Name: Unit 3 Regression and Correlation Lesson1: Bivariate Relationships PRACTICE PROBLEMS I can use appropriate vocabulary to describe the shape, strength, center, and spread of a scatterplot. Investigation Practice Problem Options Max Possible Points Total Points Earned Investigation 1: Rank Correlation #1, 2, 3 1 points Investigation 2: Shapes of Clouds of Points #4, 1 points points ** In order to earn credit for practice problems, ALL WORK must be shown.**
Applications 1 Which are the best steel roller coasters in the United States? The rankings below are from an Amusement Today annual survey of roller-coaster riders. The table and scatterplot show the top ten roller coasters from the 1999 survey and the order that those same coasters appeared in the 26 survey. Roller Coaster Rankings Roller Coaster 1999 Rank 26 Relative Rank Magnum XL-2, Cedar Point, OH 1 1 Montu, Busch Gardens, FL 2 2 Steel Force, Dorney Park, PA 3 Alpengeist, Busch Gardens, VA 4 6 Kumba, Busch Gardens, FL 8 Raptor, Cedar Point, OH 6 4 Desperado, Buffalo Bill s Resort, NV 7 1 Mind Bender, Six Flags Over Georgia, GA 8 7 Mamba, Worlds of Fun, MO 9 9 Superman, Ride of Steel, Six Flags Darien Lake, NY 1 3 Source: Amusement Today, August 2; www.amusementtoday.com6gtasteel.html Steel Coasters 26 Rank 12 1 8 6 4 2 /1 /3 2 4 6 8 1 12 1999 Rank a. Why might the ranks change from year to year? b. Is the relationship positive or negative? Strong or weak? Estimate the rank correlation by examining the scatterplot. c. Calculate the rank correlation. Compare it to your estimate. /4 Show your work. LESSON 1 Bivariate Relationships 269
2 The following are the 1 consumer products that emergency room patients in the United States most often say are related to the cause of their injuries. /4 Bathtubs and showers Beds Bicycles Cabinets, racks, and shelves Chairs Containers and packaging Knives Ladders Sofas Tables a. Ask a friend or member of your family to rank the products from 1 to 1, assigning 1 to the product he or she thinks causes the most emergency room visits in the United States, 2 to the product that he or she thinks causes the second largest number, and so on. b. The actual ranking is given below. Make a scatterplot comparing these rankings to those collected in Part a. Product Rank Bathtubs and showers 8 Beds 2 Bicycles 1 Cabinets, racks, and shelves 6 Chairs Containers and packaging 7 Knives 3 Ladders 9 Sofas 1 Tables 4 c. Compute the rank correlation between your friend s or family member s ranking and the actual ranking. Was your friend or family member relatively successful or relatively unsuccessful in matching the actual ranks? Show your work. 27 UNIT 4 Regression and Correlation
3 The population ranks for the 1 largest countries in the world for the year 2 are given in the table below. Also given is the projected rank for each country, relative to the other ten countries, for the years 22 and 2. Show your work for all calculations. Russia North Pacific Ocean United States Atlantic Ocean India China Japan Brazil Nigeria Pakistan Bangladesh Indonesia North Pacific Ocean Indian Ocean Population Rankings, Largest Countries in 2 Country 2 Population (in millions) 2 Population Rank 22 Projected Relative Rank 2 Projected Relative Rank Bangladesh 129.2 8 8 8 Brazil 17.1 6 7 China 1,277.6 1 1 2 India 1,13.7 2 2 1 Indonesia 212.1 4 4 6 Japan 126.7 9 1 1 Nigeria 111. 1 7 Pakistan 6. 6 3 Russia 146.9 7 9 9 United States 273.8 3 3 4 Source: The New York Times 21 Almanac. New York, NY: The New York Times, 2. /4 /4 a. Examine this scatterplot for the (2, 2) rankings. Write two observations that you can make from looking at the scatterplot. b. What is your estimate of the rank correlation for the (2, 2) rankings? Check your estimate by computing the rank correlation. c. Would you expect the correlation between the 2 ranking and the projected 22 ranking to be larger or smaller than the one you computed in Part b? Explain. Compute this correlation to see if you were correct. Population Rankings Relative Rank in 2 1 8 6 4 2 2 4 6 8 1 Rank in 2 LESSON 1 Bivariate Relationships 271
4 The table and scatterplot below show data for 2 countries. The variables are a measure of the carbon dioxide emissions (in metric tons) per person and the number of years a newborn can expect to live. Country Carbon Dioxide Emissions (in metric tons) per Person Life Expectancy at Birth Australia.2 8.3 Brazil. 71.4 Canada. 8. China.7 72. France 1.9 79.4 Germany 2.8 78. India.3 64 Indonesia.4 69.3 Iran 1. 69.7 Italy 2.2 79. Japan 2.6 81 Korea, South 2.6 76.7 Mexico 1. 74.9 Netherlands 4.4 78.7 Poland 2. 74.7 Russia 3. 66.8 Saudi Arabia 3.4 7.2 South Africa 2. 44.1 Spain 2.3 79.4 Taiwan 3.3 77.1 Thailand.8 71.7 Turkey.8 72.1 Ukraine 2. 68.8 United Kingdom 2.6 78.3 United States.4 77.4 Source: 26 and 27 Statistical Abstract of the U.S. Tables 1318 and 132. Life Expectancy and Carbon Dioxide Emissions by Country Life Expectancy 8 7 6 4 1 2 3 4 6 Carbon Dioxide Emissions (in metric tons) per Person 272 UNIT 4 Regression and Correlation
/4 a. What is the shape of the distribution? Describe the association between the two variables. Can you use Spearman s r s to quantify the strength? Why or why not? b. Do you think the value of one of the two variables causes or otherwise influences the value of the other? Explain your reasoning. c. Are there any outliers? If so, which type? The Places Rated Almanac ranks metropolitan areas according to a variety of categories including: crime violent crime and property crime rates health care the supply of health care services (such as number of specialists or breadth of hospital services) education the number of available educational opportunities beyond high school Some characteristics of the largest metropolitan areas in the United States are ranked in the table and following scatterplot matrix. For crime, health care, and education, a rank of 1 is best. On Your Own Philadelphia Los Angeles Boston Rankings of Metropolitan Areas Metro Area Population Crime Health Care Education Los Angeles, CA 1 14 9 12 New York, NY 2 4 Chicago, IL 3 13 7 2 Philadelphia, PA 4 4 7 Washington, DC 1 3 Detroit, MI 6 9 13 14 Houston, TX 7 7 1 13 Atlanta, GA 8 11 11 9 Boston, MA 9 3 2 1 Dallas, TX 1 12 14 6 Riverside, CA 11 1 Phoenix, AZ 12 8 12 8 Minneapolis, MN 13 1 3 4 San Diego, CA 14 6 8 1 Orange County, CA 2 6 11 Source: Savageau, David and Ralph D Agostino. Places Rated Almanac, Millennium Edition. New York: Macmillan, 2. LESSON 1 Bivariate Relationships 273
/3 /1 /1 Rankings of Metropolitan Areas 2 1 1 1 1 Population Crime Health Care Education 1 1 1 1 a. Examine the scatterplot matrix shown above. Describe the location(s) of the scatterplots for which health care is the variable graphed on the x-axis. On the y-axis. b. The open circle on each plot represents the same city. Which city is this? For which variables does this city tend to be ranked toward the best? Toward the worst? c. Which pair of variables appears to have the strongest positive correlation? Suggest some reasons why this correlation might be so strong. d. Find a pair of variables with an obvious negative correlation. Write a sentence that describes this relationship. e. Find the missing values of the rank correlation r s in the rank correlation matrix below. Population Crime Health Care Education Population 1..168.161 Crime 1..47.4 Health Care.168.47 1..7 Education.161.4.7 1. f. Why are the entries along the diagonal of the rank correlation matrix in Part e all 1s? g. How is this rank correlation matrix related to the scatterplot matrix? 2 274 UNIT 4 Regression and Correlation