(per 100,000 residents) Cancer Deaths

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1 Unit 3 Lesson 2 Investigation 2 Radioactive Waste Exposure Cancer Deaths (per 100,000 residents) Name: Index of Exposure a. Describe the direction and strength of the relationship. b. Find the equation of the regression line. What does the slope mean in the context of these data? c. Use the equation to predict the cancer death rate for Hood River. Find and interpret the residual for Hood River. d. What is the sum of the squared residuals? Can you find a different line that gives a smaller sum? Explain. Investigation 2 Behavior of the Regression Line You have seen that the least squares regression line makes the sum of the squares of the residuals as small as possible. The regression line has other important properties. As you work on the following problems, look for answers to this question: What are some of the properties of the least squares regression line? 1 The point with coordinates ( x, y ) can be thought of as the balance point for bivariate data. This point is called the centroid. The regression line always goes through the centroid. a. Shown below are the three points in Problem 4 of Investigation 1. Calculate x and y. Then verify that the point with coordinates ( x, y ) is on the regression line. x y b. Refer back to the table of data on compact car highway mileage in Problem 1 of Investigation 1 (page 282). i. Calculate the mean curb weight in 100s of pounds and the mean highway mpg of that group of cars. ii. Verify that the point (mean curb weight, mean highway mpg) lies on the regression line. 286 UNIT 4 Regression and Correlation

2 2 Now, use data analysis software to examine the idea of finding the line on a scatterplot that minimizes the sum of squared errors (residuals), or SSE. CPMP-Tools a. Using the moveable line capability of the software, visually find a line that you think best fits the compact car (curb weight in hundreds of pounds, highway mpg) data. b. Compare the line you found visually and its equation with the regression line and its equation. c. The Honda Civic Hybrid is an outlier and so may have a large effect on the location of the regression line. To investigate the effect of this point, first remove from the plot the lines you found in Part a. i. Delete the point for the Honda Civic Hybrid from the data set. How do the regression line and equation change? ii. Replace the point for the Honda Civic Hybrid and then delete the point for the Mercedes-Benz C280, which is the second heaviest car. How do the regression line and equation change in this case? iii. Does the Honda Civic Hybrid or the Mercedes-Benz C280 have more influence on the regression line and equation? An influential point is a special type of outlier. It strongly influences the equation of the regression line or the correlation. When such a point is removed from the data set, the slope or y-intercept of the regression line changes quite a bit. The interpretation of quite a bit depends on the real-life situation. You will further examine the idea of an influential point in the next two problems. LESSON 2 Least Squares Regression and Correlation 287

3 3 The data in the following table and scatterplot are from a larger set of data collected by a student for a science fair project. The student measured characteristics of a sample of horses because she wanted to see how various measurements were related to the length of the horse s stride. Horse Height (in hands) Hip Angle While Running (in degrees) Charm Hugs Otis Cosmo Gaspe Sam Pi Binky Bella Prima Bandit Blackie Hip Angle While Running (in degrees) Height (in hands) Source: AP Statistics discussion list, posted on February 03, 2006, a. Which horse is the outlier? b. What do you think will happen to the slope and intercept of the regression line if this horse is removed from the data set? c. Do the computations needed to check your conjecture. d. Is this horse influential? 288 UNIT 4 Regression and Correlation

4 4 The most famous scandal in baseball history occurred at the Eight players for the Chicago White Sox were accused of throwing the series to the Cincinnati Reds. The players were acquitted of criminal charges but banned from professional baseball for life. The 1919 season and batting averages for the nine White Sox players who had 10 or more at bats in the are given in the following table and scatterplot. These include five of the accused players. The equation of the regression line for predicting the batting average from the season batting average is y = 1.99x Chicago White Sox Player Season Accused? N=no/Y=yes Eddie Collins N Shano Collins N Happy Felsch Y Chick Gandil Y Shoeless Joe Jackson Y Nemo Leibold N Swede Risberg Y Ray Schalk N Buck Weaver Y Season Source: Accused? Yes No a. Which player did the worst in the, compared to what would be predicted? Was he one of the accused players? b. Do the accused players appear to have a different pattern than the players who were not accused? c. Select the three players who you think might have influential points on the scatterplot. For each player, predict how the regression line would change if he were removed from the data set. d. Remove one of the three players you identified in Part c, and recompute the equation. Put that player back, remove the second player, and recompute the equation. Put that player back, remove the third, and recompute the equation. Which player was most influential on the slope of the regression line? LESSON 2 Least Squares Regression and Correlation 289

5 e. How is the effect of removing the point for Nemo Leibold different from the effect of removing either Shoeless Joe Jackson or Swede Risberg? f. The scatterplot below shows all Cincinnati Reds players who had 10 or more at bats in the Series (none were accused) along with the four White Sox players who had 10 or more at bats and were not accused. It also includes Shoeless Joe Jackson who was accused (circular dot). i. Describe the relationship between season batting average and batting average for these players. Is this about what you would expect? ii. How would the slope of the regression line change if the point for Shoeless Joe were removed from this data set? s Shoeless Joe Jackson Season g. The Shoeless Joe Jackson Society is devoted to the goal of seeing Joe claim his rightful place in the Baseball Hall of Fame. From the evidence that you have seen here, make a case for Shoeless Joe that the Society could use toward their goal. Summarize the Mathematics In this investigation, you explored some properties of the regression line. a What point is always on the regression line? b What is the difference between an outlier and an influential point? Be prepared to explain your ideas to the class. 290 UNIT 4 Regression and Correlation