Watch TV 4 7 Read 5 2 Exercise 2 4 Talk to friends 7 3 Go to a movie 6 5 Go to dinner 1 6 Go to the mall 3 1

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1 Unit 3 Lesson 1 Investigation 2 Check Your Understanding Name: A couple decides to measure their compatibility by ranking their favorite leisure activities. The rankings are given below in the table. Mallisa Matt Watch TV 4 7 Read 5 2 Exercise 2 4 Talk to friends 7 3 Go to a movie 6 5 Go to dinner 1 6 Go to the mall 3 1 a. Make a scatterplot for the two rankings with appropriate scales and labels on the two axes. b. Predict whether the value of r s will be closer to -1,, or 1. Use the scatterplot to help explain your answer. c. Calculate the value of r s. Show your work. d. What would you conclude about this couple s compatibility? Investigation 2 Shapes of Clouds of Points In Investigation 1, you learned to describe the association between two rankings as shown in a scatterplot by giving its direction (positive, negative, or none) and its strength (strong, moderate, weak, none) or by reporting the correlation. In this investigation, you will examine more closely patterns in a scatterplot. As you complete the following problems, make notes of answers to this question: What are the common shapes of bivariate data displayed on a scatterplot? 1 Bivariate data are linear if they form an oval or elliptical cloud. Two examples of linear relationships are shown below a. Place a sheet of paper over each plot. Sketch the axes and an oval that contains the points. b. Describe the direction and strength of the relationship in the first plot. In the second plot. 264 UNIT 4 Regression and Correlation

2 c. When the relationship is linear, you can summarize the relationship with a straight line. However, a line would not be an appropriate summary of the relationships shown below. How could you describe the shape, trend (center), and strength (spread) of these distributions? 1, 1,3 1, d. If the points tend to fan out at one end, the relationship is said to vary in strength. Both of the plots below vary in strength. Sketch a plot where the pattern varies in strength but the points cluster about a line. That is, the shape is not curved The scatterplot below shows the time between two consecutive eruptions of the Old Faithful geyser in Yellowstone National Park plotted against the duration of the first eruption. Old Faithful Eruption Times Interruption Time (in minutes) Duration (in minutes) Source: Samprit Chatterjee, et al. A Casebook for a First Course in Statistics and Data Analysis. Wiley, a. What is the shape of this distribution? Is it appropriate to summarize the trend with a line? b. Is the relationship positive or negative? Is it strong, moderate, or weak? Does the strength of the relationship vary? LESSON 1 Bivariate Relationships 265

3 c. Can you give a reason why the duration of the first eruption might have an effect on the time until the next eruption? d. When examining a scatterplot, you should also look for clusters of points and for outliers that lie away from the main cloud of points. Do you see clusters or outliers in the scatterplot of the geyser data? 3 The state of Alaska has the largest population of black bears in the U.S. approximately,. The scatterplot below gives the weight in pounds and length in inches of a large sample of black bears. a. Describe the shape of this plot. Lengths and Weights of Black Bears Weight (in pounds) 7 6 B 4 A 3 2 D Length (in inches) E C 9 b. This scatterplot illustrates the types of outliers that can occur: an outlier for length only an outlier for weight only both an outlier for length and an outlier for weight not an outlier for length and not an outlier for weight, but an outlier when length and weight are jointly considered For each labeled point on the scatterplot, tell which type of outlier it is. Then describe the bear. 4 The plot at the top of the next page is called a scatterplot matrix, a matrix whose entries are scatterplots. In each scatterplot, one dot represents one of the 5 states, the District of Columbia, or Puerto Rico. The five variables are as follows: Dropout% percentage of 16 to 19 year olds who are not enrolled in school and have not graduated from high school Med Age PerCapIn %Poverty median age (in years) per capita (per person) income percentage of the population below the poverty level %ColGrad percentage of people at least 25 years old who have earned bachelor s degrees or higher 266 UNIT 4 Regression and Correlation

4 Population Characteristics 13 Dropout% 8 36 Med Age 3 24, PerCapIn 14, 38 %Poverty %ColGrad , 24, Source: 2 U.S. Census a. The points in the plot in the second row and fifth column have percentage college graduate plotted on the x-axis and median age plotted on the y-axis. Is there a strong positive, a strong negative, or almost no association between these two variables? b. Describe the location(s) of the scatterplots within the matrix for which percentage below poverty level is the variable graphed on the x-axis. On the y-axis. c. The state with the lowest median age is Utah. Estimate the median age in Utah. Estimate the per capita income. d. The open circle on each plot represents Puerto Rico. The scatterplot in the first row and second column shows that Puerto Rico has a relatively high dropout percentage and a relatively low median age, but it is not an outlier. What can you tell about Puerto Rico from each scatterplot identified below? If it is an outlier, give the type of outlier. i. the scatterplot in the third row and fourth column ii. the scatterplot in the fourth row and fifth column iii. the scatterplot in the first row and fifth column e. If you ignore Puerto Rico, which pair of variables has the strongest positive association? f. If you ignore Puerto Rico, which pairs of variables have negative association? g. Which scatterplot has a curved shape? Does it show varying strength? LESSON 1 Bivariate Relationships 267

5 5 Look back at the scatterplot matrix in Problem 4. a. Why are the scatterplots down the main diagonal of the matrix not included? What would they look like if they were included? b. Which pairs of scatterplots give the same information? Summarize the Mathematics In this investigation, you learned how to describe the pattern in a scatterplot. a When describing the cloud of points on a scatterplot, what information should you give? b What are the types of outliers that you may see on a scatterplot? c When would a scatterplot matrix be useful? Be prepared to share your ideas and reasoning with the class. Check Your Understanding Each point on the scatterplot below represents a state or the District of Columbia. The variables are the percentage of ninth graders who graduate from high school four years later and the percentage of people who are unemployed. (Two states are missing because their graduation rate was not available.) Unemployment Rate (in %) Graduation Rate (in %) Source: U.S. Department of Education and U.S. Bureau of Labor Statistics, 26 a. Describe the shape of this distribution. Include the direction of the relationship, the strength, and whether the strength varies. b. Do you see any unusual features? c. Nebraska has the highest graduation rate. Estimate this rate from the plot. Is Nebraska s unemployment rate about what you would expect, given its graduation rate? d. Michigan has the highest unemployment rate. Estimate this rate from the plot. Is Michigan clearly an outlier? 268 UNIT 4 Regression and Correlation