Lesson 2. Investigation. Name:

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1 Check Unit Your 3 Understanding Lesson Investigation 3 Refer to the Check Your Understanding on page 5 of the previous investigation. a. Find the mean index of exposure and the mean cancer death rate. Then verify that the centroid is on the regression line. b. Which community is a potential influential point? Remove this point, find the new regression line, and describe how the regression line changes. Investigation 3 How Strong Is the Association? Name: In previous investigations, you have learned how to describe the shape of a scatterplot and how to use the regression line as the center of an elliptical cloud of points. In this investigation, you will explore a way of describing the strength of an elliptical cloud of points by computing a correlation. A formula for calculating a measure of linear association between pairs of values (x, y) that are ranked or unranked was developed by British statistician Karl Pearson ( ). The resulting correlation, called Pearson s r, can be interpreted in the same way as Spearman s rank correlation, r s. The correlation indicates the direction of the association between the two variables by whether it is positive or negative. It indicates the strength by whether it is near 1 or -1 versus near. For ranked data with no ties, Pearson s and Spearman s formulas give the same value. As you work on the problems in this investigation, make notes of answers to this question: How can you compute and interpret Pearson s correlation? Karl Pearson 1 Since Pearson s r can be used with either ranked or unranked data, you might expect the formula to be more complex than Spearman s formula. You would be right! The formula for Pearson s correlation r is r = Σ(x - x )(y - y ). (n - 1)s x s y Here, x is the mean of the x values, y is the mean of the y values, s x is the standard deviation of the x values, s y is the standard deviation of the y values, and n is the number of data pairs. a. Examine the formula. To use it, how would you proceed? b. Now, examine the plot shown at the right. Which has the larger mean, the values of x or the values of y? Which has the larger standard deviation? What value of r would you expect for these points? 4 4 LESSON Least Squares Regression and Correlation 91

2 c. When using complex formulas, it is often helpful to organize intermediate calculations in a table. i. Compute x and y. Then complete a copy of the table below. Sum (Σ) x y x - x y - y (x - x )(y - y ) ii. Compute n - 1, s x and s y. iii. Calculate r by substituting the appropriate sums in the formula. Compare the value you calculated with your prediction in Part b. d. What value of r would you predict for the 1 points on the scatterplot at the right? Check your prediction by computing the value of r and comparing it to your prediction. e. What value do you get for r if you reverse the coordinates of each point in Part d? 4 Why does this make sense? f. Create a set of six points that have correlation -1. Create a set of six points 4 that have correlation close to. In this problem, you will explore how Pearson s formula works. Examine the formula for the correlation r. r = Σ(x - x )(y - y ) (n - 1)s x s y a. Recall that (x - x ) is called a deviation from the mean. What is Σ(x - x )? What is Σ(y - y )? Is this necessarily true of Σ(x - x )(y - y )? Explain. b. As long as the values of x are not all the same and the values of y are not all the same, the denominator of the formula for r gives a positive number. Explain why this is true. c. On the scatterplot at the right, horizontal and B A vertical lines are drawn through ( x, y ). For the points in region A: Is (x - x ) positive or negative? Is (y - y ) positive or negative? Is (x - x )(y - y ) positive or negative? C (x, y) D 9 UNIT 4 Regression and Correlation

3 d. Fill in each space on a copy of the table below with positive or negative. Region A B C D Value of x - x Value of y - y Value of (x - x )( y - y ) e. Explain why Pearson s formula will give a positive value of r for the points on the scatterplot in Part c. f. Explain why Pearson s formula will give a negative value of r for the points on the scatterplot below. B A (x, y) C D 3 For most sets of real data, calculation by hand of Pearson s r is tedious and prone to error. Thus, it is important to know how to calculate r using your graphing calculator or data analysis software. a. Refer to the scatterplot of arm span and kneeling height for the members of your class that you collected in the Think About This Situation for Lesson. Estimate the correlation. b. Learn to use your calculator or data analysis software to find the value of r for these data and check your estimate in Part a. CPMP-Tools LESSON Least Squares Regression and Correlation 93

4 Vehicles 3 4 Match each correlation with the appropriate plot. Then write a sentence that describes the association between the two variables in the plot. a. r = -.4 b. r =.5 c. r = -. d. r =.94 I Office Workers II Gas Mileage of Vehicle Weight of Worker , 1,4 1, 1,,,,4 Weight of Vehicle Age of Worker Bird Eggs 1 1 III IV High School Seniors 1 Length of Egg 14 1 Hours of Study Width of Egg Hours of Sleep 94 UNIT 4 Regression and Correlation

5 5 In this problem, you will explore the effect of a change of scale on the correlation, r. Shown below is nutritional information on fast-food hamburgers. How Hamburgers Compare Company Burger Calories Fat (in grams) Protein (in grams) Sodium (in mg) Hardee s Hamburger Thickburger ,47 Wendy s Jr. Hamburger Classic Single 4 5 Burger King Hamburger Whopper , McDonald s Hamburger Quarter Pounder Big Mac , Carl s Jr. Kid s Hamburger ,4 Famous Star Source: (December ). A scatterplot matrix of these data is shown below. 1, Calories Fat (in grams) Protein (in grams) 1, 9 3 Sodium (in mg) , a. Find and interpret the correlation between sodium and calories. Why might this correlation be so strong? LESSON Least Squares Regression and Correlation 95

6 b. Transform the amounts of sodium by converting them to grams. (Recall there are 1, milligrams in a gram.) Find r for calories and the transformed values of sodium. What do you notice? Explain why your observation makes sense. c. Now, transform the numbers of calories by subtracting from each value. Find r for sodium and the transformed values of calories. What do you notice? Explain why your observation makes sense. As with the regression equation, you can tell if an outlier is influential on the correlation by temporarily removing it from the data set and seeing how much the correlation changes. a. Which hamburger is an outlier in the (sodium, calories) data set? Is this outlier an influential point with respect to the correlation? With respect to the slope of the regression line? b. For each of the plots below, identify the outlier. Indicate whether removing the point will make the correlation stronger, weaker, or unchanged. Plot A Plot B 7 Belinda conjectured that a high correlation between two variables means that the variables are linearly related. a. Use your calculator to find the correlation for the following points. From your calculated value, would you expect the scatterplot to have a linear pattern? x y b. Produce the scatterplot on your calculator. Are the points linear? How would you respond to Belinda? c. How well does y =.97x +.x +.5 model the pattern in the points? d. Create a set of points that has a parabolic shape, but the correlation is. 9 UNIT 4 Regression and Correlation

7 The scatterplot below shows the heights of 1,7 fathers and their sons. Karl Pearson collected the data around the year 19. Son's Height (in inches) Father's Height (inches) a. Make an estimate of the correlation. Would you say this is a strong correlation? b. Does this correlation mean that a linear model is not appropriate for these data? Explain your reasoning. 9 Write a summary of what you can conclude from Problems 7 and. Compare your conclusions with that of others and resolve any differences. Summarize the Mathematics In this investigation, you learned how to compute and interpret Pearson s correlation for paired (x, y) data. a Explain why computation of Σ(x x )(y y ) determines whether r is positive or negative. How is the sign of r related to the regression equation? b Explain why it is important to examine a scatterplot of a set of data even though you know the correlation. c What is the effect on r if you: i. reverse the coordinates of the points? ii. multiply or divide each value by the same positive number? iii. add the same number to each value? iv. add an outlier to the data set? Be prepared to share your ideas and reasoning with the class. LESSON Least Squares Regression and Correlation 97