shows the winning times (in seconds) for EX0149 the Kentucky Derby races from 1950 to 2007.11 121.3 120.2 121.4 122.2 123.0 123.0 122.0 121.4 120.0 119.2† 124.0 122.2 122.1 123.3 122.2 122.1 (1950) 121.3 122.3 120.3 (1960) 122.2 124.0 122.1 (1970) 123.2 123.1 123.2 (1980) 122.0 122.0 122.4 (1990) 122.0 123.0 (2000) 121.0 119.97 121.13 121.19 124.06 122.75 121.36 122.17 †Record time set by Secretariat in 1973 Source: www.kentuckyderby.com 121.4 121.1 122.0 120.1 121.1 123.2 122.0 121.3 122.4 121.0 125.0 122.1 122.1 121.4 121.1 122.2 122.2 125.0 122.2 123.2 a. Do you think there will be a trend in the winning times over the years? Draw a line chart to verify your answer. b. Describe the distribution of winning times using an appropriate graph. Comment on the shape of the distribution and look for any unusual observations. 1.50 Computer Networks at Home As Americans become more knowledgeable about EX0150 computer hardware and software, as prices drop and installation becomes easier, home networking of PCs is expected to penetrate 27 percent of U.S. households by 2008, with wireless technology leading the way.12 U.S. Home Networks (in millions) Wireless Wired Year 2002 2003 2004 2005 2006 2007 2008 6.1 6.5 6.2 5.7 4.9 4.1 3.4 Source: Jupiter Research 1.7 4.5 8.7 13.7 19.1 24.0 28.2 a. What graphical methods could you use to describe the data? b. Before you draw a graph, look at the predicted number of wired and wireless households in |
the table. What trends do you expect to see in the graphs? c. Use a line chart to describe the predicted number of wired households for the years 2002 to 2008. d. Use a bar chart to describe the predicted number of wireless households for the years 2002 to 2008. 1.51 Election Results The 2004 election was a race in which the incumbent, George EX0151 W. Bush, defeated John Kerry, Ralph Nader, and other candidates, receiving 50.7% of the popular vote. The popular vote (in thousands) for George W. Bush in each of the 50 states is listed below:8 194 MA 1071 HI 1176 AL 409 MI 2314 ID 191 AK 2346 MN 1347 IL 1104 AZ 1479 MS 685 IN AR 573 572 MO 1456 IA CA 5510 266 736 MT KS CO 1101 513 NE 1069 KY 694 CT 419 NV 1102 172 DE LA 331 NH FL 330 3965 ME 1670 NJ GA 1914 MD 1025 NM 377 SD 2962 NY TN 1961 NC TX 197 ND UT 2860 OH VT VA 960 OK 867 WA OR 2794 WV PA 169 WI RI 938 WY SC 233 1384 4527 664 121 1717 1305 424 1478 168 a. By just looking at the table, what shape do you think the data distribution for the popular vote by state will have? b. Draw a relative frequency histogram to describe the distribution of the popular vote for President Bush in the 50 states. c. Did the histogram in part b confirm your guess in part a? Are there any outliers? How can you explain them? SUPPLEMENTARY EXERCISES ❍ 45 EX0152 1.52 Election Results, continued Refer to Exercise 1.51. Listed here is the percentage of the popular vote received by President Bush in each of the 50 states:8 45 HI 62 AL 68 ID 61 AK 44 IL 55 AZ 60 IN 54 AR 50 IA 44 CA 62 KS 52 CO 60 KY 44 CT 57 LA 46 DE ME 52 FL 45 MD 43 58 GA NM 50 40 NY 56 NC 63 ND 51 OH 66 OK 47 OR 48 PA 39 RI 58 SC SD TN TX UT VT VA WA WV WI WY MA MI MN MS MO MT NE NV NH NJ 37 48 48 59 53 59 66 51 49 46 60 57 61 73 39 54 46 56 49 69 a. By just looking at the table, what shape do you think the data distribution for |
the percentage of the popular vote by state will have? b. Draw a relative frequency histogram to describe the distribution. Describe the shape of the distribution and look for outliers. Did the graph confirm your answer to part a? 1.53 Election Results, continued Refer to Exercises 1.51 and 1.52. The accompanying stem and leaf plots were generated using MINITAB for the variables named “Popular Vote” and “Percent Vote.” Stem-and-Leaf Display: Popular Vote, Percent Vote Stem-and-leaf of Popular Vote N = 50 Leaf Unit = 100 Stem-and-leaf of Percent Vote N = 50 Leaf Unit = 1.0 3 799 4 03444 4 55666788899 3 8 19 (9) 5 001122344 5 566778899 22 6 00011223 13 6 6689 5 7 3 1 7 12 18 22 25 25 18 15 12 10 8 8 6 6 5 0 1111111 0 22333 0 444555 0 6667 0 899 1 0001111 1 333 1 444 1 67 1 99 2 2 33 2 2 7 2 89 HI 39, 45, 55 a. Describe the shapes of the two distributions. Are there any outliers? b. Do the stem and leaf plots resemble the relative frequency histograms constructed in Exercises 1.51 and 1.52? c. Explain why the distribution of the popular vote for President Bush by state is skewed while the 46 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS percentage of popular votes by state is moundshaped. 1.54 Student Heights The self-reported heights of 105 students in a biostatistics class EX0154 are described in the relative frequency histogram below. 10/105 5/105. Superimpose another line chart on the one drawn in part a to describe the percentage that do not approve. c. The following line chart was created using MINITAB. Does it differ from the graph that you drew? Use the line chart to summarize changes in the polls just after the terrorist attacks in Spain on March 11, 2004 and in England in July of 2005. d. A plot to bring down domestic flights from England to the United States was foiled by British undercover agents, and the arrest of 12 suspects followed on August 9, 2006. Summarize any changes in approval rating that may have been brought about following the August 9 |
th arrests. 0 60 63 66 Heights 69 72 75 Approve or Disapprove of the President’s Handling of Terrorism and Homeland Security? a. Describe the shape of the distribution. b. Do you see any unusual feature in this histogram? c. Can you think of an explanation for the two peaks in the histogram? Is there some other factor that is causing the heights to mound up in two separate peaks? What is it? EX0155 1.55 Fear of Terrorism Many opinion polls have tracked opinions regarding the fear of terrorist attacks following the September 11, 2001, attacks on the World Trade Center. A Newsweek poll conducted by Princeton Survey Research Associates International presented the results of several polls over a two-year period that asked, “Do you approve or disapprove of the way Bush is handling terrorism and homeland security?” The data are shown in the table below:13 Date Approve % Disapprove % Unsure % 8/10–11/06 5/11–12/06 3/16–17/06 11/10–11/05 9/29–30/05 9/8–9/05 8/2–4/05 3/17–18/05 4/8–9/04 3/25–26/04 2/19–20/04 55 44 44 45 51 46 51 57 59 57 65 40 50 50 49 44 48 41 35 35 38 28. Draw a line chart to describe the percentage that approve of Bush’s handling of terrorism and homeland security. Use time as the horizontal axis. 70 60 50 40 30 e s n o p s e R Opinion Approve Disapprove Feb ’04 M ar 25, ’04 A pr ’04 M ar’05 Sept 8, ’05 A ug ’05 Sept 29, ’05 N ov’05 M ar’06 A ug 10, ’06 M ay’06 Date 1.56 Pulse Rates A group of 50 biomedical students recorded their pulse rates by EX0156 counting the number of beats for 30 seconds and multiplying by 2. 80 52 60 84 84 70 72 82 84 72 88 90 88 60 62 70 70 54 84 90 84 96 66 88 72 66 84 66 58 84 84 96 80 72 72 82 86 88 84 110 66 62 56 68 100 42 78 104 74 58 a. Why are all of the measurements even numbers? b. Draw a stem and leaf plot to describe the data, splitting each |
stem into two lines. c. Construct a relative frequency histogram for the data. d. Write a short paragraph describing the distribution of the student pulse rates. 1.57 Internet On-the-Go The mobile Internet is growing, with users accessing sites such as Yahoo! Mail, the Weather Channel, ESPN, Google, Hotmail, and Mapquest from their cell phones. The most popular web browsers are shown in the table below, along with the percentage market share for each.14 Browser Openwave Motorola Nokia Access Net Front Source: www.clickz.com Market Share 27% 24% 13% 9% Browser Teleca AU Sony Ericsson RIM Blazer Market Share 6% 5% 5% 4% a. Do the percentages add up to 100%? If not, create a category called “Other” to account for the missing percentages. b. Use a pie chart to describe the market shares for the various mobile web browsers. 1.58 How Much Can You Save? An advertisement in a recent Time magazine EX0158 claimed that Geico Insurance will help you save an average of $200 per year on your automobile insurance.15 WA $178 OR $180 ID $189 NV $239 WY $189 CA $144 UT $191 AZ $188 WI $189 NY $237 NE $189 IN $203 PA $194 CT $268 IL $149 OH $208 MO $174 VA $215 MD $240 NC $127 NM $146 OK $189 TX $183 TN $235 AL $189 GA $209 SUPPLEMENTARY EXERCISES ❍ 47 Llanederyn Caldicot Island Thorns Ashley Rails 14.4 13.8 14.6 11.5 13.8 10.9 10.1 11.6 11.1 13.4 12.4 13.1 12.7 12.5 11.8 11.6 18.3 15.8 18.0 18.0 20.8 17.7 18.3 16.7 14.8 19.1 a. Construct a relative frequency histogram to describe the aluminum oxide content in the 26 pottery samples. b. What unusual feature do you see in this graph? Can you think of an explanation for this feature? c. Draw a dotplot for the data, using a letter (L, C, I, or A) to locate the data point on the horizontal scale. Does this help explain the unusual feature in part b? 1.60 The Great Calorie Debate Want |
to lose weight? You can do it by cutting calories, as long as you get enough nutritional value from the foods that you do eat! Below you will see a visual representation of the number of calories in some of America’s favorite foods adapted from an article in The Press-Enterprise.17 Number of calories A SAMPLING OF SAVINGS FL $130 26 Hershey's Kiss 53 Oreo cookie 140 12-ounce can of Coke 145 12-ounce bottle of Budweiser beer 330 Slice of a large Papa John's pepperoni pizza 800 Burger King Whopper with cheese a. Construct a relative frequency histogram to describe the average savings for the 27 states shown on the United States map. Do you see any unusual features in the histogram? b. Construct a stem and leaf plot for the data provided by Geico Insurance. a. Comment on the accuracy of the graph shown above. Do the sizes, heights, and volumes of the six items accurately represent the number of calories in the item? b. Draw an actual bar chart to describe the number of c. How do you think that Geico selected the 27 states calories in these six food favorites. for inclusion in this advertisement? EX0159 1.59 An Archeological Find An article in Archaeometry involved an analysis of 26 samples of Romano-British pottery, found at four different kiln sites in the United Kingdom.16 The samples were analyzed to determine their chemical composition, and the percentage of aluminum oxide in each of the 26 samples is shown in the following table. EX0161 1.61 Laptops and Learning An informal experiment was conducted at McNair Academic High School in Jersey City, New Jersey, to investigate the use of laptop computers as a learning tool in the study of algebra.18 A freshman class of 20 students was given laptops to use at school and at home, while another freshman class of 27 students was not given laptops; however, many of these students 48 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS were able to use computers at home. The final exam scores for the two classes are shown below. Laptops No Laptops 98 97 88 100 100 78 68 47 90 94 84 93 57 84 81 83 84 93 57 83 63 83 93 52 83 63 86 81 99 91 80 81 78 29 74 72 67 89 97 74 88 84 49 89 64 89 70 The histograms below show the distribution of final exam scores for the two groups. |
Laptops No laptops 30 40 50 60 70 80 90 100 40.30.20.10 0 30 40 50 60 70 80 90 100 Write a summary paragraph describing and comparing the distribution of final exam scores for the two groups of students. 1.62 Old Faithful The data below are the waiting times between eruptions of the Old EX0162 Faithful geyser in Yellowstone National Park.19 Use one of the graphical methods from this chapter to describe the distribution of waiting times. If there are any unusual features in your graph, see if you can think of any practical explanation for them. 56 69 55 59 76 79 75 65 68 93 89 75 87 86 94 72 78 75 87 50 51 77 53 78 75 78 64 77 61 87 79 53 85 71 50 77 80 69 81 77 58 80 61 77 83 79 49 92 55 74 82 54 93 89 82 72 49 91 93 89 52 79 54 45 72 82 88 53 53 87 88 74 76 93 77 74 51 86 84 76 52 65 80 72 75 80 78 49 70 59 78 78 81 71 65 49 85 79 73 80 1.63 Gasoline Tax The following are the 2006 state gasoline tax rates in cents per gallon EX0163 for the 50 United States and the District of Columbia.5 HI AL 18.0 ID AK 8.0 IL AZ 18.0 IN 18.0 MS AR 20.0 21.0 MO IA CA 18.0 24.0 MT KS CO 22.0 NE KY 19.0 CT 25.0 NV DE 23.0 LA 20.0 NH DC 20.0 ME 25.9 NJ 14.9 MD 23.5 FL GA 10.0 21.0 16.0 MA 25.0 MI 19.0 19.0 MN 20.0 18.0 17.0 27.0 26.1 23.0 18.0 10.5 SD NM 17.0 TN 23.9 NY TX NC 29.9 UT ND 23.0 VT OH 28.0 20.0 OK VA 24.0 WA OR 32.0 WV PA 30.0 WI RI 16.0 WY SC 20.0 20.0 20.0 24.5 19.0 17.5 31.0 20.5 32.9 14.0 Source: The World Almanac and Book of Facts 2007 a. Construct a stem and leaf display for the data. b. How would you describe the shape of this distribution? c. Are there states with unusually high or low gasoline |
taxes? If so, which states are they? 1.64 Hydroelectric Plants The following data represent the planned rated capacities in EX0164 megawatts (millions of watts) for the world’s 20 largest hydroelectric plants.5 18,200 14,000 10,000 8,370 6,400 6,300 6,000 4,500 4,200 4,200 3,840 3,230 3,300 3,100 3,000 2,940 2,715 2,700 2,541 2,512 Source: The World Almanac and Book of Facts 2007 a. Construct a stem and leaf display for the data. b. How would you describe the shape of this distribution? 1.65 Car Colors The most popular colors for compact and sports cars in a recent year are EX0165 given in the table.5 Color Percentage Silver Gray Blue Black White 20 17 16 14 10 Color Percentage Red Green Light Brown Yellow/Gold Other 9 6 5 1 2 Source: The World Almanac and Book of Facts 2007 Use an appropriate graphical display to describe these data. 1.66 Starbucks The number of Starbucks coffee shops in cities within 20 miles of the EX0166 University of California, Riverside is shown in the following table.20 City Starbucks City Starbucks Riverside Grand Terrace Rialto Colton San Bernardino Redlands Corona Yucaipa Chino 16 1 3 2 5 7 7 2 1 Source: www.starbucks.com Ontario Norco Fontana Mira Loma Perris Highland Rancho Cucamonga Lake Elsinore Moreno Valley 11 4 6 1 1 1 12 1 4 a. Draw a dotplot to describe the data. b. Describe the shape of the distribution. c. Is there another variable that you could measure that might help to explain why some cities have more Starbucks than others? Explain. 1.67 What’s Normal? The 98.6 degree standard for human body temperature was EX0167 derived by a German doctor in 1868. In an attempt to verify his claim, Mackowiak, Wasserman, and Levine21 took temperatures from 148 healthy people over a three-day period. A data set closely matching the one in Mackowiak’s article was derived by Allen Shoemaker, and appears in the Journal of Statistics Education.22 The body temperatures for these 130 individuals are shown in the relative frequency histogram that follows. Exercises 1.68 If you have not yet done so, use the first applet |
in Building a Dotplot to create a dotplot for the following data set: 2, 3, 9, 6, 7, 6. 1.69 Cheeseburgers Use the second applet in Building a Dotplot to create a dotplot for the number of cheeseburgers consumed in a given week by 10 college students. How would you describe the shape of the distribution? b. What proportion of the students ate more than 4 cheeseburgers that week? 1.70 Social Security Numbers A group of 70 students were asked to record the last digit EX0170 of their Social Security number. MYAPPLET EXERCISES ❍ 49 25.20.15.10.05 0 96.8 97.6 98.4 99.2 100.0 100.8 Temperature a. Describe the shape of the distribution of temperatures. b. Are there any unusual observations? Can you think of any explanation for these? c. Locate the 98.6-degree standard on the horizontal axis of the graph. Does it appear to be near the center of the distribution. Before graphing the data, use your common sense to guess the shape of the data distribution. Explain your reasoning. b. Use the second applet in Building a Dotplot to create a dotplot to describe the data. Was your intuition correct in part a? 1.71 If you have not yet done so, use the first applet in Building a Histogram to create a histogram for the data in Example 1.11, the number of visits to Starbucks during a typical week. 50 ❍ CHAPTER 1 DESCRIBING DATA WITH GRAPHS 1.72 The United Fund The following data set records the yearly charitable contributions EX0172 (in dollars) to the United Fund for a group of employees at a public university. 41 28 77 42 81 51 75 78 80 112 59 81 65 71 63 90 47 83 63 103 56 84 80 125 80 82 101 92 69 103 115 79 79 80 99 24 63 70 67 93 Use the second applet in Building a Histogram to construct a relative frequency histogram for the data. What is the shape of the distribution? Can you see any obvious outliers? 1.73 Survival Times Altman and Bland report the survival times for patients with EX0173 active hepatitis, half treated with prednisone and half receiving no treatment.23 The data that follow are adapted from their data for those treated with prednisone. The survival times |
are recorded to the nearest month: 8 11 52 57 65 87 93 97 109 120 127 133 139 142 144 147 148 157 162 165 a. Look at the data. Can you guess the approximate shape of the data distribution? b. Use the second applet in Building a Histogram to construct a relative frequency histogram for the data. What is the shape of the distribution? c. Are there any outliers in the set? If so, which survival times are unusually short? CASE STUDY Blood Pressure How Is Your Blood Pressure? Blood pressure is the pressure that the blood exerts against the walls of the arteries. When physicians or nurses measure your blood pressure, they take two readings. The systolic blood pressure is the pressure when the heart is contracting and therefore pumping. The diastolic blood pressure is the pressure in the arteries when the heart is relaxing. The diastolic blood pressure is always the lower of the two readings. Blood pressure varies from one person to another. It will also vary for a single individual from day to day and even within a given day. If your blood pressure is too high, it can lead to a stroke or a heart attack. If it is too low, blood will not get to your extremities and you may feel dizzy. Low blood pressure is usually not serious. So, what should your blood pressure be? A systolic blood pressure of 120 would be considered normal. One of 150 would be high. But since blood pressure varies with gender and increases with age, a better gauge of the relative standing of your blood pressure would be obtained by comparing it with the population of blood pressures of all persons of your gender and age in the United States. Of course, we cannot supply you with that data set, but we can show you a very large sample selected from it. The blood pressure data on 1910 persons, 965 men and 945 women between the ages of 15 and 20, are found at the Student Companion Website. The data are part of a health survey conducted by the National Institutes of Health (NIH). Entries for each person include that person’s age and systolic and diastolic blood pressures at the time the blood pressure was recorded. 1. Describe the variables that have been measured in this survey. Are the variables quantitative or qualitative? Discrete or continuous? Are the data univariate, bivariate, or multivariate? 2. What types of graphical methods are available for describing this data set? What types of questions could be answered using |
various types of graphical techniques? CASE STUDY ❍ 51 3. Using the systolic blood pressure data set, construct a relative frequency histogram for the 965 men and another for the 945 women. Use a statistical software package if you have access to one. Compare the two histograms. 4. Consider the 965 men and 945 women as the entire population of interest. Choose a sample of n 50 men and n 50 women, recording their systolic blood pressures and their ages. Draw two relative frequency histograms to graphically display the systolic blood pressures for your two samples. Do the shapes of the histograms resemble the population histograms from part 3? 5. How does your blood pressure compare with that of others of your same gender? Check your systolic blood pressure against the appropriate histogram in part 3 or 4 to determine whether your blood pressure is “normal” or whether it is unusually high or low. 2 Describing Data with Numerical Measures © Joe Sohm-VisionsofAmerica/Photodisc/Getty GENERAL OBJECTIVES Graphs are extremely useful for the visual description of a data set. However, they are not always the best tool when you want to make inferences about a population from the information contained in a sample. For this purpose, it is better to use numerical measures to construct a mental picture of the data. CHAPTER INDEX ● Box plots (2.7) ● Measures of center: mean, median, and mode (2.2) ● Measures of relative standing: z-scores, percentiles, quartiles, and the interquartile range (2.6) ● Measures of variability: range, variance, and standard deviation (2.3) ● Tchebysheff’s Theorem and the Empirical Rule (2.4) How Do I Calculate Sample Quartiles? The Boys of Summer Are the baseball champions of today better than those of “yesteryear”? Do players in the National League hit better than players in the American League? The case study at the end of this chapter involves the batting averages of major league batting champions. Numerical descriptive measures can be used to answer these and similar questions. 52 2.2 MEASURES OF CENTER ❍ 53 DESCRIBING A SET OF DATA WITH NUMERICAL MEASURES 2.1 Graphs can help you describe the basic shape of a data distribution; “a picture is worth a |
thousand words.” There are limitations, however, to the use of graphs. Suppose you need to display your data to a group of people and the bulb on the data projector blows out! Or you might need to describe your data over the telephone—no way to display the graphs! You need to find another way to convey a mental picture of the data to your audience. A second limitation is that graphs are somewhat imprecise for use in statistical inference. For example, suppose you want to use a sample histogram to make inferences about a population histogram. How can you measure the similarities and differences between the two histograms in some concrete way? If they were identical, you could say “They are the same!” But, if they are different, it is difficult to describe the “degree of difference.” One way to overcome these problems is to use numerical measures, which can be calculated for either a sample or a population of measurements. You can use the data to calculate a set of numbers that will convey a good mental picture of the frequency distribution. These measures are called parameters when associated with the population, and they are called statistics when calculated from sample measurements. Definition Numerical descriptive measures associated with a population of measurements are called parameters; those computed from sample measurements are called statistics. MEASURES OF CENTER 2.2 In Chapter 1, we introduced dotplots, stem and leaf plots, and histograms to describe the distribution of a set of measurements on a quantitative variable x. The horizontal axis displays the values of x, and the data are “distributed” along this horizontal line. One of the first important numerical measures is a measure of center—a measure along the horizontal axis that locates the center of the distribution. The birth weight data presented in Table 1.9 ranged from a low of 5.6 to a high of 9.4, with the center of the histogram located in the vicinity of 7.5 (see Figure 2.1). Let’s consider some rules for locating the center of a distribution of measurements. FI GUR E 2.1 Center of the birth weight data ● /30 7/30 6/30 5/30 4/30 3/30 2/30 1/30 0 5.6 6.1 6.6 7.1 7.6 Center Birth Weights 8.1 8.6 9.1 9.6 54 ❍ CHAPTER 2 DESCRI |
BING DATA WITH NUMERICAL MEASURES The arithmetic average of a set of measurements is a very common and useful measure of center. This measure is often referred to as the arithmetic mean, or simply the mean, of a set of measurements. To distinguish between the mean for the sample and the mean for the population, we will use the symbol x (x-bar) for a sample mean and the symbol m (Greek lowercase mu) for the mean of a population. Definition The arithmetic mean or average of a set of n measurements is equal to the sum of the measurements divided by n. Since statistical formulas often involve adding or “summing” numbers, we use a shorthand symbol to indicate the process of summing. Suppose there are n measurements on the variable x—call them x1, x2,..., xn. To add the n measurements together, we use this shorthand notation: n i1 xi which means x1 x2 x3 xn The Greek capital sigma (S) tells you to add the items that appear to its right, beginning with the number below the sigma (i 1) and ending with the number above (i n). However, since the typical sums in statistical calculations are almost always made on the total set of n measurements, you can use a simpler notation: Sxi which means “the sum of all the x measurements” Using this notation, we write the formula for the sample mean: NOTATION Sample mean: S xi x n Population mean: m EXAMPLE 2.1 Draw a dotplot for the n 5 measurements 2, 9, 11, 5, 6. Find the sample mean and compare its value with what you might consider the “center” of these observations on the dotplot. Solution The dotplot in Figure 2.2 seems to be centered between 6 and 8. To find the sample mean, calculate S xi x n 2 9 11 5 6 5 6.6 F IG URE 2. 2 Dotplot for Example 2.1 ● 2 4 6 8 10 Measurements The statistic x 6.6 is the balancing point or fulcrum shown on the dotplot. It does seem to mark the center of the data. mean balancing point or fulcrum Remember that samples are measurements drawn from a larger population that is usually unknown. An important use of the sample mean x is as an estimator of the unknown population mean m. The birth weight data in Table 1. |
9 are a sample from a larger population of birth weights, and the distribution is shown in Figure 2.1. The mean of the 30 birth weights is 2.2 MEASURES OF CENTER ❍ 55 S xi 22.2 7.57 7 x 0 3 30 shown in Figure 2.1; it marks the balancing point of the distribution. The mean of the entire population of newborn birth weights is unknown, but if you had to guess its value, your best estimate would be 7.57. Although the sample mean x changes from sample to sample, the population mean m stays the same. A second measure of central tendency is the median, which is the value in the mid- dle position in the set of measurements ordered from smallest to largest. Definition The median m of a set of n measurements is the value of x that falls in the middle position when the measurements are ordered from smallest to largest. EXAMPLE 2.2 Find the median for the set of measurements 2, 9, 11, 5, 6. Solution Rank the n 5 measurements from smallest to largest: 2 5 6 9 11 The middle observation, marked with an arrow, is in the center of the set, or m 6. EXAMPLE 2.3 Find the median for the set of measurements 2, 9, 11, 5, 6, 27. Solution Rank the measurements from smallest to largest: 2 5 6 9 11 27 Roughly 50% of the measurements are smaller, 50% are larger than the median. EXAMPLE 2.4 Now there are two “middle” observations, shown in the box. To find the median, choose a value halfway between the two middle observations: m 6 9 7.5 2 The value.5(n 1) indicates the position of the median in the ordered data set. If the position of the median is a number that ends in the value.5, you need to average the two adjacent values. For the n 5 ordered measurements from Example 2.2, the position of the median is.5(n 1).5(6) 3, and the median is the 3rd ordered observation, or m 6. For the n 6 ordered measurements from Example 2.3, the position of the median is.5(n 1).5(7) 3.5, and the median is the average of the 3rd and 4th ordered observations, or m (6 9)/2 7.5. 56 ❍ CHAPTER 2 DES |
CRIBING DATA WITH NUMERICAL MEASURES symmetric: mean median skewed right: mean median skewed left: mean median Although both the mean and the median are good measures of the center of a distribution, the median is less sensitive to extreme values or outliers. For example, the value x 27 in Example 2.3 is much larger than the other five measurements. The median, m 7.5, is not affected by the outlier, whereas the sample average, S xi 6 0 10 x n 6 is affected; its value is not representative of the remaining five observations. When a data set has extremely small or extremely large observations, the sample mean is drawn toward the direction of the extreme measurements (see Figure 2.3). ● F IG URE 2. 3 Relative frequency distributions showing the effect of extreme values on the mean and median 25.19.12.06 0 (a) (b25.19.12.06 0 Mean Median Mean Median If a distribution is skewed to the right, the mean shifts to the right; if a distribution is skewed to the left, the mean shifts to the left. The median is not affected by these extreme values because the numerical values of the measurements are not used in its calculation. When a distribution is symmetric, the mean and the median are equal. If a distribution is strongly skewed by one or more extreme values, you should use the median rather than the mean as a measure of center. You can see the effect of extreme values on both the mean and the median using the How Extreme Values Affect the Mean and Median applet. The first of three applets (Figure 2.4) shows a dotplot of the data in Example 2.2. Use your mouse to move the largest observation (x 11) even further to the right. How does this larger observation affect the mean? How does it affect the median? We will use this applet again for the MyApplet Exercises at the end of the chapter. ● F IG URE 2. 4 How Extreme Values Affect the Mean and Median applet 2.2 MEASURES OF CENTER ❍ 57 Another way to locate the center of a distribution is to look for the value of x that occurs with the highest frequency. This measure of the center is called the mode. Definition The mode is the category that occurs most frequently, or the most frequently occurring value of x. When measurements on a continuous variable have been grouped as a |
frequency or relative frequency histogram, the class with the highest peak or frequency is called the modal class, and the midpoint of that class is taken to be the mode. The mode is generally used to describe large data sets, whereas the mean and median are used for both large and small data sets. From the data in Example 1.11, the mode of the distribution of the number of reported weekly visits to Starbucks for 30 Starbucks customers is 5. The modal class and the value of x occurring with the highest frequency are the same, as shown in Figure 2.5(a). For the data in Table 1.9, a birth weight of 7.7 occurs four times, and therefore the mode for the distribution of birth weights is 7.7. Using the histogram to find the modal class, you find that the class with the highest peak is the fifth class, from 7.6 to 8.1. Our choice for the mode would be the midpoint of this class, or 7.85. See Figure 2.5(b). It is possible for a distribution of measurements to have more than one mode. These modes would appear as “local peaks” in the relative frequency distribution. For example, if we were to tabulate the length of fish taken from a lake during one season, we might get a bimodal distribution, possibly reflecting a mixture of young and old fish in the population. Sometimes bimodal distributions of sizes or weights reflect a mixture of measurements taken on males and females. In any case, a set or distribution of measurements may have more than one mode. Remember that there can be several modes or no mode (if each observation occurs only once). (a) (b/30 7/30 6/30 5/30 4/30 3/30 2/30 1/30 0 1 2 3 4 5 Visits 6 7 8 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6 Birth Weights ● FI GUR E 2.5 Relative frequency histograms for the Starbucks and birth weight data /25 6/25 4/25 2/25 0 2.2 EXERCISES BASIC TECHNIQUES 2.1 You are given n 5 measurements: 0, 5, 1, 1, 3. a. Draw a |
dotplot for the data. (HINT: If two measurements are the same, place one dot above the other.) Guess the approximate “center.” b. Find the mean, median, and mode. c. Locate the three measures of center on the dotplot in part a. Based on the relative positions of the mean and median, are the measurements symmetric or skewed? 58 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES 2.2 You are given n 8 measurements: 3, 2, 5, 6, 4, 4, 3, 5. a. Find x. b. Find m. c. Based on the results of parts a and b, are the measurements symmetric or skewed? Draw a dotplot to confirm your answer. 2.3 You are given n 10 measurements: 3, 5, 4, 6, 10, 5, 6, 9, 2, 8. a. Calculate x. b. Find m. c. Find the mode. APPLICATIONS 2.4 Auto Insurance The cost of automobile insurance has become a sore subject in California because insurance rates are dependent on so many different variables, such as the city in which you live, the number of cars you insure, and the company with which you are insured. The website www.insurance.ca.gov reports the annual 2006–2007 premium for a single male, licensed for 6–8 years, who drives a Honda Accord 12,600 to 15,000 miles per year and has no violations or accidents.1 City Allstate 21st Century Long Beach Pomona San Bernardino Moreno Valley $2617 2305 2286 2247 Source: www.insurance.ca.gov $2228 2098 2064 1890 a. What is the average premium for Allstate Insurance? b. What is the average premium for 21st Century Insurance? c. If you were a consumer, would you be interested in the average premium cost? If not, what would you be interested in? EX0205 2.5 DVD Players The DVD player is a common fixture in most American households. In fact, most American households have DVDs, and many have more than one. A sample of 25 households produced the following measurements on x, the number of DVDs in the household. Is the distribution of x, the number of DVDs in a household, symmetric or skewed? Explain. b. Guess the value of the mode, the value of x that |
occurs most frequently. c. Calculate the mean, median, and mode for these measurements. d. Draw a relative frequency histogram for the data set. Locate the mean, median, and mode along the horizontal axis. Are your answers to parts a and b correct? 2.6 Fortune 500 Revenues Ten of the 50 largest businesses in the United States, randomly selected from the Fortune 500, are listed below along with their revenues (in millions of dollars):2 Company Revenues Company Revenues General Motors IBM Bank of America Home Depot Boeing $192,604 91,134 83,980 81,511 54,848 Source: Time Almanac 2007 Target Morgan Stanley Johnson & Johnson Intel Safeway $52,620 52,498 50,514 38,826 38,416 a. Draw a stem and leaf plot for the data. Are the data skewed? b. Calculate the mean revenue for these 10 businesses. Calculate the median revenue. c. Which of the two measures in part b best describes the center of the data? Explain. 2.7 Birth Order and Personality Does birth order have any effect on a person’s personality? A report on a study by an MIT researcher indicates that later-born children are more likely to challenge the establishment, more open to new ideas, and more accepting of change.3 In fact, the number of later-born children is increasing. During the Depression years of the 1930s, families averaged 2.5 children (59% later born), whereas the parents of baby boomers averaged 3 to 4 children (68% later born). What does the author mean by an average of 2.5 children? 2.8 Tuna Fish An article in Consumer Reports gives the price—an estimated average EX0208 for a 6-ounce can or a 7.06-ounce pouch—for 14 different brands of water-packed light tuna, based on prices paid nationally in supermarkets:4.99 1.12 1.92.63 1.23.67.85.69.65.60.53.60 1.41.66 a. Find the average price for the 14 different brands of tuna. b. Find the median price for the 14 different brands of tuna. c. Based on your findings in parts a and b, do you think that the distribution of prices is skewed? Explain. 2.9 Sports Salaries As professional sports teams become a more and more lucrative business for their owners, the salaries paid to the |
players have also increased. In fact, sports superstars are paid astronomical salaries for their talents. If you were asked by a sports management firm to describe the distribution of players’ salaries in several different categories of professional sports, what measure of center would you choose? Why? 2.10 Time on Task In a psychological experiment, the time on task was recorded for 10 subjects under a 5-minute time constraint. These measurements are in seconds: 175 200 190 185 250 190 230 225 240 265 2.2 MEASURES OF CENTER ❍ 59 2.11 Starbucks The number of Starbucks coffee shops in 18 cities within 20 miles of the EX0211 University of California, Riverside is shown in the following table (www.starbucks.com).5 16 1 3 5 7 7 2 1 2 1 11 4 6 1 1 12 4 1 a. Find the mean, the median, and the mode. b. Compare the median and the mean. What can you say about the shape of this distribution? c. Draw a dotplot for the data. Does this confirm your conclusion about the shape of the distribution from part b? 2.12 HDTVs The cost of televisions exhibits huge variation—from $100–200 for a standard EX0212 TV to $8,000–10,000 for a large plasma screen TV. Consumer Reports gives the prices for the top 10 LCD high definition TVs (HDTVs) in the 30- to 40-inch category:6 Brand JVC LT-40FH96 Sony Bravia KDL-V32XBR1 Sony Bravia KDL-V40XBR1 Toshiba 37HLX95 Sharp Aquos LC-32DA5U Sony Bravia KLV-S32A10 Panasonic Viera TC-32LX50 JVC LT-37X776 LG 37LP1D Samsung LN-R328W Price $2900 1800 2600 3000 1300 1500 1350 2000 2200 1200 a. Find the average time on task. b. Find the median time on task. c. If you were writing a report to describe these data, which measure of central tendency would you use? Explain. a. What is the average price of these 10 HDTVs? b. What is the median price of these 10 HDTVs? c. As a consumer, would you be interested in the average cost of an HDTV? What other variables would be important to you? 60 ❍ |
CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES MEASURES OF VARIABILITY 2.3 Data sets may have the same center but look different because of the way the numbers spread out from the center. Consider the two distributions shown in Figure 2.6. Both distributions are centered at x 4, but there is a big difference in the way the measurements spread out, or vary. The measurements in Figure 2.6(a) vary from 3 to 5; in Figure 2.6(b) the measurements vary from 0 to 8. F IG URE 2. 6 Variability or dispersion of data ● (a) (b Variability or dispersion is a very important characteristic of data. For example, if you were manufacturing bolts, extreme variation in the bolt diameters would cause a high percentage of defective products. On the other hand, if you were trying to discriminate between good and poor accountants, you would have trouble if the examination always produced test grades with little variation, making discrimination very difficult. Measures of variability can help you create a mental picture of the spread of the data. We will present some of the more important ones. The simplest measure of variation is the range. Definition The range, R, of a set of n measurements is defined as the difference between the largest and smallest measurements. For the birth weight data in Table 1.9, the measurements vary from 5.6 to 9.4. Hence, the range is 9.4 5.6 3.8. The range is easy to calculate, easy to interpret, and is an adequate measure of variation for small sets of data. But, for large data sets, the range is not an adequate measure of variability. For example, the two relative frequency distributions in Figure 2.7 have the same range but very different shapes and variability. ● F IG URE 2. 7 Distributions with equal range and unequal variability a) (b.3 MEASURES OF VARIABILITY ❍ 61 Is there a measure of variability that is more sensitive than the range? Consider, as an example, the sample measurements 5, 7, 1, 2, 4, displayed as a dotplot in Figure 2.8. The mean of these five measurements is ● FI GUR E 2.8 Dotplot showing the deviations of points from the mean S xi 1 9 3.8 x n 5 x = 3.8 (xi – x as indicated on the dotplot |
. The horizontal distances between each dot (measurement) and the mean x will help you to measure the variability. If the distances are large, the data are more spread out or variable than if the distances are small. If xi is a particular dot (measurement), then the deviation of that measurement from the mean is (xi x). Measurements to the right of the mean produce positive deviations, and those to the left produce negative deviations. The values of x and the deviations for our example are listed in the first and second columns of Table 2.1. TABLE 2.1 ● Computation of S(xi x )2 (xi xx)2 (xi xx) x 5 7 1 2 4 19 1.2 3.2 2.8 1.8.2 0.0 1.44 10.24 7.84 3.24.04 22.80 Because the deviations in the second column of the table contain information on variability, one way to combine the five deviations into one numerical measure is to average them. Unfortunately, the average will not work because some of the deviations are positive, some are negative, and the sum is always zero (unless round-off errors have been introduced into the calculations). Note that the deviations in the second column of Table 2.1 sum to zero. Another possibility might be to disregard the signs of the deviations and calculate the average of their absolute values.† This method has been used as a measure of variability in exploratory data analysis and in the analysis of time series data. We prefer, however, to overcome the difficulty caused by the signs of the deviations by working †The absolute value of a number is its magnitude, ignoring its sign. For example, the absolute value of 2, represented by the symbol 2, is 2. The absolute value of 2—that is, 2—is 2. 62 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES with their sum of squares. From the sum of squared deviations, a single measure called the variance is calculated. To distinguish between the variance of a sample and the variance of a population, we use the symbol s2 for a sample variance and s 2 (Greek lowercase sigma) for a population variance. The variance will be relatively large for highly variable data and relatively small for less variable data. Definition The variance of a population of N measurements is the average of the squares of the deviations of the measurements about their mean m. |
The population variance is denoted by s 2 and is given by the formula S(xi m)2 s 2 N Most often, you will not have all the population measurements available but will need to calculate the variance of a sample of n measurements. Definition The variance of a sample of n measurements is the sum of the squared deviations of the measurements about their mean x divided by (n 1). The sample variance is denoted by s 2 and is given by the formula S( x)2 xi s 2 n 1 For the set of n 5 sample measurements presented in Table 2.1, the square of the deviation of each measurement is recorded in the third column. Adding, we obtain The variance and the standard deviation cannot be negative numbers. S(xi x)2 22.80 and the sample variance is S( x)2 xi.80 5.70 22 s 2 n 4 1 The variance is measured in terms of the square of the original units of measurement. If the original measurements are in inches, the variance is expressed in square inches. Taking the square root of the variance, we obtain the standard deviation, which returns the measure of variability to the original units of measurement. Definition The standard deviation of a set of measurements is equal to the positive square root of the variance. NOTATION n: number of measurements in the N: number of measurements in the sample s 2: sample variance s s 2: sample standard deviation population s 2: population variance s s 2: population standard deviation If you are using your calculator, make sure to choose the correct key for the sample standard deviation. 2.3 MEASURES OF VARIABILITY ❍ 63 For the set of n 5 sample measurements in Table 2.1, the sample variance is s2 5.70, so the sample standard deviation is s s2 5.70 2.39. The more variable the data set is, the larger the value of s. For the small set of measurements we used, the calculation of the variance is not too difficult. However, for a larger set, the calculations can become very tedious. Most scientific calculators have built-in programs that will calculate x and s or m and s, so that your computational work will be minimized. The sample or population mean key is usually marked with x. The sample standard deviation key is usually marked with s, sx, or sxn1, and the population standard deviation key with s, sx, or sxn. In using any |
calculator with these built-in function keys, be sure you know which calculation is being carried out by each key! If you need to calculate s2 and s by hand, it is much easier to use the alternative computing formula given next. This computational form is sometimes called the shortcut method for calculating s2. THE COMPUTING FORMULA FOR CALCULATING s2 i (S xi)2 Sx2 n n 1 s2 The symbols (Sxi)2 and Sx 2 i in the computing formula are shortcut ways to indicate the arithmetic operation you need to perform. You know from the formula for the sample mean that Sxi is the sum of all the measurements. To find Sx 2 i, you square each individual measurement and then add them together. Sx 2 i Sum of the squares of the individual measurements (Sxi)2 Square of the sum of the individual measurements The sample standard deviation, s, is the positive square root of s 2. EXAMPLE 2.5 Calculate the variance and standard deviation for the five measurements in Table 2.2, which are 5, 7, 1, 2, 4. Use the computing formula for s 2 and compare your results with those obtained using the original definition of s 2. TABLE 2.2 ● Table for Simplified Calculation of s2 and s xi 5 7 1 2 4 19 x 2 i 25 49 1 4 16 95 64 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES s 2 Solution The entries in Table 2.2 are the individual measurements, xi, and their i, together with their sums. Using the computing formula for s 2, you have squares, x 2 i (S xi)2 Sx 2 n n 1 9)2 95 (1 5 4.80 5.70 22 4 and s s 2 5.70 2.39, as before. You may wonder why you need to divide by (n 1) rather than n when computing the sample variance. Just as we used the sample mean x to estimate the population mean m, you may want to use the sample variance s 2 to estimate the population variance s 2. It turns out that the sample variance s 2 with (n 1) in the denominator provides better estimates of s 2 than would an estimator calculated with n in the denominator. For this reason, we always divide by (n 1) when computing the sample variance |
s 2 and the sample standard deviation s. You can compare the accuracy of estimators of the population variance s 2 using the Why Divide by n 1? applet. The applet selects samples from a population with standard deviation s 29.2. It then calculates the standard deviation s using (n 1) in the denominator as well as a standard deviation calculated using n in the denominator. You can choose to compare the estimators for a single new sample, for 10 samples, or for 100 samples. Notice that each of the 10 samples shown in Figure 2.9 has a different sample standard deviation. However, when the 10 standard deviations are averaged at the bottom of the applet, one of the two estimators is closer to the population standard deviation, s 29.2. Which one is it? We will use this applet again for the MyApplet Exercises at the end of the chapter. Don’t round off partial results as you go along! F IG URE 2. 9 Why Divide by n 1? applet ● At this point, you have learned how to compute the variance and standard deviation of a set of measurements. Remember these points: 2.3 MEASURES OF VARIABILITY ❍ 65 • The value of s is always greater than or equal to zero. • The larger the value of s 2 or s, the greater the variability of the data set. • • If s 2 or s is equal to zero, all the measurements must have the same value. In order to measure the variability in the same units as the original observations, we compute the standard deviation s s2. This information allows you to compare several sets of data with respect to their locations and their variability. How can you use these measures to say something more specific about a single set of data? The theorem and rule presented in the next section will help answer this question. 2.3 EXERCISES BASIC TECHNIQUES 2.13 You are given n 5 measurements: 2, 1, 1, 3, 5. a. Calculate the sample mean, x. b. Calculate the sample variance, s 2, using the formula 2.16 You are given n 8 measurements: 3, 1, 5, 6, 4, 4, 3, 5. a. Calculate the range. b. Calculate the sample mean. c. Calculate the sample variance and standard given by the definition. deviation. c. |
Find the sample standard deviation, s. d. Find s 2 and s using the computing formula. Compare the results with those found in parts b and c. d. Compare the range and the standard deviation. The range is approximately how many standard deviations? 2.14 Refer to Exercise 2.13. a. Use the data entry method in your scientific calculator to enter the five measurements. Recall the proper memories to find the sample mean and standard deviation. b. Verify that the calculator provides the same values for x and s as in Exercise 2.13, parts a and c. 2.15 You are given n 8 measurements: 4, 1, 3, 1, 3, 1, 2, 2. a. Find the range. b. Calculate x. c. Calculate s 2 and s using the computing formula. d. Use the data entry method in your calculator to find x, s, and s 2. Verify that your answers are the same as those in parts b and c. APPLICATIONS 2.17 An Archeological Find, again An article in Archaeometry involved an analysis of 26 samples of Romano-British pottery found at four different kiln sites in the United Kingdom.7 The samples were analyzed to determine their chemical composition. The percentage of iron oxide in each of five samples collected at the Island Thorns site was: 1.28, 2.39, 1.50, 1.88, 1.51 a. Calculate the range. b. Calculate the sample variance and the standard deviation using the computing formula. c. Compare the range and the standard deviation. The range is approximately how many standard deviations? 66 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES 2.18 Utility Bills in Southern California The monthly utility bills for a household in EX0218 Riverside, California, were recorded for 12 consecutive months starting in January 2006: a. Calculate the range of the utility bills for the year 2006. b. Calculate the average monthly utility bill for the year 2006. Month January February March April May June Amount ($) Month Amount ($) c. Calculate the standard deviation for the 2006 utility $266.63 163.41 219.41 162.64 187.16 289.17 July August September October November December $306.55 335.48 343.50 226.80 208.99 230.46 bills. ON THE PRACTICAL SIGNIFICANCE |
OF THE STANDARD DEVIATION 2.4 We now introduce a useful theorem developed by the Russian mathematician Tchebysheff. Proof of the theorem is not difficult, but we are more interested in its application than its proof. Tchebysheff’s Theorem Given a number k greater than or equal to 1 and a set of n measurements, at least [1 (1/k 2)] of the measurements will lie within k standard deviations of their mean. Tchebysheff’s Theorem applies to any set of measurements and can be used to describe either a sample or a population. We will use the notation appropriate for populations, but you should realize that we could just as easily use the mean and the standard deviation for the sample. The idea involved in Tchebysheff’s Theorem is illustrated in Figure 2.10. An interval is constructed by measuring a distance ks on either side of the mean m. The number k can be any number as long as it is greater than or equal to 1. Then Tchebysheff’s Theorem states that at least 1 (1/k 2) of the total number n measurements lies in the constructed interval. F IG URE 2. 10 Illustrating Tchebysheff’s Theorem ● At least 1 – (1/k2) µ kσ kσ x 2.4 ON THE PRACTICAL SIGNIFICANCE OF THE STANDARD DEVIATION ❍ 67 In Table 2.3, we choose a few numerical values for k and compute [1 (1/k2)]. TABLE 2.3 ● Illustrative Values of [1 (1/k2)] k 1 2 3 1 (1/k 2) 1 1 0 1 1/4 3/4 1 1/9 8/9 EXAMPLE 2.6 From the calculations in Table 2.3, the theorem states: • At least none of the measurements lie in the interval m s to m s. • At least 3/4 of the measurements lie in the interval m 2s to m 2s. • At least 8/9 of the measurements lie in the interval m 3s to m 3s. Although the first statement is not at all helpful, the other two values of k provide valuable information about the proportion of measurements that fall in certain intervals. The values k 2 and k 3 are not the only values of k you can use; for example, the proportion of |
measurements that fall within k 2.5 standard deviations of the mean is at least 1 [1/(2.5)2].84. The mean and variance of a sample of n 25 measurements are 75 and 100, respectively. Use Tchebysheff’s Theorem to describe the distribution of measurements. Solution You are given x 75 and s 2 100. The standard deviation is s 100 10. The distribution of measurements is centered about x 75, and Tchebysheff’s Theorem states: • At least 3/4 of the 25 measurements lie in the interval x 2s 75 2(10) —that is, 55 to 95. • At least 8/9 of the measurements lie in the interval x 3s 75 3(10)—that is, 45 to 105. Since Tchebysheff’s Theorem applies to any distribution, it is very conservative. This is why we emphasize “at least 1 (1/k 2)” in this theorem. Another rule for describing the variability of a data set does not work for all data sets, but it does work very well for data that “pile up” in the familiar mound shape shown in Figure 2.11. The closer your data distribution is to the mound-shaped curve in Figure 2.11, the more accurate the rule will be. Since mound-shaped data distributions occur quite frequently in nature, the rule can often be used in practical applications. For this reason, we call it the Empirical Rule. FI GUR E 2.1 1 Mound-shaped distribution ● 68 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES Remember these three numbers: 68—95—99.7 EXAMPLE 2.7 Empirical Rule Given a distribution of measurements that is approximately mound-shaped: The interval (m s) contains approximately 68% of the measurements. The interval (m 2s) contains approximately 95% of the measurements. The interval (m 3s) contains approximately 99.7% of the measurements. The mound-shaped distribution shown in Figure 2.11 is commonly known as the normal distribution and will be discussed in detail in Chapter 6. In a time study conducted at a manufacturing plant, the length of time to complete a specified operation is measured for each of n 40 workers. The mean and standard deviation are found to be 12.8 and 1.7, respectively. Describe the |
sample data using the Empirical Rule. Solution To describe the data, calculate these intervals: (x s) 12.8 1.7 (x 2s) 12.8 2(1.7) or (x 3s) 12.8 3(1.7) or or 11.1 to 14.5 9.4 to 16.2 7.7 to 17.9 EXAMPLE 2.8 TABLE 2.4 According to the Empirical Rule, you expect approximately 68% of the measurements to fall into the interval from 11.1 to 14.5, approximately 95% to fall into the interval from 9.4 to 16.2, and approximately 99.7% to fall into the interval from 7.7 to 17.9. If you doubt that the distribution of measurements is mound-shaped, or if you wish for some other reason to be conservative, you can apply Tchebysheff’s Theorem and be absolutely certain of your statements. Tchebysheff’s Theorem tells you that at least 3/4 of the measurements fall into the interval from 9.4 to 16.2 and at least 8/9 into the interval from 7.7 to 17.9. Student teachers are trained to develop lesson plans, on the assumption that the written plan will help them to perform successfully in the classroom. In a study to assess the relationship between written lesson plans and their implementation in the classroom, 25 lesson plans were scored on a scale of 0 to 34 according to a Lesson Plan Assessment Checklist. The 25 scores are shown in Table 2.4. Use Tchebysheff’s Theorem and the Empirical Rule (if applicable) to describe the distribution of these assessment scores. ● Lesson Plan Assessment Scores 29.3 31.9 17.8 22.1 22.1 19.7 25.0 13.3 13.8 10.2 14.5 26.6 20.2 15.7 23.5 26.0 21.2 20.8 26.5 21.3 26.1 22.1 15.9 25.6 29.0 Solution Use your calculator or the computing formulas to verify that x 21.6 and s 5.5. The appropriate intervals are calculated and listed in Table 2.5. We have also referred back to the original 25 measurements and counted the actual number of measurements that fall into each of these intervals. These frequencies and relative frequencies are shown in Table |
2.5. 2.4 ON THE PRACTICAL SIGNIFICANCE OF THE STANDARD DEVIATION ❍ 69 TABLE 2.5 ● Intervals x ks for the Data of Table 2.4 Interval x ks Frequency in Interval Relative Frequency 16.1–27.1 10.6–32.6 5.1–38.1 16 24 25.64.96 1.00 k 1 2 3 Empirical Rule ⇔ mound-shaped data Tchebysheff ⇔ any shaped data Is Tchebysheff’s Theorem applicable? Yes, because it can be used for any set of data. According to Tchebysheff’s Theorem, • • at least 3/4 of the measurements will fall between 10.6 and 32.6. at least 8/9 of the measurements will fall between 5.1 and 38.1. You can see in Table 2.5 that Tchebysheff’s Theorem is true for these data. In fact, the proportions of measurements that fall into the specified intervals exceed the lower bound given by this theorem. Is the Empirical Rule applicable? You can check for yourself by drawing a graph— either a stem and leaf plot or a histogram. The MINITAB histogram in Figure 2.12 shows that the distribution is relatively mound-shaped, so the Empirical Rule should work relatively well. That is, • • • approximately 68% of the measurements will fall between 16.1 and 27.1. approximately 95% of the measurements will fall between 10.6 and 32.6. approximately 99.7% of the measurements will fall between 5.1 and 38.1. The relative frequencies in Table 2.5 closely approximate those specified by the Empirical Rule. F IG URE 2. 12 MINITAB histogram for Example 2.8 ● 6/25 /25 2/25 0 8.5 14.5 20.5 Scores 26.5 32.5 USING TCHEBYSHEFF’S THEOREM AND THE EMPIRICAL RULE Tchebysheff’s Theorem can be proven mathematically. It applies to any set of measurements—sample or population, large or small, mound-shaped or skewed. Tchebysheff’s Theorem gives a lower bound to the fraction of measurements to be found in an interval |
constructed as x ks. At least 1 (1/k 2) of the measurements will fall into this interval, and probably more! 70 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES The Empirical Rule is a “rule of thumb” that can be used as a descriptive tool only when the data tend to be roughly mound-shaped (the data tend to pile up near the center of the distribution). When you use these two tools for describing a set of measurements, Tchebysheff’s Theorem will always be satisfied, but it is a very conservative estimate of the fraction of measurements that fall into a particular interval. If it is appropriate to use the Empirical Rule (mound-shaped data), this rule will give you a more accurate estimate of the fraction of measurements that fall into the interval. A CHECK ON THE CALCULATION OF s 2.5 Tchebysheff’s Theorem and the Empirical Rule can be used to detect gross errors in the calculation of s. Roughly speaking, these two tools tell you that most of the time, measurements lie within two standard deviations of their mean. This interval is marked off in Figure 2.13, and it implies that the total range of the measurements, from smallest to largest, should be somewhere around four standard deviations. This is, of course, a very rough approximation, but it can be very useful in checking for large errors in your calculation of s. If the range, R, is about four standard deviations, or 4s, you can write R 4s or s R 4 The computed value of s using the shortcut formula should be of roughly the same order as the approximation. F IG URE 2. 13 Range approximation to s ● 2s + x 2s x – 2s x + 2s EXAMPLE 2.9 Use the range approximation to check the calculation of s for Table 2.2. Solution The range of the five measurements—5, 7, 1, 2, 4—is R 7 1 6 Then s R 6 1.5 4 4 This is the same order as the calculated value s 2.4. s R/4 gives only an approximate value for s. The range approximation is not intended to provide an accurate value for s. Rather, its purpose is to detect gross errors in calculating, such as the failure to divide the sum of squares of deviations by (n |
1) or the failure to take the square root of s 2. If you make one of these mistakes, your answer will be many times larger than the range approximation of s. 2.5 A CHECK ON THE CALCULATION OF s ❍ 71 EXAMPLE 2.10 Use the range approximation to determine an approximate value for the standard deviation for the data in Table 2.4. Solution The range R 31.9 10.2 21.7. Then s R.7 5.4 21 4 4 Since the exact value of s is 5.5 for the data in Table 2.4, the approximation is very close. The range for a sample of n measurements will depend on the sample size, n. For larger values of n, a larger range of the x values is expected. The range for large samples (say, n 50 or more observations) may be as large as 6s, whereas the range for small samples (say, n 5 or less) may be as small as or smaller than 2.5s. The range approximation for s can be improved if it is known that the sample is drawn from a mound-shaped distribution of data. Thus, the calculated s should not differ substantially from the range divided by the appropriate ratio given in Table 2.6. TABLE 2.6 ● Divisor for the Range Approximation of s Expected Ratio of Range to s Number of Measurements 5 10 25 2.5 3 4 2.5 EXERCISES BASIC TECHNIQUES 2.19 A set of n 10 measurements consists of the values 5, 2, 3, 6, 1, 2, 4, 5, 1, 3. a. Use the range approximation to estimate the value of s for this set. (HINT: Use the table at the end of Section 2.5.) b. Use your calculator to find the actual value of s. Is the actual value close to your estimate in part a? c. Draw a dotplot of this data set. Are the data mound- shaped? that the mean and standard deviation of the data set are 36 and 3, respectively. a. If you are fairly certain that the relative frequency distribution of the data is mound-shaped, how might you picture the relative frequency distribution? (HINT: Use the Empirical Rule.) b. If you have no prior information concerning the shape of the relative frequency distribution, what can you say about the relative frequency histogram? (HINT |
: Construct intervals x ks for several choices of k.) d. Can you use Tchebysheff’s Theorem to describe this data set? Why or why not? e. Can you use the Empirical Rule to describe this data set? Why or why not? 2.20 Suppose you want to create a mental picture of the relative frequency histogram for a large data set consisting of 1000 observations, and you know 2.21 A distribution of measurements is relatively mound-shaped with mean 50 and standard deviation 10. a. What proportion of the measurements will fall between 40 and 60? b. What proportion of the measurements will fall between 30 and 70? 72 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES c. What proportion of the measurements will fall between 30 and 60? d. If a measurement is chosen at random from this distribution, what is the probability that it will be greater than 60? 2.22 A set of data has a mean of 75 and a standard deviation of 5. You know nothing else about the size of the data set or the shape of the data distribution. a. What can you say about the proportion of measurements that fall between 60 and 90? b. What can you say about the proportion of measurements that fall between 65 and 85? c. What can you say about the proportion of measurements that are less than 65? APPLICATIONS 2.23 Driving Emergencies The length of time required for an automobile driver to respond to a particular emergency situation was recorded for n 10 drivers. The times (in seconds) were.5,.8, 1.1,.7,.6,.9,.7,.8,.7,.8. a. Scan the data and use the procedure in Section 2.5 to find an approximate value for s. Use this value to check your calculations in part b. b. Calculate the sample mean x and the standard deviation s. Compare with part a. 2.25 Breathing Rates Is your breathing rate normal? Actually, there is no standard breathing rate for humans. It can vary from as low as 4 breaths per minute to as high as 70 or 75 for a person engaged in strenuous exercise. Suppose that the resting breathing rates for college-age students have a relative frequency distribution that is mound-shaped, with a mean equal to 12 and a standard deviation of 2.3 breaths per minute. What fraction of all students would have breathing rates in the |
following intervals? a. 9.7 to 14.3 breaths per minute b. 7.4 to 16.6 breaths per minute c. More than 18.9 or less than 5.1 breaths per minute EX0226 2.26 Ore Samples A geologist collected 20 different ore samples, all the same weight, and randomly divided them into two groups. She measured the titanium (Ti) content of the samples using two different methods. Method 1 Method 2.011.013.013.015.014.013.010.013.011.012.011.016.013.012.015.012.017.013.014.015 a. Construct stem and leaf plots for the two data sets. Visually compare their centers and their ranges. b. Calculate the sample means and standard deviations for the two sets. Do the calculated values confirm your visual conclusions from part a? 2.24 Packaging Hamburger Meat The data listed here are the weights (in pounds) of EX0224 27 packages of ground beef in a supermarket meat display: 2.27 Social Security Numbers The data from Exercise 1.70 (see data set EX0170), reproduced below, show the last digit of the Social Security number for a group of 70 students. 1.08 1.06.89.89.99 1.14.89.98.97 1.38.96 1.14 1.18.75 1.12.92 1.41.96 1.12 1.18 1.28 1.08.93 1.17.83.87 1.24 a. Construct a stem and leaf plot or a relative frequency histogram to display the distribution of weights. Is the distribution relatively moundshaped? b. Find the mean and standard deviation of the data set. c. Find the percentage of measurements in the intervals x s, x 2s, and x 3s. d. How do the percentages obtained in part c compare with those given by the Empirical Rule? Explain. e. How many of the packages weigh exactly 1 pound? Can you think of any explanation for this. You found in Exercise 1.70 that the distribution of this data was relatively “flat,” with each different value from 0 to 9 occurring with nearly equal frequency. Using this fact, what would be your best estimate for the mean of the data set? b. Use the range approximation to guess the value of s for this set. c. Use |
your calculator to find the actual values of x and s. Compare with your estimates in parts a and b. 2.28 Social Security Numbers, continued Refer to the data set in Exercise 2.27. a. Find the percentage of measurements in the intervals x s, x 2s, and x 3s. b. How do the percentages obtained in part a compare with those given by the Empirical Rule? Should they be approximately the same? Explain. 2.29 Survival Times A group of experimental animals is infected with a particular form of bacteria, and their survival time is found to average 32 days, with a standard deviation of 36 days. a. Visualize the distribution of survival times. Do you think that the distribution is relatively moundshaped, skewed right, or skewed left? Explain. b. Within what limits would you expect at least 3/4 of the measurements to lie? 2.30 Survival Times, continued Refer to Exercise 2.29. You can use the Empirical Rule to see why the distribution of survival times could not be moundshaped. a. Find the value of x that is exactly one standard deviation below the mean. b. If the distribution is in fact mound-shaped, approximately what percentage of the measurements should be less than the value of x found in part a? c. Since the variable being measured is time, is it possible to find any measurements that are more than one standard deviation below the mean? d. Use your answers to parts b and c to explain why the data distribution cannot be mound-shaped. 2.31 Timber Tracts To estimate the amount of lumber in a tract of timber, an owner EX0231 decided to count the number of trees with diameters exceeding 12 inches in randomly selected 50-by-50foot squares. Seventy 50-by-50-foot squares were chosen, and the selected trees were counted in each tract. The data are listed here: 7 9 3 10 9 6 10 8 6 9 2 6 11 8 7 4 5 7 8 9 8 10 9 9 4 8 11 5 4 10 10 8 11 10 9 8 7 8 13 8 a. Construct a relative frequency histogram to describe the data. b. Calculate the sample mean x as an estimate of m, the mean number of timber trees for all 50-by-50-foot squares in the tract. 2.5 A CHECK ON THE CALCULATION OF s ❍ 73 c. Calculate s for the data. Construct |
the intervals x s, x 2s, and x 3s. Calculate the percentage of squares falling into each of the three intervals, and compare with the corresponding percentages given by the Empirical Rule and Tchebysheff’s Theorem. 2.32 Tuna Fish, again Refer to Exercise 2.8 and data set EX0208. The prices of a 6-ounce can or a 7.06 pouch for 14 different brands of water-packed light tuna, based on prices paid nationally in supermarkets are reproduced here.4.99 1.12 1.92.63 1.23.67.85.69.65.60.53.60 1.41.66 a. Use the range approximation to find an estimate of s. b. How does it compare to the computed value of s? 2.33 Old Faithful The data below are 30 waiting times between eruptions of the Old EX0233 Faithful geyser in Yellowstone National Park.8 56 55 89 87 51 53 79 85 58 61 82 93 52 54 88 76 52 80 78 81 69 59 75 86 77 78 72 71 71 77 a. Calculate the range. b. Use the range approximation to approximate the standard deviation of these 30 measurements. c. Calculate the sample standard deviation s. d. What proportion of the measurements lie within two standard deviations of the mean? Within three standard deviations of the mean? Do these proportions agree with the proportions given in Tchebysheff’s Theorem? EX0234 2.34 The President’s Kids The table below shows the names of the 42 presidents of the United States along with the number of their children.2 Washington Adams Jefferson Madison Monroe J.Q. Adams Jackson 0 5 6 0 2 4 0 5 Cleveland 3 B. Harrison* McKinley 2 T. Roosevelt* 6 3 Taft 3 Wilson* 0 Harding Van Buren W.H. Harrison Tyler* Polk Taylor Fillmore* Pierce Coolidge Hoover F.D. Roosevelt Truman Eisenhower Kennedy L.B. Johnson 4 10 15 Buchanan Lincoln A. Johnson Grant Hayes Garfield Arthur 0 4 5 4 8 7 3 2 Nixon 4 Ford 4 Carter Reagan* 4 G.H.W. Bush 6 1 Clinton 2 G.W. Bush *Married twice Source: Time Almanac 2007 74 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES a. Construct a relative frequency histogram to describe the data. |
How would you describe the shape of this distribution? b. Calculate the mean and the standard deviation for the data set. c. Construct the intervals x s, x 2s, and x 3s. Find the percentage of measurements falling into these three intervals and compare with the corresponding percentages given by Tchebysheff’s Theorem and the Empirical Rule. 2.35 An Archeological Find, again Refer to Exercise 2.17. The percentage of iron oxide in each of five pottery samples collected at the Island Thorns site was: 1.28 2.39 1.50 1.88 1.51 a. Use the range approximation to find an estimate of s, using an appropriate divisor from Table 2.6. b. Calculate the standard deviation s. How close did your estimate come to the actual value of s? 2.36 Brett Favre The number of passes completed by Brett Favre, quarterback for EX0236 the Green Bay Packers, was recorded for each of the 16 regular season games in the fall of 2006 (www.espn.com).9 15 17 22 31 28 20 25 24 26 22 5 21 22 22 19 24 a. Draw a stem and leaf plot to describe the data. b. Calculate the mean and standard deviation for Brett Favre’s per game pass completions. c. What proportion of the measurements lie within two standard deviations of the mean? CALCULATING THE MEAN AND STANDARD DEVIATION FOR GROUPED DATA (OPTIONAL) 2.37 Suppose that some measurements occur more than once and that the data x1, x2,..., xk are arranged in a frequency table as shown here: Observations Frequency fi x1 x2... xk f1 f2... fk The formulas for the mean and variance for grouped data are Sx i fi, x n and where n Sfi i fi (Sx i fi)2 Sx2 n n 1 s 2 Notice that if each value occurs once, these formulas reduce to those given in the text. Although these formulas for grouped data are primarily of value when you have a large number of measurements, demonstrate their use for the sample 1, 0, 0, 1, 3, 1, 3, 2, 3, 0, 0, 1, 1, 3, 2. a. Calculate x and s 2 directly, using the formulas for ungrouped |
data. b. The frequency table for the n 15 measurements is as follows Calculate x and s 2 using the formulas for grouped data. Compare with your answers to part a. 2.38 International Baccalaureate The International Baccalaureate (IB) program is an accelerated academic program offered at a growing number of high schools throughout the country. Students enrolled in this program are placed in accelerated or advanced courses and must take IB examinations in each of six subject areas at the end of their junior or senior year. Students are scored on a scale of 1–7, with 1–2 being poor, 3 mediocre, 4 average, and 5–7 excellent. During its first year of operation at John W. North High School in Riverside, California, 17 juniors attempted the IB economics exam, with these results: Exam Grade Number of Students Calculate the mean and standard deviation for these scores. 2.39 A Skewed Distribution To illustrate the utility of the Empirical Rule, consider a distribution that is heavily skewed to the right, as shown in the accompanying figure. a. Calculate x and s for the data shown. (NOTE: There are 10 zeros, 5 ones, and so on.) b. Construct the intervals x s, x 2s, and x 3s and locate them on the frequency distribution. c. Calculate the proportion of the n 25 measurements that fall into each of the three intervals. Compare with Tchebysheff’s Theorem and the Empirical Rule. Note that, although the proportion that falls into the interval x s does not agree closely with the Empirical Rule, the 2.6 MEASURES OF RELATIVE STANDING ❍ 75 proportions that fall into the intervals x 2s and x 3s agree very well. Many times this is true, even for non-mound-shaped distributions of data. Distribution for Exercise 2.39 y c n e u q e r F 10 10 8 6 4 2 0 2 4 6 n 25 8 10 MEASURES OF RELATIVE STANDING 2.6 Sometimes you need to know the position of one observation relative to others in a set of data. For example, if you took an examination with a total of 35 points, you might want to know how your score of 30 compared to the scores of the other students in the class. The mean and standard deviation of the scores can be used to calculate a z-score, which measures the relative standing |
of a measurement in a data set. Definition The sample z-score is a measure of relative standing defined by Positive z-score ⇔ x is above the mean. Negative z-score ⇔ x is below the mean. z-score x x s A z-score measures the distance between an observation and the mean, measured in units of standard deviation. For example, suppose that the mean and standard deviation of the test scores (based on a total of 35 points) are 25 and 4, respectively. The z-score for your score of 30 is calculated as follows: x 30 z-score x 25 1.25 s 4 Your score of 30 lies 1.25 standard deviations above the mean (30 x 1.25s). 76 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES The z-score is a valuable tool for determining whether a particular observation is likely to occur quite frequently or whether it is unlikely and might be considered an outlier. According to Tchebysheff’s Theorem and the Empirical Rule, • • at least 75% and more likely 95% of the observations lie within two standard deviations of their mean: their z-scores are between 2 and 2. Observations with z-scores exceeding 2 in absolute value happen less than 5% of the time and are considered somewhat unlikely. at least 89% and more likely 99.7% of the observations lie within three standard deviations of their mean: their z-scores are between 3 and 3. Observations with z-scores exceeding 3 in absolute value happen less than 1% of the time and are considered very unlikely. z-scores above 3 in absolute value are very unusual. You should look carefully at any observation that has a z-score exceeding 3 in absolute value. Perhaps the measurement was recorded incorrectly or does not belong to the population being sampled. Perhaps it is just a highly unlikely observation, but a valid one nonetheless! EXAMPLE 2.11 Consider this sample of n 10 measurements: 1, 1, 0, 15, 2, 3, 4, 0, 1, 3 The measurement x 15 appears to be unusually large. Calculate the z-score for this observation and state your conclusions. Solution Calculate x 3.0 and s 4.42 for the n 10 measurements. Then the z-score for the suspected outlier, x 15, is calculated as z-score x x 15 3 2.71 s |
42 4. Hence, the measurement x 15 lies 2.71 standard deviations above the sample mean, x 3.0. Although the z-score does not exceed 3, it is close enough so that you might suspect that x 15 is an outlier. You should examine the sampling procedure to see whether x 15 is a faulty observation. A percentile is another measure of relative standing and is most often used for large data sets. (Percentiles are not very useful for small data sets.) Definition A set of n measurements on the variable x has been arranged in order of magnitude. The pth percentile is the value of x that is greater than p% of the measurements and is less than the remaining (100 p)%. EXAMPLE 2.12 Suppose you have been notified that your score of 610 on the Verbal Graduate Record Examination placed you at the 60th percentile in the distribution of scores. Where does your score of 610 stand in relation to the scores of others who took the examination? Solution Scoring at the 60th percentile means that 60% of all the examination scores were lower than your score and 40% were higher. 2.6 MEASURES OF RELATIVE STANDING ❍ 77 In general, the 60th percentile for the variable x is a point on the horizontal axis of the data distribution that is greater than 60% of the measurements and less than the others. That is, 60% of the measurements are less than the 60th percentile and 40% are greater (see Figure 2.14). Since the total area under the distribution is 100%, 60% of the area is to the left and 40% of the area is to the right of the 60th percentile. Remember that the median, m, of a set of data is the middle measurement; that is, 50% of the measurements are smaller and 50% are larger than the median. Thus, the median is the same as the 50th percentile! ● FI GUR E 2.1 4 The 60th percentile shown on the relative frequency histogram for a data set 60% 40% 60th percentile x The 25th and 75th percentiles, called the lower and upper quartiles, along with the median (the 50th percentile), locate points that divide the data into four sets, each containing an equal number of measurements. Twenty-five percent of the measurements will be less than the lower (first) quartile, 50% will be less than the median (the second quartile |
), and 75% will be less than the upper (third) quartile. Thus, the median and the lower and upper quartiles are located at points on the x-axis so that the area under the relative frequency histogram for the data is partitioned into four equal areas, as shown in Figure 2.15. FI GUR E 2.1 5 Location of quartiles ● 25% 25% 25% 25% Median, m x Lower quartile, Q1 Upper quartile, Q3 78 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES Definition A set of n measurements on the variable x has been arranged in order of magnitude. The lower quartile (first quartile), Q1, is the value of x that is greater than one-fourth of the measurements and is less than the remaining three-fourths. The second quartile is the median. The upper quartile (third quartile), Q3, is the value of x that is greater than three-fourths of the measurements and is less than the remaining one-fourth. For small data sets, it is often impossible to divide the set into four groups, each of which contains exactly 25% of the measurements. For example, when n 10, you would need to have 21 measurements in each group! Even when you can perform this 2 task (for example, if n 12), there are many numbers that would satisfy the preceding definition, and could therefore be considered “quartiles.” To avoid this ambiguity, we use the following rule to locate sample quartiles. CALCULATING SAMPLE QUARTILES • When the measurements are arranged in order of magnitude, the lower quartile, Q1, is the value of x in position.25(n 1), and the upper quartile, Q3, is the value of x in position.75(n 1). • When.25(n 1) and.75(n 1) are not integers, the quartiles are found by interpolation, using the values in the two adjacent positions.† EXAMPLE 2.13 Find the lower and upper quartiles for this set of measurements: 16, 25, 4, 18, 11, 13, 20, 8, 11, 9 Solution Rank the n 10 measurements from smallest to largest: 4, 8, 9, 11, 11, 13, 16, 18, 20, 25 Calculate Position of Q1. |
25(n 1).25(10 1) 2.75 Position of Q3.75(n 1).75(10 1) 8.25 Since these positions are not integers, the lower quartile is taken to be the value 3/4 of the distance between the second and third ordered measurements, and the upper quartile is taken to be the value 1/4 of the distance between the eighth and ninth ordered measurements. Therefore, Q1 8.75(9 8) 8.75 8.75 and Q3 18.25(20 18) 18.5 18.5 Because the median and the quartiles divide the data distribution into four parts, each containing approximately 25% of the measurements, Q1 and Q3 are the upper and lower boundaries for the middle 50% of the distribution. We can measure the range of this “middle 50%” of the distribution using a numerical measure called the interquartile range. †This definition of quartiles is consistent with the one used in the MINITAB package. Some textbooks use ordinary rounding when finding quartile positions, whereas others compute sample quartiles as the medians of the upper and lower halves of the data set. 2.6 MEASURES OF RELATIVE STANDING ❍ 79 Definition The interquartile range (IQR) for a set of measurements is the difference between the upper and lower quartiles; that is, IQR Q3 Q1. For the data in Example 2.13, IQR Q3 Q1 18.50 8.75 9.75. We will use the IQR along with the quartiles and the median in the next section to construct another graph for describing data sets. How Do I Calculate Sample Quartiles? 1. Arrange the data set in order of magnitude from smallest to largest. 2. Calculate the quartile positions: • • Position of Q1:.25(n 1) Position of Q3:.75(n 1) 3. 4. If the positions are integers, then Q1 and Q3 are the values in the ordered data set found in those positions. If the positions in step 2 are not integers, find the two measurements in positions just above and just below the calculated position. Calculate the quartile by finding a value either one-fourth, one-half, or three-fourths of the way between these two measurements. Exercise Reps A. |
Below you will find two practice data sets. Fill in the blanks to find the neces- sary quartiles. The first data set is done for you. Data Set Sorted 2, 5, 7, 1, 1, 2, 8 1, 1, 2, 2, 5, 7, 8 5, 0, 1, 3, 1, 5, 5, 2, 4, 4, 1 Position of Q1 2nd n 7 Position of Q3 6th Lower Quartile, Q1 Upper Quartile, Q3 1 7 B. Below you will find three data sets that have already been sorted. The positions of the upper and lower quartiles are shown in the table. Find the measurements just above and just below the quartile position. Then find the upper and lower quartiles. The first data set is done for you. Sorted Data Set Position Measurements of Q1 Above and Below 0, 1, 4, 4, 5, 9 1.75 0 and 1 Q1 Position Measurements of Q3 0.75(1) 5.25.75 5 and 9 Above and Below Q3 5.25(4) 6 0, 1, 3, 3, 4, 7, 7, 8 2.25 1, 1, 2, 5, 6, 6, 7, 9, 9 2.5 and and 6.75 7.5 and and Progress Report • Still having trouble? Try again using the Exercise Reps at the end of this section. • Mastered sample quartiles? You can skip the Exercise Reps at the end of this section! Answers are located on the perforated card at the back of this book. 80 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES Many of the numerical measures that you have learned are easily found using computer programs or even graphics calculators. The MINITAB command Stat Basic Statistics Display Descriptive Statistics (see the section “My MINITAB ” at the end of this chapter) produces output containing the mean, the standard deviation, the median, and the lower and upper quartiles, as well as the values of some other statistics that we have not discussed yet. The data from Example 2.13 produced the MINITAB output shown in Figure 2.16. Notice that the quartiles are identical to the handcalculated values in that example. F IG U |
RE 2. 16 MINITAB output for the data in Example 2.13 ● Descriptive Statistics: x Variable X N N* 10 0 13.50 Mean SE Mean StDev Minimum Q1 Median 1.98 6.28 4.00 8.75 12.00 18.50 Q3 Maximum 25.00 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT 2.7 The median and the upper and lower quartiles shown in Figure 2.15 divide the data into four sets, each containing an equal number of measurements. If we add the largest number (Max) and the smallest number (Min) in the data set to this group, we will have a set of numbers that provide a quick and rough summary of the data distribution. The five-number summary consists of the smallest number, the lower quartile, the median, the upper quartile, and the largest number, presented in order from smallest to largest: Min Q1 Median Q3 Max By definition, one-fourth of the measurements in the data set lie between each of the four adjacent pairs of numbers. The five-number summary can be used to create a simple graph called a box plot to visually describe the data distribution. From the box plot, you can quickly detect any skewness in the shape of the distribution and see whether there are any outliers in the data set. An outlier may result from transposing digits when recording a measurement, from incorrectly reading an instrument dial, from a malfunctioning piece of equipment, or from other problems. Even when there are no recording or observational errors, a data set may contain one or more valid measurements that, for one reason or another, differ markedly from the others in the set. These outliers can cause a marked distortion in commonly used numerical measures such as x and s. In fact, outliers may themselves contain important information not shared with the other measurements in the set. Therefore, isolating outliers, if they are present, is an important step in any preliminary analysis of a data set. The box plot is designed expressly for this purpose. 2.7 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT ❍ 81 TO CONSTRUCT A BOX PLOT • Calculate the median, the upper and lower quartiles, and the IQR for the data set. • Draw a horizontal line representing the scale of measurement. Form a box just above the horizontal line with the right and left ends at |
Q1 and Q3. Draw a vertical line through the box at the location of the median. A box plot is shown in Figure 2.17. FI GUR E 2.1 7 Box plot ● Lower fence Q1 m Q3 Upper fence In Section 2.6, the z-score provided boundaries for finding unusually large or small measurements. You looked for z-scores greater than 2 or 3 in absolute value. The box plot uses the IQR to create imaginary “fences” to separate outliers from the rest of the data set: DETECTING OUTLIERS—OBSERVATIONS THAT ARE BEYOND: • Lower fence: Q1 1.5(IQR) • Upper fence: Q3 1.5(IQR) The upper and lower fences are shown with broken lines in Figure 2.17, but they are not usually drawn on the box plot. Any measurement beyond the upper or lower fence is an outlier; the rest of the measurements, inside the fences, are not unusual. Finally, the box plot marks the range of the data set using “whiskers” to connect the smallest and largest measurements (excluding outliers) to the box. TO FINISH THE BOX PLOT • Mark any outliers with an asterisk (*) on the graph. • Extend horizontal lines called “whiskers” from the ends of the box to the smallest and largest observations that are not outliers. EXAMPLE 2.14 As American consumers become more careful about the foods they eat, food processors try to stay competitive by avoiding excessive amounts of fat, cholesterol, and sodium in the foods they sell. The following data are the amounts of sodium per slice (in milligrams) for each of eight brands of regular American cheese. Construct a box plot for the data and look for outliers. 340, 300, 520, 340, 320, 290, 260, 330 82 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES Solution The n 8 measurements are first ranked from smallest to largest: 260, 290, 300, 320, 330, 340, 340, 520 The positions of the median, Q1, and Q3 are.5(n 1).5(9) 4.5.25(n 1).25(9) 2.25.75(n 1).75(9) 6.75 so that m (320 |
330)/2 325, Q1 290.25(10) 292.5, and Q3 340. The interquartile range is calculated as IQR Q3 Q1 340 292.5 47.5 Calculate the upper and lower fences: Lower fence: 292.5 1.5(47.5) 221.25 Upper fence: 340 1.5(47.5) 411.25 The value x 520, a brand of cheese containing 520 milligrams of sodium, is the only outlier, lying beyond the upper fence. The box plot for the data is shown in Figure 2.18. The outlier is marked with an asterisk (*). Once the outlier is excluded, we find (from the ranked data set) that the smallest and largest measurements are x 260 and x 340. These are the two values that form the whiskers. Since the value x 340 is the same as Q3, there is no whisker on the right side of the box. F IG URE 2. 18 Box plot for Example 2.14 ● * 250 300 350 400 Sodium 450 500 550 Now would be a good time to try the Building a Box Plot applet. The applet in Figure 2.19 shows a dotplot of the data in Example 2.14. Using the button, you will see a step-by-step description explaining how the box plot is constructed. We will use this applet again for the MyApplet Exercises at the end of the chapter. FI GUR E 2.1 9 Building a Box Plot applet ● 2.7 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT ❍ 83 You can use the box plot to describe the shape of a data distribution by looking at the position of the median line compared to Q1 and Q3, the left and right ends of the box. If the median is close to the middle of the box, the distribution is fairly symmetric, providing equal-sized intervals to contain the two middle quarters of the data. If the median line is to the left of center, the distribution is skewed to the right; if the median is to the right of center, the distribution is skewed to the left. Also, for most skewed distributions, the whisker on the skewed side of the box tends to be longer than the whisker on the other side. We used the MINITAB command Graph Boxplot to draw two box plots, one for the sodium contents of the |
eight brands of cheese in Example 2.14, and another for five brands of fat-free cheese with these sodium contents: 300, 300, 320, 290, 180 The two box plots are shown together in Figure 2.20. Look at the long whisker on the left side of both box plots and the position of the median lines. Both distributions are skewed to the left; that is, there are a few unusually small measurements. The regular cheese data, however, also show one brand (x 520) with an unusually large amount of sodium. In general, it appears that the sodium content of the fat-free brands is lower than that of the regular brands, but the variability of the sodium content for regular cheese (excluding the outlier) is less than that of the fat-free brands. FI GUR E 2.2 0 MINITAB output for regular and fat-free cheese ● Fat-Free Type Regular * 200 250 300 350 400 450 500 550 Sodium 84 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES 2.7 EXERCISES EXERCISE REPS These exercises refer back to the MyPersonal Trainer section on page 79. 2.40 Below you will find two practice data sets. Fill in the blanks to find the necessary quartiles. Data Set Sorted n Position of Q1 Position of Q3 Lower Quartile, Q1 Upper Quartile, Q3.13,.76,.34,.88,.21,.16,.28 2.3, 1.0, 2.1, 6.5, 2.8, 8.8, 1.7, 2.9, 4.4, 5.1, 2.0 2.41 Below you will find three data sets that have already been sorted. Fill in the blanks to find the upper and lower quartiles. Sorted Data Set Position of Q1 Measurements Above and Below Q1 Position of Q3 Measurements Above and Below Q3 1, 1.5, 2, 2, 2.2 0, 1.7, 1.8, 3.1, 3.2, 7, 8, 8.8, 8.9, 9, 10.23,.30,.35,.41,.56,.58,.76,.80 and and and and and and BASIC TECHNIQUES APPLICATIONS 2.42 |
Given the following data set: 8, 7, 1, 4, 6, 6, 4, 5, 7, 6, 3, 0 a. Find the five-number summary and the IQR. b. Calculate x and s. c. Calculate the z-score for the smallest and largest observations. Is either of these observations unusually large or unusually small? 2.43 Find the five-number summary and the IQR for these data: 19, 12, 16, 0, 14, 9, 6, 1, 12, 13, 10, 19, 7, 5, 8 2.44 Construct a box plot for these data and identify any outliers: 25, 22, 26, 23, 27, 26, 28, 18, 25, 24, 12 2.45 Construct a box plot for these data and identify any outliers: 3, 9, 10, 2, 6, 7, 5, 8, 6, 6, 4, 9, 22 2.46 If you scored at the 69th percentile on a placement test, how does your score compare with others? 2.47 Mercury Concentration in Dolphins Environmental scientists are increasingly EX0247 concerned with the accumulation of toxic elements in marine mammals and the transfer of such elements to the animals’ offspring. The striped dolphin (Stenella coeruleoalba), considered to be the top predator in the marine food chain, was the subject of one such study. The mercury concentrations (micrograms/gram) in the livers of 28 male striped dolphins were as follows: 1.70 1.72 8.80 5.90 101.00 85.40 118.00 183.00 168.00 218.00 180.00 264.00 481.00 485.00 221.00 406.00 252.00 329.00 316.00 445.00 278.00 286.00 315.00 241.00 397.00 209.00 314.00 318.00 a. Calculate the five-number summary for the data. b. Construct a box plot for the data. c. Are there any outliers? d. If you knew that the first four dolphins were all less than 3 years old, while all the others were more than 8 years old, would this information help explain the difference in the magnitude of those four observations? Explain. 2.48 Hamburger Meat The weights (in pounds) of the 27 packages |
of ground beef from Exercise 2.24 (see data set EX0224) are listed here in order from smallest to largest:.75.93 1.08 1.18.83.96 1.08 1.18.87.96 1.12 1.24.89.97 1.12 1.28.89.98 1.14 1.38.89.99 1.14 1.41.92 1.06 1.17 a. Confirm the values of the mean and standard deviation, calculated in Exercise 2.24 as x 1.05 and s.17. b. The two largest packages of meat weigh 1.38 and 1.41 pounds. Are these two packages unusually heavy? Explain. c. Construct a box plot for the package weights. What does the position of the median line and the length of the whiskers tell you about the shape of the distribution? 2.49 Comparing NFL Quarterbacks How does Brett Favre, quarterback for the Green EX0249 Bay Packers, compare to Peyton Manning, quarterback for the Indianapolis Colts? The table below shows the number of completed passes for each athlete during the 2006 NFL football season:9 Brett Favre Peyton Manning 22 20 26 21 15 31 25 22 22 19 17 28 24 5 22 24 25 29 21 22 25 26 14 21 20 25 32 30 27 20 14 21 a. Calculate five-number summaries for the number of passes completed by both Brett Favre and Peyton Manning. b. Construct box plots for the two sets of data. Are there any outliers? What do the box plots tell you about the shapes of the two distributions? c. Write a short paragraph comparing the number of pass completions for the two quarterbacks. 2.7 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT ❍ 85 2.50 Presidential Vetoes The set of presidential vetoes in Exercise 1.47 and data set EX0147 is listed here, along with a box plot generated by MINITAB. Use the box plot to describe the shape of the distribution and identify any outliers. Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren W. H. Harrison Tyler Polk Taylor Fillmore Pierce Buchanan Lincoln A. Johnson Grant Hayes Garfield Arthur Cleveland 21 45 12 0 4 304 B. Harrison Cleveland McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. D. Roosevelt Truman Eisenhower Kennedy L. Johnson Nixon Ford Carter |
Reagan G. H. W. Bush Clinton G. W. Bush 19 42 6 42 30 33 5 20 21 372 180 73 12 16 26 48 13 39 29 36 1 Source: The World Almanac and Book of Facts 2007 Box plot for Exercise 2.50 * * * 0 100 200 Vetoes 300 400 2.51 Survival Times Altman and Bland report the survival times for patients with active hepatitis, half treated with prednisone and half receiving no treatment.10 The survival times (in months) (Exercise 1.73 and EX0173) are adapted from their data for those treated with prednisone. 86 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES 8 11 52 57 65 87 93 97 109 120 127 133 139 142 144 147 148 157 162 165 a. Can you tell by looking at the data whether it is roughly symmetric? Or is it skewed? b. Calculate the mean and the median. Use these measures to decide whether or not the data are symmetric or skewed. c. Draw a box plot to describe the data. Explain why the box plot confirms your conclusions in part b. 2.52 Utility Bills in Southern California, again The monthly utility bills for a house- EX0252 hold in Riverside, California, were recorded for 12 consecutive months starting in January 2006: Month Amount ($) Month Amount ($) January February March April May June $266.63 163.41 219.41 162.64 187.16 289.17 July August September October November December $306.55 335.48 343.50 226.80 208.99 230.46 a. Construct a box plot for the monthly utility costs. b. What does the box plot tell you about the distribution of utility costs for this household? 2.53 What’s Normal? again Refer to Exercise 1.67 and data set EX0167. In addition to the normal body temperature in degrees Fahrenheit for the 130 individuals, the data record the gender of the individuals. Box plots for the two groups, male and female, are shown below:11 Box plots for Exercise 2.53 Male r e d n e G Female * * * 96 97 98 99 100 101 Temperature How would you describe the similarities and differences between male and female temperatures in this data set? CHAPTER REVIEW Key Concepts and Formulas I. Measures of the Center of a Data Distribution 1. Arithmetic mean (mean) or average a. Population: m S xi b. |
Sample of n measurements: x n 2. Median; position of the median.5(n 1) 3. Mode 4. The median may be preferred to the mean if the data are highly skewed. II. Measures of Variability 1. Range: R largest smallest 2. Variance a. Population of N measurements: S(xi m)2 s 2 N b. Sample of n measurements: i (S xi)2 Sx 2 n n 1 S( x)2 xi s 2 n 1 3. Standard deviation a. Population: s s 2 b. Sample: s s 2 4. A rough approximation for s can be calculated as s R/4. The divisor can be adjusted depending on the sample size. III. Tchebysheff’s Theorem and the Empirical Rule 1. Use Tchebysheff’s Theorem for any data set, regardless of its shape or size. a. At least 1 (1/k2) of the measurements lie within k standard deviations of the mean. b. This is only a lower bound; there may be more measurements in the interval. 2. The Empirical Rule can be used only for relatively mound-shaped data sets. Approximately CHAPTER REVIEW ❍ 87 68%, 95%, and 99.7% of the measurements are within one, two, and three standard deviations of the mean, respectively. IV. Measures of Relative Standing 1. Sample z-score: z x x s 2. pth percentile; p% of the measurements are smaller, and (100 p)% are larger. 3. Lower quartile, Q1; position of Q1.25 (n 1) 4. Upper quartile, Q3; position of Q3.75 (n 1) 5. Interquartile range: IQR Q3 Q1 V. The Five-Number Summary and Box Plots 1. The five-number summary: Min Q1 Median Q3 Max One-fourth of the measurements in the data set lie between each of the four adjacent pairs of numbers. 2. Box plots are used for detecting outliers and shapes of distributions. 3. Q1 and Q3 form the ends of the box. The median line is in the interior of the box. 4. Upper and lower fences are used to find outliers, observations that lie outside these fences. a. Lower fence: Q1 1.5(IQR) b. |
Upper fence: Q3 1.5(IQR) 5. Outliers are marked on the box plot with an asterisk (*). 6. Whiskers are connected to the box from the smallest and largest observations that are not outliers. 7. Skewed distributions usually have a long whisker in the direction of the skewness, and the median line is drawn away from the direction of the skewness. 88 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES Numerical Descriptive Measures MINITAB provides most of the basic descriptive statistics presented in Chapter 2 using a single command in the drop-down menus. Once you are on the Windows desktop, double-click on the MINITAB icon or use the Start button to start MINITAB. Practice entering some data into the Data window, naming the columns appropriately in the gray cell just below the column number. When you have finished entering your data, you will have created a MINITAB worksheet, which can be saved either singly or as a MINITAB project for future use. Click on File Save Current Worksheet or File Save Project. You will need to name the worksheet (or project)—perhaps “test data”—so that you can retrieve it later. The following data are the floor lengths (in inches) behind the second and third seats in nine different minivans:12 Second seat: Third seat: 62.0, 62.0, 64.5, 48.5, 57.5, 61.0, 45.5, 47.0, 33.0 27.0, 27.0, 24.0, 16.5, 25.0, 27.5, 14.0, 18.5, 17.0 Since the data involve two variables, we enter the two rows of numbers into columns C1 and C2 in the MINITAB worksheet and name them “2nd Seat” and “3rd Seat,” respectively. Using the drop-down menus, click on Stat Basic Statistics Display Descriptive Statistics. The Dialog box is shown in Figure 2.21. F IG URE 2. 21 ● Now click on the Variables box and select both columns from the list on the left. (You can click on the Graphs option and choose one of several graphs if you like. You may also click on the Statistics option to select the |
statistics you would like to see displayed.) Click OK. A display of descriptive statistics for both columns will appear in the Session window (see Figure 2.22). You may print this output using File Print Session Window if you choose. To examine the distribution of the two variables and look for outliers, you can create box plots using the command Graph Boxplot One Y Simple. Click OK. Select the appropriate column of measurements in the Dialog box (see Figure 2.23). You can change the appearance of the box plot in several ways. Scale Axes and Ticks will allow you to transpose the axes and orient the box plot horizontally, when you check the box marked “Transpose value and category scales.” Multiple Graphs MY MINITAB ❍ 89 provides printing options for multiple box plots. Labels will let you annotate the graph with titles and footnotes. If you have entered data into the worksheet as a frequency distribution (values in one column, frequencies in another), the Data Options will allow the data to be read in that format. The box plot for the third seat lengths is shown in Figure 2.24. You can use the MINITAB commands from Chapter 1 to display stem and leaf plots or histograms for the two variables. How would you describe the similarities and differences in the two data sets? Save this worksheet in a file called “Minivans” before exiting MINITAB. We will use it again in Chapter 3. FI GUR E 2.2 2 ● FI GUR E 2.2 3 ● FI GUR E 2.2 4 ● 90 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES Supplementary Exercises 2.54 Raisins The number of raisins in each of 14 miniboxes (1/2-ounce size) was counted EX0254 for a generic brand and for Sunmaid brand raisins. The two data sets are shown here: c. Find the percentage of the viewing hours per household that falls into the interval x 2s. Compare with the corresponding percentage given by the Empirical Rule. Generic Brand Sunmaid 25 26 26 26 26 28 27 26 25 28 24 28 27 25 25 28 25 28 29 24 28 24 24 28 30 24 22 27 a. What are the mean and standard deviation for the generic brand? b. What are the mean and standard deviation for the Sunmaid brand? c. Compare the centers and variabilities |
of the two brands using the results of parts a and b. 2.55 Raisins, continued Refer to Exercise 2.54. a. Find the median, the upper and lower quartiles, and the IQR for each of the two data sets. 2.57 A Recurring Illness Refer to Exercise 1.26 and data set EX0126. The lengths of time (in months) between the onset of a particular illness and its recurrence were recorded: 2.1 9.0 14.7 19.2 4.1 7.4 14.1 8.7 1.6 3.7 4.4 2.0 9.6 6.9 18.4.2 1.0 24.0 3.5 12.6 2.7 6.6 16.7 4.3.2 8.3 2.4 1.4 11.4 23.1 32.3 3.9 7.4 3.3 6.1.3 2.4 8.2 18.0 5.6 9.9 1.6 8.2 1.2 13.5 1.3 18.0 5.8 26.7.4 a. Find the range. b. Use the range approximation to find an approximate b. Construct two box plots on the same horizontal value for s. scale to compare the two sets of data. c. Draw two stem and leaf plots to depict the shapes of the two data sets. Do the box plots in part b verify these results? d. If we can assume that none of the boxes of raisins are being underfilled (that is, they all weigh approximately 1/2 ounce), what do your results say about the average number of raisins for the two brands? EX0256 2.56 TV Viewers The number of television viewing hours per household and the prime viewing times are two factors that affect television advertising income. A random sample of 25 households in a particular viewing area produced the following estimates of viewing hours per household: 3.0 6.5 5.0 7.5 9.0 6.0 8.0 12.0 5.0 2.0 7.5 4.0 1.0 10.0 6.5 15.0 5.5 3.5 8.0 1.0 12.0 6.0 3.0 3.5 5.0 a. Scan the data and use the range to find an approximate value for |
s. Use this value to check your calculations in part b. b. Calculate the sample mean x and the sample standard deviation s. Compare s with the approximate value obtained in part a. c. Compute s for the data and compare it with your approximation from part b. 2.58 A Recurring Illness, continued Refer to Exercise 2.57. a. Examine the data and count the number of observations that fall into the intervals x s, x 2s, and x 3s. b. Do the percentages that fall into these intervals agree with Tchebysheff’s Theorem? With the Empirical Rule? c. Why might the Empirical Rule be unsuitable for describing these data? 2.59 A Recurring Illness, again Find the median and the lower and upper quartiles for the data on times until recurrence of an illness in Exercise 2.57. Use these descriptive measures to construct a box plot for the data. Use the box plot to describe the data distribution. 2.60 Tuna Fish, again Refer to Exercise 2.8. The prices of a 6-ounce can or a 7.06-ounce pouch for 14 different brands of water-packed light tuna, based on prices paid nationally in supermarkets, are reproduced here.4.99 1.12 1.92.63 1.23.67.85.69.65.60.53.60 1.41.66 SUPPLEMENTARY EXERCISES ❍ 91 a. Calculate the five-number summary. b. Construct a box plot for the data. Are there any hours that they slept on the previous night with the following results: outliers? c. The value x 1.92 looks large in comparison to the other prices. Use a z-score to decide whether this is an unusually expensive brand of tuna. 2.61 Electrolysis An analytical chemist wanted to use electrolysis to determine the number of moles of cupric ions in a given volume of solution. The solution was partitioned into n 30 portions of.2 milliliter each, and each of the portions was tested. The average number of moles of cupric ions for the n 30 portions was found to be.17 mole; the standard deviation was.01 mole. a. Describe the distribution of the measurements for the n 30 portions of the solution using Tchebysheff’s Theorem. b. Describe the distribution of the measurements for |
the n 30 portions of the solution using the Empirical Rule. (Do you expect the Empirical Rule to be suitable for describing these data?) c. Suppose the chemist had used only n 4 portions of the solution for the experiment and obtained the readings.15,.19,.17, and.15. Would the Empirical Rule be suitable for describing the n 4 measurements? Why? 2.62 Chloroform According to the EPA, chloroform, which in its gaseous form is suspected of being a cancer-causing agent, is present in small quantities in all of the country’s 240,000 public water sources. If the mean and standard deviation of the amounts of chloroform present in the water sources are 34 and 53 micrograms per liter, respectively, describe the distribution for the population of all public water sources. 2.63 Aptitude Tests In contrast to aptitude tests, which are predictive measures of what one can accomplish with training, achievement tests tell what an individual can do at the time of the test. Mathematics achievement test scores for 400 students were found to have a mean and a variance equal to 600 and 4900, respectively. If the distribution of test scores was mound-shaped, approximately how many of the scores would fall into the interval 530 to 670? Approximately how many scores would be expected to fall into the interval 460 to 740? 2.64 Sleep and the College Student How much sleep do you get on a typical school night? A group of 10 college students were asked to report the number of 7, 6, 7.25, 7, 8.5, 5, 8, 7, 6.75, 6 a. Find the mean and the standard deviation of the number of hours of sleep for these 10 students. b. Calculate the z-score for the largest value (x 8.5). Is this an unusually sleepy college student? c. What is the most frequently reported measurement? What is the name for this measure of center? d. Construct a box plot for the data. Does the box plot confirm your results in part b? [HINT: Since the z-score and the box plot are two unrelated methods for detecting outliers, and use different types of statistics, they do not necessarily have to (but usually do) produce the same results.] 2.65 Gas Mileage The miles per gallon (mpg) for each of 20 medium-sized cars selected from EX0265 a production |
line during the month of March follow. 23.1 20.2 24.7 25.9 24.9 21.3 24.4 22.7 24.7 22.2 23.6 25.3 26.2 24.4 22.9 23.7 27.0 23.2 24.2 24.6 a. What are the maximum and minimum miles per gallon? What is the range? b. Construct a relative frequency histogram for these data. How would you describe the shape of the distribution? c. Find the mean and the standard deviation. d. Arrange the data from smallest to largest. Find the z-scores for the largest and smallest observations. Would you consider them to be outliers? Why or why not? e. What is the median? f. Find the lower and upper quartiles. 2.66 Gas Mileage, continued Refer to Exercise 2.65. Construct a box plot for the data. Are there any outliers? Does this conclusion agree with your results in Exercise 2.65? 2.67 Polluted Seawater Petroleum pollution in seas and oceans stimulates the growth of some types of bacteria. A count of petroleumlytic micro-organisms (bacteria per 100 milliliters) in ten portions of seawater gave these readings: 49, 70, 54, 67, 59, 40, 61, 69, 71, 52 92 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES a. Guess the value for s using the range approximation. b. Calculate x and s and compare with the range approximation of part a. c. Construct a box plot for the data and use it to describe the data distribution. 2.68 Basketball Attendances at a high school’s basketball games were recorded and found to have a sample mean and variance of 420 and 25, respectively. Calculate x s, x 2s, and x 3s and then state the approximate fractions of measurements you would expect to fall into these intervals according to the Empirical Rule. 2.69 SAT Tests The College Board’s verbal and mathematics scholastic aptitude tests are scored on a scale of 200 to 800. Although the tests were originally designed to produce mean scores of approximately 500, the mean verbal and math scores in recent years have been as low as 463 and 493, respectively, and have been trending downward. It seems reasonable to assume that a distribution of all test scores, either verbal or math, |
is mound-shaped. If s is the standard deviation of one of these distributions, what is the largest value (approximately) that s might assume? Explain. 2.70 Summer Camping A favorite summer pastime for many Americans is camping. In fact, camping has become so popular at the California beaches that reservations must sometimes be made months in advance! Data from a USA Today Snapshot is shown below.13 Favorite Camping Activity 50% 40% 30% 20% 10% 0% Gathering at campfire Enjoying scenery Being outside The Snapshot also reports that men go camping 2.9 times a year, women go 1.7 times a year; and men are more likely than women to want to camp more often. What does the magazine mean when they talk about 2.9 or 1.7 times a year? 2.71 Long-Stemmed Roses A strain of longstemmed roses has an approximate normal distribution with a mean stem length of 15 inches and standard deviation of 2.5 inches. a. If one accepts as “long-stemmed roses” only those roses with a stem length greater than 12.5 inches, what percentage of such roses would be unacceptable? b. What percentage of these roses would have a stem length between 12.5 and 20 inches? 2.72 Drugs for Hypertension A pharmaceutical company wishes to know whether an EX0272 experimental drug being tested in its laboratories has any effect on systolic blood pressure. Fifteen randomly selected subjects were given the drug, and their systolic blood pressures (in millimeters) are recorded. 172 140 123 130 115 148 108 129 137 161 123 152 133 128 142 a. Guess the value of s using the range approximation. b. Calculate x and s for the 15 blood pressures. c. Find two values, a and b, such that at least 75% of the measurements fall between a and b. 2.73 Lumber Rights A company interested in lumbering rights for a certain tract of slash pine trees is told that the mean diameter of these trees is 14 inches with a standard deviation of 2.8 inches. Assume the distribution of diameters is roughly mound-shaped. a. What fraction of the trees will have diameters between 8.4 and 22.4 inches? b. What fraction of the trees will have diameters greater than 16.8 inches? 2.74 Social Ambivalence The following data represent the social ambivalence scores EX0274 for 15 people as measured by a |
psychological test. (The higher the score, the stronger the ambivalence.) 9 14 10 8 11 13 15 4 19 17 12 11 10 13 9 a. Guess the value of s using the range approximation. b. Calculate x and s for the 15 social ambivalence scores. c. What fraction of the scores actually lie in the inter- val x 2s? 2.75 TV Commercials The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed. a. What is the approximate probability that a commercial will last less than 35 seconds? b. What is the approximate probability that a commercial will last longer than 55 seconds? 2.76 Parasites in Foxes A random sample of 100 foxes was examined by a team of veterinarians to determine the prevalence of a particular type of parasite. Counting the number of parasites per fox, the veterinarians found that 69 foxes had no parasites, 17 had one parasite, and so on. A frequency tabulation of the data is given here: SUPPLEMENTARY EXERCISES ❍ 93 Mark McGwire’s record of 70 home runs hit in a single season. At the end of the 2003 major league baseball season, the number of home runs hit per season by each of four major league superstars over each player’s career were recorded, and are shown in the box plots below:14 r e y a l P Ruth McGwire Sosa Bonds 0 10 20 30 40 50 60 70 80 Homers Number of Parasites, x Number of Foxes, f 0 1 69 17 Write a short paragraph comparing the home run hitting patterns of these four players. a. Construct a relative frequency histogram for x, the number of parasites per fox. b. Calculate x and s for the sample. c. What fraction of the parasite counts fall within two standard deviations of the mean? Within three standard deviations? Do these results agree with Tchebysheff’s Theorem? With the Empirical Rule? 2.77 College Teachers Consider a population consisting of the number of teachers per college at small 2-year colleges. Suppose that the number of teachers per college has an average m 175 and a standard deviation s 15. a. Use Tchebysheff’s Theorem to make a statement about the percentage of colleges that have between 145 and 205 teachers. What fraction of colleges have more than 190 teachers? 2 |
.78 Is It Accurate? From the following data, a student calculated s to be.263. On what EX0278 grounds might we doubt his accuracy? What is the correct value (to the nearest hundredth)? 17.2 17.1 17.1 17.0 17.0 17.1 17.1 16.9 16.9 17.0 17.0 17.1 17.1 17.3 17.0 17.2 17.3 17.4 17.2 17.1 2.79 Homerun Kings In the summer of 2001, Barry Bonds began his quest to break EX0279 2.80 Barry Bonds In the seasons that followed his 2001 record-breaking season, EX0280 Barry Bonds hit 46, 45, 45, 5, and 26 homers, respectively (www.espn.com).14 Two boxplots, one of Bond’s homers through 2001, and a second including the years 2002–2006, follow. 2001 2006 * 60 70 80 30 40 Homers by Barry Bonds 50 The statistics used to construct these boxplots are given in the table. Years Min Q1 Median Q3 IQR Max 2001 2006 16 5 25.00 25.00 34.00 34.00 41.50 45.00 16.5 20.0 73 73 n 16 21 a. Calculate the upper fences for both of these boxplots. b. Can you explain why the record number of homers is an outlier in the 2001 boxplot, but not in the 2006 boxplot? b. Assume that the population is normally distributed. 0 10 20 94 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES 2.81 Ages of Pennies Here are the ages of 50 pennies from Exercise 1.45 and data set EX0145. The data have been sorted from smallest to largest. Stem-and-Leaf Display: Liters Stem-and-leaf of Liters N 30 Leaf Unit 0.10 0 0 2 6 19 0 0 3 8 20 0 1 3 9 20 0 1 3 9 21 0 1 4 10 22 0 1 4 16 23 0 1 5 17 25 0 1 5 17 25 0 2 5 19 28 0 2 5 19 36 a. What is the average age of the pennies? b. What is the median age of the pennies? c. Based on the results of parts a and b, how would you describe the age distribution of these 50 pennies? |
d. Construct a box plot for the data set. Are there any outliers? Does the box plot confirm your description of the distribution’s shape? 2.82 Snapshots Here are a few facts reported as Snapshots in USA Today. • The median hourly pay for salespeople in the building supply industry is $10.41.15 • Sixty-nine percent of U.S. workers ages 16 and older work at least 40 hours per week.16 • Seventy-five percent of all Associate Professors of Mathematics in the U.S. earn $91,823 or less.17 Identify the variable x being measured, and any percentiles you can determine from this information. 2.83 Breathing Patterns Research psychologists are interested in finding out EX0283 whether a person’s breathing patterns are affected by a particular experimental treatment. To determine the general respiratory patterns of the n 30 people in the study, the researchers collected some baseline measurements—the total ventilation in liters of air per minute adjusted for body size—for each person before the treatment. The data are shown here, along with some descriptive tools generated by MINITAB. 5.23 5.92 4.67 4.79 5.38 5.77 5.83 6.34 5.84 5.37 5.12 6.19 4.35 5.14 5.58 5.54 4.72 5.72 6.04 5.17 5.16 5.48 4.99 5.32 6.58 4.51 4.96 4.82 5.70 5.63 Descriptive Statistics: Liters N Variable 30 Liters N* 0 Mean 5.3953 SE Mean 0.0997 StDev 0.5462 Minimum Q1 Median 4.3500 4.9825 5.3750 5.7850 Liters Q3 Variable Maximum 6.5800 4 3 4 5 4 677 4 899 1 2 5 8 12 5 1111 (4) 5 2333 14 5 455 11 5 6777 7 4 2 1 5 889 6 01 6 3 6 5 a. Summarize the characteristics of the data distribution using the MINITAB output. b. Does the Empirical Rule provide a good description of the proportion of measurements that fall within two or three standard deviations of the mean? Explain. c. How large or small does a ventilation measurement have to be before it is considered unusual? |
2.84 Arranging Objects The following data are the response times in seconds for n 25 EX0284 first graders to arrange three objects by size. 5.2 4.2 3.1 3.6 4.7 3.8 4.1 2.5 3.9 3.3 5.7 4.3 3.0 4.8 4.2 3.9 4.7 4.4 5.3 3.8 3.7 4.3 4.8 4.2 5.4 a. Find the mean and the standard deviation for these 25 response times. b. Order the data from smallest to largest. c. Find the z-scores for the smallest and largest response times. Is there any reason to believe that these times are unusually large or small? Explain. 2.85 Arranging Objects, continued Refer to Exercise 2.84. a. Find the five-number summary for this data set. b. Construct a box plot for the data. c. Are there any unusually large or small response times identified by the box plot? d. Construct a stem and leaf display for the response times. How would you describe the shape of the distribution? Does the shape of the box plot confirm this result? Exercises 2.86 Refer to Data Set #1 in the How Extreme Values Affect the Mean and Median applet. This applet loads with a dotplot for the following n 5 observations: 2, 5, 6, 9, 11. a. What are the mean and median for this data set? b. Use your mouse to change the value x 11 (the moveable green dot) to x 13. What are the mean and median for the new data set? c. Use your mouse to move the green dot to x 33. When the largest value is extremely large compared to the other observations, which is larger, the mean or the median? d. What effect does an extremely large value have on the mean? What effect does it have on the median? 2.87 Refer to Data Set #2 in the How Extreme Values Affect the Mean and Median applet. This applet loads with a dotplot for the following n 5 observations: 2, 5, 10, 11, 12. a. Use your mouse to move the value x 12 to the left until it is smaller than the value x 11. b. As the value of x gets smaller, what happens to the |
sample mean? c. As the value of x gets smaller, at what point does the value of the median finally change? d. As you move the green dot, what are the largest and smallest possible values for the median? 2.88 Refer to Data Set #3 in the How Extreme Values Affect the Mean and Median applet. This applet loads with a dotplot for the following n 5 observations: 27, 28, 32, 34, 37. a. What are the mean and median for this data set? b. Use your mouse to change the value x 27 (the moveable green dot) to x 25. What are the mean and median for the new data set? c. Use your mouse to move the green dot to x 5. When the smallest value is extremely small compared to the other observations, which is larger, the mean or the median? d. At what value of x does the mean equal the median? e. What are the smallest and largest possible values for the median? f. What effect does an extremely small value have on the mean? What effect does it have on the median? 2.89 Refer to the Why Divide by n 1 applet. The first applet on the page randomly selects sample of MYAPPLET EXERCISES ❍ 95 n 3 from a population in which the standard deviation is s 29.2. a. Click. A sample consisting of n 3 observations will appear. Use your calculator to verify the values of the standard deviation when dividing by n 1 and n as shown in the applet. b. Click again. Calculate the average of the two standard deviations (dividing by n 1) from parts a and b. Repeat the process for the two standard deviations (dividing by n). Compare your results to those shown in red on the applet. c. You can look at how the two estimators in part a or behave “in the long run” by clicking a number of times, until the average of all the standard deviations begins to stabilize. Which of the two methods gives a standard deviation closer to s 29.2? d. In the long run, how far off is the standard deviation when dividing by n? 2.90 Refer to Why Divide by n 1 applet. The second applet on the page randomly selects sample of n 10 from the same population in which the standard deviation is s 29.2. a. Repeat the instructions in part c |
and d of Exer- cise 2.89. b. Based on your simulation, when the sample size is larger, does it make as much difference whether you divide by n or n 1 when computing the sample standard deviation? 2.91 If you have not yet done so, use the first Building a Box Plot applet to construct a box plot for the data in Example 2.14. a. Compare the finished box plot to the plot shown in Figure 2.18. b. How would you describe the shape of the data distribution? c. Are there any outliers? If so, what is the value of the unusual observation? 2.92 Use the second Building a Box Plot applet to construct a box plot for the data in Example 2.13. a. How would you describe the shape of the data distribution? b. Use the box plot to approximate the values of the median, the lower quartile, and the upper quartile. Compare your results to the actual values calculated in Example 2.13. 96 ❍ CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES CASE STUDY Batting The Boys of Summer Which baseball league has had the best hitters? Many of us have heard of baseball greats like Stan Musial, Hank Aaron, Roberto Clemente, and Pete Rose of the National League and Ty Cobb, Babe Ruth, Ted Williams, Rod Carew, and Wade Boggs of the American League. But have you ever heard of Willie Keeler, who batted.432 for the Baltimore Orioles, or Nap Lajoie, who batted.422 for the Philadelphia A’s? The batting averages for the batting champions of the National and American Leagues are given on the Student Companion Website. The batting averages for the National League begin in 1876 with Roscoe Barnes, whose batting average was.403 when he played with the Chicago Cubs. The last entry for the National League is for the year 2006, when Freddy Sanchez of the Pittsburgh Pirates averaged.344. The American League records begin in 1901 with Nap Lojoie of the Philadelphia A’s, who batted.422, and end in 2006 with Joe Mauer of the Minnesota Twins, who batted.347.18 How can we summarize the information in this data set? 1. Use MINITAB or another statistical software package to describe the batting averages for the American and National League batting champions. Generate any graphics that may help you in interpreting these |
data sets. 2. Does one league appear to have a higher percentage of hits than the other? Do the batting averages of one league appear to be more variable than the other? 3. Are there any outliers in either league? 4. Summarize your comparison of the two baseball leagues. 3 Describing Bivariate Data GENERAL OBJECTIVES Sometimes the data that are collected consist of observations for two variables on the same experimental unit. Special techniques that can be used in describing these variables will help you identify possible relationships between them. CHAPTER INDEX ● The best-fitting line (3.4) ● Bivariate data (3.1) ● Covariance and the correlation coefficient (3.4) ● Scatterplots for two quantitative variables (3.3) ● Side-by-side pie charts, comparative line charts (3.2) ● Side-by-side bar charts, stacked bar charts (3.2) How Do I Calculate the Correlation Coefficient? How Do I Calculate the Regression Line? © Janis Christie/Photodisc/Getty Images Do You Think Your Dishes Are Really Clean? Does the price of an appliance, such as a dishwasher, convey something about its quality? In the case study at the end of this chapter, we rank 20 different brands of dishwashers according to their prices, and then we rate them on various characteristics, such as how the dishwasher performs, how much noise it makes, its cost for either gas or electricity, its cycle time, and its water use. The techniques presented in this chapter will help to answer our question. 97 98 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA BIVARIATE DATA 3.1 Very often researchers are interested in more than just one variable that can be measured during their investigation. For example, an auto insurance company might be interested in the number of vehicles owned by a policyholder as well as the number of drivers in the household. An economist might need to measure the amount spent per week on groceries in a household and also the number of people in that household. A real estate agent might measure the selling price of a residential property and the square footage of the living area. When two variables are measured on a single experimental unit, the resulting data are called bivariate data. How should you display these data? Not only are both variables important when studied separately, but you also may want to explore the relationship between the two variables. Methods for |
graphing bivariate data, whether the variables are qualitative or quantitative, allow you to study the two variables together. As with univariate data, you use different graphs depending on the type of variables you are measuring. “Bi” means “two.” Bivariate data generate pairs of measurements. GRAPHS FOR QUALITATIVE VARIABLES 3.2 When at least one of the two variables is qualitative, you can use either simple or more intricate pie charts, line charts, and bar charts to display and describe the data. Sometimes you will have one qualitative and one quantitative variable that have been measured in two different populations or groups. In this case, you can use two sideby-side pie charts or a bar chart in which the bars for the two populations are placed side by side. Another option is to use a stacked bar chart, in which the bars for each category are stacked on top of each other. Are professors in private colleges paid more than professors at public colleges? The data in Table 3.1 were collected from a sample of 400 college professors whose rank, type of college, and salary were recorded.1 The number in each cell is the average salary (in thousands of dollars) for all professors who fell into that category. Use a graph to answer the question posed for this sample. EXAMPLE 3.1 TABLE 3.1 ● Salaries of Professors by Rank and Type of College Full Professor Associate Professor Assistant Professor Public Private 94.8 118.1 65.9 76.0 56.4 65.1 Source: Digest of Educational Statistics Solution To display the average salaries of these 400 professors, you can use a side-by-side bar chart, as shown in Figure 3.1. The height of the bars is the average salary, with each pair of bars along the horizontal axis representing a different professorial rank. Salaries are substantially higher for full professors in private colleges, however, there are less striking differences at the lower two ranks. 3.2 GRAPHS FOR QUALITATIVE VARIABLES ❍ 99 School Public Private FI GUR E 3. 1 Comparative bar charts for Example 3.1 ● 120 100 80 60 40 20 ) School Rank Public Private Full Public Private Associate Public Private Assistant EXAMPLE 3.2 Along with the salaries for the 400 college professors in Example 3.1, the researcher recorded two qualitative variables for each professor: rank and type of college. Table 3.2 shows the number of professors in |
each of the 2 3 6 categories. Use comparative charts to describe the data. Do the private colleges employ as many highranking professors as the public colleges do? TABLE 3.2 ● Number of Professors by Rank and Type of College Full Professor Associate Professor Assistant Professor Public Private 24 60 57 78 69 112 Total 150 250 Solution The numbers in the table are not quantitative measurements on a single experimental unit (the professor). They are frequencies, or counts, of the number of professors who fall into each category. To compare the numbers of professors at public and private colleges, you might draw two pie charts and display them side by side, as in Figure 3.2. FI GUR E 3. 2 Comparative pie charts for Example 3.2 ● Private Public Category Full Professor Associate Professor Assistant Professor 24.0% 16.0% 44.8% 46.0% 31.2% 38.0% 100 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA Alternatively, you could draw either a stacked or a side-by-side bar chart. The stacked bar chart is shown in Figure 3.3. F IG URE 3. 3 Stacked bar chart for Example 3.2 ● 200 150 100 50 Rank Full Associate Assistant School Public Private Although the graphs are not strikingly different, you can see that public colleges have fewer full professors and more associate professors than private colleges. The reason for these differences is not clear, but you might speculate that private colleges, with their higher salaries, are able to attract more full professors. Or perhaps public colleges are not as willing to promote professors to the higher-paying ranks. In any case, the graphs provide a means for comparing the two sets of data. You can also compare the distributions for public versus private colleges by creating conditional data distributions. These conditional distributions are shown in Table 3.3. One distribution shows the proportion of professors in each of the three ranks under the condition that the college is public, and the other shows the proportions under the condition that the college is private. These relative frequencies are easier to compare than the actual frequencies and lead to the same conclusions: • The proportion of assistant professors is roughly the same for both public and private colleges. • Public colleges have a smaller proportion of full professors and a larger proportion of associate professors. TABLE 3.3 ● Proportions of Professors by Rank for Public and Private Colleges Assistant Professor Associate Professor Full Professor Total Public Private 2 4.16 0 5 1 0 6.24 5 0 2 5 7.38 |
0 5 1 8 7.31 5 0 2 6 9.46 0 5 1 2 1 1.45 5 0 2 1.00 1.00 3.2 GRAPHS FOR QUALITATIVE VARIABLES ❍ 101 3.2 EXERCISES BASIC TECHNIQUES APPLICATIONS 3.1 Gender Differences Male and female respondents to a questionnaire about gender differences are categorized into three groups according to their answers to the first question: 3.4 M&M’S The color distributions for two snacksize bags of M&M’S® candies, one plain and one peanut, are displayed in the table. Choose an appropriate graphical method and compare the distributions. Group 1 Group 2 Group 3 Brown Yellow Red Orange Green Blue Men Women 37 7 49 50 72 31 a. Create side-by-side pie charts to describe these data. b. Create a side-by-side bar chart to describe these data. c. Draw a stacked bar chart to describe these data. d. Which of the three charts best depicts the difference or similarity of the responses of men and women? 3.2 State-by-State A group of items are categorized according to a certain attribute—X, Y, Z—and according to the state in which they are produced: New York California X 20 10 Y 5 10 Z 5 5 a. Create a comparative (side-by-side) bar chart to compare the numbers of items of each type made in California and New York. b. Create a stacked bar chart to compare the numbers of items of each type made in the two states. c. Which of the two types of presentation in parts a and b is more easily understood? Explain. d. What other graphical methods could you use to describe the data? 3.3 Consumer Spending The table below shows the average amounts spent per week by men and women in each of four spending categories: A $54 21 B $27 85 C $105 100 D $22 75 Men Women a. What possible graphical methods could you use to compare the spending patterns of women and men? b. Choose two different methods of graphing and dis- play the data in graphical form. Plain Peanut 15 6 14 2 12 2 4 3 5 3 6 5 3.5 How Much Free Time? When you were growing up, did you feel that you did not have enough free time? Parents and children have differing opinions on this subject. A research group surveyed 198 parents and 200 children and |
recorded their responses to the question, “How much free time does your child have?” or “How much free time do you have?” The responses are shown in the table below:2 Just the Right Amount Not Enough Much Too Don’t Know Parents Children 138 130 14 48 40 16 6 6 a. Define the sample and the population of interest to the researchers. b. Describe the variables that have been measured in this survey. Are the variables qualitative or quantitative? Are the data univariate or bivariate? c. What do the entries in the cells represent? d. Use comparative pie charts to compare the responses for parents and children. e. What other graphical techniques could be used to describe the data? Would any of these techniques be more informative than the pie charts constructed in part d? 3.6 Consumer Price Index The price of living in the United States has increased EX0306 dramatically in the past decade, as demonstrated by the consumer price indexes (CPIs) for housing and transportation. These CPIs are listed in the table for the years 1996 through the first five months of 2007.3 Year 1996 1997 1998 1999 2000 2001 Housing Transportation 152.8 143.0 2002 180.3 152.9 156.8 144.3 2003 184.8 157.6 160.4 141.6 2004 189.5 163.1 163.9 144.4 2005 195.7 173.9 169.6 153.3 2006 203.2 180.9 176.4 154.3 2007 207.8 181.0 c. What can you say about the similarities or differ- Year ences in the spending patterns for men and women? d. Which of the two methods used in part b provides a better descriptive graph? Housing Transportation Source: www.bls.gov 102 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA a. Create side-by-side comparative bar charts to c. What conclusions can you draw using the graphs in describe the CPIs over time. parts a and b? b. Draw two line charts on the same set of axes to describe the CPIs over time. c. What conclusions can you draw using the two graphs in parts a and b? Which is the most effective? EX0308 3.8 Charitable Contributions Charitable organizations count on support from both private donations and other sources. Here are the sources of income in a recent year for several well-known charitable |
organizations in the United States.4 3.7 How Big Is the Household? A local chamber of commerce surveyed 126 house- EX0307 holds within its city and recorded the type of residence and the number of family members in each of the households. The data are shown in the table. Family Members Apartment Duplex Single Residence Type of Residence 1 2 3 4 or more 8 15 9 6 10 4 5 1 2 14 24 28 a. Use a side-by-side bar chart to compare the number of family members living in each of the three types of residences. b. Use a stacked bar chart to compare the number of family members living in each of the three types of residences. Amounts ($ millions) Organization Salvation Army YMCA American Red Cross American Cancer Society American Heart Association Total Private $1545 773 557 868 436 $4179 Source: The World Almanac and Book of Facts 2007 Other $1559 4059 2509 58 157 $8342 Total $3104 4832 3066 926 593 $12,521 a. Construct a stacked bar chart to display the sources of income given in the table. b. Construct two comparative pie charts to display the sources of income given in the table. c. Write a short paragraph summarizing the information that can be gained by looking at these graphs. Which of the two types of comparative graphs is more effective? SCATTERPLOTS FOR TWO QUANTITATIVE VARIABLES 3.3 When both variables to be displayed on a graph are quantitative, one variable is plotted along the horizontal axis and the second along the vertical axis. The first variable is often called x and the second is called y, so that the graph takes the form of a plot on the (x, y) axes, which is familiar to most of you. Each pair of data values is plotted as a point on this two-dimensional graph, called a scatterplot. It is the twodimensional extension of the dotplot we used to graph one quantitative variable in Section 1.4. You can describe the relationship between two variables, x and y, using the pat- terns shown in the scatterplot. • What type of pattern do you see? Is there a constant upward or downward trend that follows a straight-line pattern? Is there a curved pattern? Is there no pattern at all, but just a random scattering of points? • How strong is the pattern? Do all of the points follow the pattern exactly, |
or is the relationship only weakly visible? • Are there any unusual observations? An outlier is a point that is far from the cluster of the remaining points. Do the points cluster into groups? If so, is there an explanation for the observed groupings? 3.3 SCATTERPLOTS FOR TWO QUANTITATIVE VARIABLES ❍ 103 EXAMPLE 3.3 The number of household members, x, and the amount spent on groceries per week, y, are measured for six households in a local area. Draw a scatterplot of these six data points95.75 $110.19 $118.33 $150.92 $85.86 $180.62 Solution Label the horizontal axis x and the vertical axis y. Plot the points using the coordinates (x, y) for each of the six pairs. The scatterplot in Figure 3.4 shows the six pairs marked as dots. You can see a pattern even with only six data pairs. The cost of weekly groceries increases with the number of household members in an apparent straight-line relationship. Suppose you found that a seventh household with two members spent $165 on groceries. This observation is shown as an X in Figure 3.4. It does not fit the linear pattern of the other six observations and is classified as an outlier. Possibly these two people were having a party the week of the survey! FI GUR E 3. 4 Scatterplot for Example 3.3 ● y 180 160 140 120 100 80 1 2 3 x 4 5 EXAMPLE 3.4 A distributor of table wines conducted a study of the relationship between price and demand using a type of wine that ordinarily sells for $10.00 per bottle. He sold this wine in 10 different marketing areas over a 12-month period, using five different price levels—from $10 to $14. The data are given in Table 3.4. Construct a scatterplot for the data, and use the graph to describe the relationship between price and demand. TABLE 3.4 ● Cases of Wine Sold at Five Price Levels Cases Sold per 10,000 Population Price per Bottle 23, 21 19, 18 15, 17 19, 20 25, 24 $10 11 12 13 14 104 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA Solution The 10 data points are plotted in Figure 3.5. As the price increases from $10 to $12 the demand decreases. However, as |
the price continues to increase, from $12 to $14, the demand begins to increase. The data show a curved pattern, with the relationship changing as the price changes. How do you explain this relationship? Possibly, the increased price is a signal of increased quality for the consumer, which causes the increase in demand once the cost exceeds $12. You might be able to think of other reasons, or perhaps some other variable, such as the income of people in the marketing areas, that may be causing the change. F IG URE 3. 5 Scatterplot for Example 3.4 ● 25.0 22.5 s e s a C 20.0 17.5 15.0 10 11 12 Price 13 14 Now would be a good time for you to try creating a scatterplot on your own. Use the applets in Building a Scatterplot to create the scatterplots that you see in Figures 3.5 and 3.7. You will find step-by-step instructions on the left-hand side of the applet (Figure 3.6), and you will be corrected if you make a mistake! F IG URE 3. 6 Building a Scatterplot applet ● 3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA ❍ 105 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA 3.4 A constant rate of increase or decrease is perhaps the most common pattern found in bivariate scatterplots. The scatterplot in Figure 3.4 exhibits this linear pattern—that is, a straight line with the data points lying both above and below the line and within a fixed distance from the line. When this is the case, we say that the two variables exhibit a linear relationship. EXAMPLE 3.5 The data in Table 3.5 are the size of the living area (in square feet), x, and the selling price, y, of 12 residential properties. The MINITAB scatterplot in Figure 3.7 shows a linear pattern in the data. TABLE 3.5 ● Living Area and Selling Price of 12 Properties Residence x (sq. ft.) y (in thousands) 1360 1940 1750 1550 1790 1750 2230 1600 1450 1870 2210 1480 $278.5 375.7 339.5 329.8 295.6 310.3 460.5 305.2 288.6 365.7 425.3 268. 10 |
11 12 450 400 y 350 300 250 FI GUR E 3. 7 Scatterplot of x versus y for Example 3.5 ● 1400 1600 1800 x 2000 2200 For the data in Example 3.5, you could describe each variable, x and y, individually using descriptive measures such as the means (x and y) or the standard deviations (sx and sy). However, these measures do not describe the relationship between x and y for a particular residence—that is, how the size of the living space affects the selling price of the home. A simple measure that serves this purpose is called the correlation coefficient, denoted by r, and is defined as r sxy sxsy 106 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA The quantities sx and sy are the standard deviations for the variables x and y, respectively, which can be found by using the statistics function on your calculator or the computing formula in Section 2.3. The new quantity sxy is called the covariance between x and y and is defined as yi y) x )( sxy 1 S(xi n There is also a computing formula for the covariance: (Syi) Sxiyi (Sxi) n n 1 sxy where Sxiyi is the sum of the products xiyi for each of the n pairs of measurements. How does this quantity detect and measure a linear pattern in the data? Look at the signs of the cross-products (xi x)(yi y) in the numerator of r, or sxy. When a data point (x, y) is in either area I or III in the scatterplot shown in Figure 3.8, the cross-product will be positive; when a data point is in area II or IV, the cross-product will be negative. We can draw these conclusions: • • • If most of the points are in areas I and III (forming a positive pattern), sxy and r will be positive. If most of the points are in areas II and IV (forming a negative pattern), sxy and r will be negative. If the points are scattered across all four areas (forming no pattern), sxy and r will be close to 0. ● F IG URE 3. 8 The signs of the crossproducts (xi x)(yi y) in the covariance formula y y II : – I : + II : – I : + y y y y |
II : – I : + III : + IV : – III : + IV : – III : + IV : – x x x x x x (a) Positive pattern (b) Negative pattern (c) No pattern The applet called Exploring Correlation will help you to visualize how the pattern of points affects the correlation coefficient. Use your mouse to move the slider at the bottom of the scatterplot (Figure 3.9). You will see the value of r change as the pattern of the points changes. Notice that a positive pattern (a) results in a positive value of r; no pattern (c) gives a value of r close to zero; and a negative pattern (b) results in a negative value of r. What pattern do you see when r 1? When r 1? You will use this applet again for the MyApplet Exercises section at the end of the chapter. 3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA ❍ 107 FI GUR E 3. 9 Exploring Correlation applet ● r ⇔ positive linear relationship r 0 ⇔ negative linear relationship r 0 ⇔ no relationship Most scientific and graphics calculators can compute the correlation coefficient, r, when the data are entered in the proper way. Check your calculator manual for the proper sequence of entry commands. Computer programs such as MINITAB are also programmed to perform these calculations. The MINITAB output in Figure 3.10 shows the covariance and correlation coefficient for x and y in Example 3.5. In the covariance table, you will find these values: sxy 15,545.20 x 79,233.33 s2 y 3571.16 s2 and in the correlation output, you find r.924. However you decide to calculate the correlation coefficient, it can be shown that the value of r always lies between 1 and 1. When r is positive, x increases when y increases, and vice versa. When r is negative, x decreases when y increases, or x increases when y decreases. When r takes the value 1 or 1, all the points lie exactly on a straight line. If r 0, then there is no apparent linear relationship between the two variables. The closer the value of r is to 1 or 1, the stronger the linear relationship between the two variables. ● FI GUR E 3. 10 MINITAB output of covariance and correlation for Example 3 |
.5 Covariances: x, y Correlations: x, y x y x 79233.33 y 15545.20 3571.16 Pearson correlation of x and y = 0.924 P-Value = 0.000 EXAMPLE 3.6 Find the correlation coefficient for the number of square feet of living area and the selling price of a home for the data in Example 3.5. Solution Three quantities are needed to calculate the correlation coefficient. The standard deviations of the x and y variables are found using a calculator with a statistical function. You can verify that sx 281.4842 and sy 59.7592. Finally, 108 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA (Syi) Sxiyi (Sxi) n n 1 4043.5) )( 7,240,383 (20,980 1 2 11 sxy 15,545.19697 This agrees with the value given in the MINITAB printout in Figure 3.10. Then r sxy sxsy 15,545.19697 (281.4842)(59.7592).9241 which also agrees with the value of the correlation coefficient given in Figure 3.10. (You may wish to verify the value of r using your calculator.) This value of r is fairly close to 1, which indicates that the linear relationship between these two variables is very strong. Additional information about the correlation coefficient and its role in analyzing linear relationships, along with alternative calculation formulas, can be found in Chapter 12. x “explains” y or y “depends on” x. x is the explanatory or independent variable. y is the response or dependent variable. Sometimes the two variables, x and y, are related in a particular way. It may be that the value of y depends on the value of x; that is, the value of x in some way explains the value of y. For example, the cost of a home (y) may depend on its amount of floor space (x); a student’s grade point average (x) may explain her score on an achievement test (y). In these situations, we call y the dependent variable, while x is called the independent variable. If one of the two variables can be classified as the dependent variable y and the other as x, and if the data exhibit a straight-line pattern, it is possible to describe the relationship relating |
y to x using a straight line given by the equation y a bx as shown in Figure 3.11. F IG URE 3. 11 The graph of a straight line ● y y = a + bx As you can see, a is where the line crosses or intersects the y-axis: a is called the y-intercept. You can also see that for every one-unit increase in x, y increases by an amount b. The quantity b determines whether the line is increasing (b 0), decreasing (b 0), or horizontal (b 0) and is appropriately called the slope of the line. 3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA ❍ 109 You can see the effect of changing the slope and the y-intercept of a line using the applet called How a Line Works. Use your mouse to move the slider on the right side of the scatterplot. As you move the slider, the slope of the line, shown as the vertical side of the green triangle (light gray in Figure 3.12), will change. Moving the slider on the left side of the applet causes the y-intercept, shown in red (blue in Figure 3.12), to change. What is the slope and y-intercept for the line shown in the applet in Figure 3.12? You will use this applet again for the MyApplet Exercises section at the end of the chapter. FI GUR E 3. 12 How a Line Works applet ● Our points (x, y) do not all fall on a straight line, but they do show a trend that could be described as a linear pattern. We can describe this trend by fitting a line as best we can through the points. This best-fitting line relating y to x, often called the regression or least-squares line, is found by minimizing the sum of the squared differences between the data points and the line itself, as shown in Figure 3.13. The formulas for computing b and a, which are derived mathematically, are shown below. COMPUTING FORMULAS FOR THE LEAST-SQUARES REGRESSION LINE b r and the least-squares regression line is: y a bx and sy sx a y bx FI GUR E 3. 13 The best-fitting line ● y 3 2 1 0 y = a + |
bx 1 2 3 4 5 x 110 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA Remember that r and b have the same sign! EXAMPLE 3.7 Since sx and sy are both positive, b and r have the same sign, so that: • When r is positive, so is b, and the line is increasing with x. • When r is negative, so is b, and the line is decreasing with x. • When r is close to 0, then b is close to 0. Find the best-fitting line relating y starting hourly wage to x number of years of work experience for the following data. Plot the line and the data points on the same graph6.00 7.50 8.00 12.00 13.00 15.50 Solution Use the data entry method for your calculator to find these descriptive statistics for the bivariate data set: x 4.5 y 10.333 sx 1.871 sy 3.710 r.980 Use the regression line to predict y for a given value of x. Then b r sy sx and.9803 1.9432389 1.943.. 7 8 1 7 0 1 1 a y bx 10.333 1.943(4.5) 1.590 Therefore, the best-fitting line is y 1.590 1.943x. The plot of the regression line and the actual data points are shown in Figure 3.14. The best-fitting line can be used to estimate or predict the value of the variable y when the value of x is known. For example, if a person applying for a job has 3 years of work experience (x), what would you predict his starting hourly wage (y) to be? From the best-fitting line in Figure 3.14, the best estimate would be y a bx 1.590 1.943(3) 7.419 F IG URE 3. 14 Fitted line and data points for Example 3.7 ● 15.0 12.5 y 10.0 7.5 5.0 y 1.590 1.943x 2 3 4 5 6 7 x 3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA ❍ 111 How Do I Calculate the Correlation Coefficient? 1. First, create a table or use your |
calculator to find Sx, Sy, and Sxy. 2. Calculate the covariance, sxy. 3. Use your calculator or the computing formula from Chapter 2 to calculate sx and sy. 4. Calculate r sxy sxsy. How Do I Calculate the Regression Line? 1. First, calculate y and x. Then, calculate r sxy sxsy. 2. Find the slope, b r sy sx and the y-intercept, a y bx. 3. Write the regression line by substituting the values for a and b into the equa- tion: y a bx. Exercise Reps A. Below you will find a simple set of bivariate data. Fill in the blanks to find the correlation coefficient. xy x 0 2 4 y 1 5 2 Calculate: n sx sy Covariance (Sy) Sxy (Sx) n n 1 sxy Correlation Coefficient Sx Sy Sxy r sxy sxsy B. Use the information from part A and find the regression line. From Part A Sx Sy From Part A sx sy r Calculate: x y Slope b r sy sx y-intercept a y bx Regression Line: y Answers are located on the perforated card at the back of this book. When should you describe the linear relationship between x and y using the correlation coefficient r, and when should you use the regression line y a bx? The regression approach is used when the values of x are set in advance and then the corresponding value of y is measured. The correlation approach is used when an experimental unit is selected at random and then measurements are made on both 112 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA variables x and y. This technical point will be taken up in Chapter 12, which addresses regression analysis. Most data analysts begin any data-based investigation by examining plots of the variables involved. If the relationship between two variables is of interest, data analysts can also explore bivariate plots in conjunction with numerical measures of location, dispersion, and correlation. Graphs and numerical descriptive measures are only the first of many statistical tools you will soon have at your disposal. 3.4 EXERCISES EXERCISE REPS These questions refer to the MyPersonal Trainer section on page 111. 3.9 Below you will find a simple |
set of bivariate data. Fill in the blanks to find the correlation coefficient. xy x 1 3 2 y 6 2 4 Calculate: n sx sy Covariance (Sy) Sxy (Sx) n n 1 sxy Correlation Coefficient Sx Sy Sxy r sxy sxsy 3.10 Use the information from Exercise 3.9 and find the regression line. From Part A Sx Sy From Part A sx sy r Calculate: x y Slope b r sy sx y-intercept a y bx Regression Line: y BASIC TECHNIQUES 3.11 A set of bivariate data consists of these measurements on two variables, x and y: EX0311 (3, 6) (5, 8) (2, 6) (1, 4) (4, 7) (4, 6) a. Draw a scatterplot to describe the data. b. Does there appear to be a relationship between x and y? If so, how do you describe it? 3.12 Refer to Exercise 3.11. a. Use the data entry method in your scientific calculator to enter the six pairs of measurements. Recall the proper memories to find the correlation coefficient, r, the y-intercept, a, and the slope, b, of the line. b. Verify that the calculator provides the same values for r, a, and b as in Exercise 3.11. c. Calculate the correlation coefficient, r, using the 3.13 Consider this set of bivariate data: computing formula given in this section. EX0313 d. Find the best-fitting line using the computing for- mulas. Graph the line on the scatterplot from part a. Does the line pass through the middle of the points? x y 1 5.6 2 4.6 3 4.5 4 3.7 5 3.2 6 2.7 a. Draw a scatterplot to describe the data. 3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA ❍ 113 b. Does there appear to be a relationship between x and y? If so, how do you describe it? c. Calculate the correlation coefficient, r. Does the value of r confirm your conclusions in part b? Explain. 3.14 The value of a quantitative variable is measured once a year |
for a 10-year period: EX0314 price of 12 residential properties given in Example 3.5 are reproduced here. First, find the best-fitting line that describes these data, and then plot the line and the data points on the same graph. Comment on the goodness of the fitted line in describing the selling price of a residential property as a linear function of the square feet of living area. Year Measurement Year Measurement Residence x (sq. ft.) y (in thousands) 1 2 3 4 5 61.5 62.3 60.7 59.8 58.0 6 7 8 9 10 58.2 57.5 57.5 56.1 56.0 a. Draw a scatterplot to describe the variable as it changes over time. b. Describe the measurements using the graph con- structed in part a. c. Use this MINITAB output to calculate the correlation 1 2 3 4 5 6 7 8 9 10 11 12 1360 1940 1750 1550 1790 1750 2230 1600 1450 1870 2210 1480 $278.5 375.7 339.5 329.8 295.6 310.3 460.5 305.2 288.6 365.7 425.3 268.8 coefficient, r: MINITAB output for Exercise 3.14 C ovarianc es x y x 9.16667 y -6.42222 4.84933 d. Find the best-fitting line using the results of part c. Verify your answer using the data entry method in your calculator. e. Plot the best-fitting line on your scatterplot from part a. Describe the fit of the line. APPLICATIONS 3.15 Grocery Costs These data relating the amount spent on groceries per week and the EX0315 number of household members are from Example 3.3: 3.17 Disabled Students A social skills training program, reported in Psychology in EX0317 the Schools, was implemented for seven students with mild handicaps in a study to determine whether the program caused improvement in pre/post measures and behavior ratings.5 For one such test, these are the pretest and posttest scores for the seven students: Student Pretest Posttest Earl Ned Jasper Charlie Tom Susie Lori 101 89 112 105 90 91 89 113 89 121 99 104 94 99 a. Draw a scatterplot relating the posttest score to the pretest score. 2 2 |
3 4 1 5 b. Describe the relationship between pretest and $95.75 $110.19 $118.33 $150.92 $85.86 $180.62 a. Find the best-fitting line for these data. b. Plot the points and the best-fitting line on the same graph. Does the line summarize the information in the data points? c. What would you estimate a household of six to spend on groceries per week? Should you use the fitted line to estimate this amount? Why or why not? 3.16 Real Estate Prices The data relating the square feet of living space and the selling EX0316 posttest scores using the graph in part a. Do you see any trend? c. Calculate the correlation coefficient and interpret its value. Does it reinforce any relationship that was apparent from the scatterplot? Explain. EX0318 3.18 Lexus, Inc. The makers of the Lexus automobile have steadily increased their sales since their U.S. launch in 1989. However, the rate of increase changed in 1996 when Lexus introduced a line of trucks. The sales of Lexus from 1996 to 2005 are shown in the table.6 x y 114 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Sales (thousands 80 of vehicles) 100 155 180 210 224 234 260 288 303 Source: Adapted from: Automotive News, January 26, 2004, and May 22, 2006. a. Plot the data using a scatterplot. How would you describe the relationship between year and sales of Lexus? b. Find the least-squares regression line relating the sales of Lexus to the year being measured. c. If you were to predict the sales of Lexus in the year 2015, what problems might arise with your prediction? 3.19 HDTVs, again In Exercise 2.12, Consumer Reports gave the prices for the top EX0319 10 LCD high definition TVs (HDTVs) in the 30- to 40-inch category. Does the price of an LCD TV depend on the size of the screen? The table below shows the 10 costs again, along with the screen size.6 a. Which of the two variables (price and size) is the independent variable, and which is the dependent variable? b. Construct a scatterplot for the data. Does the rela |
- tionship appear to be linear? 3.20 HDTVs, continued Refer to Exercise 3.19. Suppose we assume that the relationship between x and y is linear. a. Find the correlation coefficient, r. What does this value tell you about the strength and direction of the relationship between size and price? b. What is the equation of the regression line used to predict the price of the TV based on the size of the screen? c. The Sony Corporation is introducing a new 37" LCD TV. What would you predict its price to be? d. Would it be reasonable to try to predict the price of a 45" LCD TV? Explain. Brand JVC LT-40FH96 Sony Bravia KDL-V32XBR1 Sony Bravia KDL-V40XBR1 Toshiba 37HLX95 Sharp Aquos LC-32DA5U Sony Bravia KLV-S32A10 Panasonic Viera TC-32LX50 JVC LT-37X776 LG 37LP1D Samsung LN-R328W Price $2900 1800 2600 3000 1300 1500 1350 2000 2200 1200 Size 40" 32" 40" 37" 32" 32" 32" 37" 37" 32" CHAPTER REVIEW Key Concepts I. Bivariate Data 1. Both qualitative and quantitative variables 2. Describing each variable separately 3. Comparative bar charts a. Side-by-side b. Stacked 3. Describing the relationship between the two 4. Relative frequencies to describe the relation- variables ship between the two variables II. Describing Two Qualitative III. Describing Two Quantitative Variables 1. Side-by-side pie charts 2. Comparative line charts Variables 1. Scatterplots a. Linear or nonlinear pattern b. Strength of relationship c. Unusual observations: clusters and outliers 2. Covariance and correlation coefficient 3. The best-fitting regression line a. Calculating the slope and y-intercept b. Graphing the line c. Using the line for prediction MY MINITAB ❍ 115 Describing Bivariate Data MINITAB provides different graphical techniques for qualitative and quantitative bivariate data, as well as commands for obtaining bivariate descriptive measures when the data are quantitative. To explore both types of bivariate procedures, you need to enter two different sets of bivariate data into a MINITAB worksheet. Once you are on the Windows desktop, double-click on the MIN |
ITAB icon or use the Start button to start MINITAB. Start a new project using File New Minitab Project. Then open the existing project called “Chapter 1.” We will use the college student data, which should be in Worksheet 1. Suppose that the 105 students already tabulated were from the University of California, Riverside, and that another 100 students from an introductory statistics class at UC Berkeley were also interviewed. Table 3.6 shows the status distribution for both sets of students. Create another variable in C3 of the worksheet called “College” and enter UCR for the first five rows. Now enter the UCB data in columns C1–C3. You can use the familiar Windows cut-and-paste icons if you like. TABLE 3.6 ● Freshman Sophomore Junior Senior Grad Student Frequency (UCR) Frequency (UCB) 5 10 23 35 32 24 35 25 10 6 The other worksheet in “Chapter 1” is not needed and can be deleted by clicking on the X in the top right corner of the worksheet. We will use the worksheet called “Minivans” from Chapter 2, which you should open using File Open Worksheet and selecting “Minivans.mtw.” Now save this new project as “Chapter 3.” To graphically describe the UCR/UCB student data, you can use comparative pie charts—one for each school (see Chapter 1). Alternatively, you can use either stacked or side-by-side bar charts. Use Graph Bar Chart. In the “Bar Charts” Dialog box (Figure 3.15), select Values from a Table in the drop-down list and click either Stack or Cluster in the row marked “One Column of Values.” Click OK. In the next Dialog box (Figure 3.16), select “Frequency” for the Graph variables box and “Status” and “College” for the Categorical variable for grouping box. Click OK. Once the bar chart is displayed (Figure 3.17), you can right-click on various items in the bar chart to edit. If you right-click on the bars and select Update Graph Automatically, the bar chart will automatically update when you change the data in the Minitab worksheet. 116 ❍ CHAPTER 3 DESCRIBING BIVARIATE |
DATA F IG URE 3. 15 ● F IG URE 3. 16 ● FI GUR E 3. 17 ● MY MINITAB ❍ 117 Turn to Worksheet 2, in which the bivariate minivan data from Chapter 2 are located. To examine the relationship between the second and third car seat lengths, you can plot the data and numerically describe the relationship with the correlation coefficient and the best-fitting line. Use Stat Regression Fitted Line Plot, and select “2nd Seat” and “3rd Seat” for Y and X, respectively (see Figure 3.18). Make sure that the dot next to Linear is selected, and click OK. The plot of the nine data points and the best-fitting line will be generated as in Figure 3.19. FI GUR E 3. 18 ● 118 ❍ CHAPTER 3 DESCRIBING BIVARIATE DATA F IG URE 3. 19 ● To calculate the correlation coefficient, use Stat Basic Statistics Correlation, selecting “2nd Seat” and “3rd Seat” for the Variables box. To select both variables at once, hold the Shift key down as you highlight the variables and then click Select. Click OK, and the correlation coefficient will appear in the Session window (see Figure 3.20). Notice the relatively strong positive correlation and the positive slope of the regression line, indicating that a minivan with a long floor length behind the second seat will also tend to have a long floor length behind the third seat. Save “Chapter 3” before you exit MINITAB! F IG URE 3. 20 ● Supplementary Exercises 3.21 Professor Asimov Professor Isaac Asimov was one of the most prolific writers of all time. He wrote nearly 500 books during a 40-year career prior to his death in 1992. In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time.8 These data are the times (in months) required to write his books, in increments of 100: Number of Books Time (in months) 100 237 200 350 300 419 400 465 490 507 a. Plot the accumulated number of books as a func- tion of time using a scatterplot. b. Describe the productivity of Professor Asimov in light of the data set graphed |
in part a. Does the relationship between the two variables seem to be linear? EX0322 3.22 Cheese, Please! Health-conscious Americans often consult the nutritional information on food packages in an attempt to avoid foods with large amounts of fat, sodium, or cholesterol. The following information was taken from eight different brands of American cheese slices: Brand Fat (g) Saturated Fat (g) Cholesterol (mg) Sodium (mg) Calories Kraft Deluxe American Kraft Velveeta Slices Private Selection Ralphs Singles Kraft 2% Milk Singles Kraft Singles American Borden Singles Lake to Lake American 7 5 8 4 3 5 5 5 4.5 3.5 5.0 2.5 2.0 3.5 3.0 3.5 20 15 25 15 10 15 15 15 340 300 520 340 320 290 260 330 80 70 100 60 50 70 60 70 a. Which pairs of variables do you expect to be strongly related? b. Draw a scatterplot for fat and saturated fat. De- scribe the relationship. c. Draw a scatterplot for fat and calories. Compare the pattern to that found in part b. d. Draw a scatterplot for fat versus sodium and another for cholesterol versus sodium. Compare the patterns. Are there any clusters or outliers? e. For the pairs of variables that appear to be linearly related, calculate the correlation coefficients. SUPPLEMENTARY EXERCISES ❍ 119 f. Write a paragraph to summarize the relationships you can see in these data. Use the correlations and the patterns in the four scatterplots to verify your conclusions. 3.23 Army versus Marine Corps Who are the men and women who serve in our armed forces? Are they male or female, officers or enlisted? What is their ethnic origin and their average age? An article in Time magazine provided some insight into the demographics of the U.S. armed forces.9 Two of the bar charts are shown below. U.S. Army Enlisted Officers 50% 40% 30% 20% 10% 0% 17 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 50 and over 45 to 49 Age U.S. Marine Corps Enlisted Officers 50% 40% 30% 20% 10% 0% 17 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 50 and over 45 to 49 Age a. What variables have been measured in this study? Are the variables qualitative or quantitative? b. Describe the |
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