Definition

1. Definition The linear correlation coefficientr measures the strength of the linear relationship between the paired quantitative x- and y-values in a sample.

2. Requirements 1. The sample of paired (x, y) data is a simple random sample of quantitative data. 2. Visual examination of the scatterplot must confirm that the points approximate a straight-line pattern. 3. The outliers must be removed if they are known to be errors. The effects of any other outliers should be considered by calculating r with and without the outliers included.

3. Notation for the Linear Correlation Coefficient n = number of pairs of sample data  denotes the addition of the items indicated. x denotes the sum of all x-values. x2 indicates that each x-value should be squared and then those squares added. (x)2 indicates that the x-values should be added and then the total squared.

4. Notation for the Linear Correlation Coefficient xy indicates that each x-value should be first multiplied by its corresponding y-value. After obtaining all such products, find their sum. r = linear correlation coefficient for sample data.  = linear correlation coefficient for population data.

5. nxy–(x)(y) r= n(x2)– (x)2n(y2)– (y)2 Formula The linear correlation coefficientr measures the strength of a linear relationship between the paired values in a sample. Computer software or calculators can compute r

6. Properties of the Linear Correlation Coefficient r 1. –1 r 1 2. if all values of either variable are converted to a different scale, the value of r does not change. 3. The value of r is not affected by the choice of x and y. Interchange all x- and y-valuesand the value of r will not change. 4. r measures strength of a linear relationship. 5. r is very sensitive to outliers, they can dramatically affect its value.

7. Example: Using software or a calculator, r is automatically calculated:

8. Example: Using the pizza subway fare costs, we have found that the linear correlation coefficient is r = 0.988. What proportion of the variation in the subway fare can be explained by the variation in the costs of a slice of pizza? With r = 0.988, we get r2 = 0.976. We conclude that 0.976 (or about 98%) of the variation in the cost of a subway fares can be explained by the linear relationship between the costs of pizza and subway fares. This implies that about 2% of the variation in costs of subway fares cannot be explained by the costs of pizza.

9. Common Errors Involving Correlation 1. Causation: It is wrong to conclude that correlation implies causality. 2. Averages: Averages suppress individual variation and may inflate the correlation coefficient. 3. Linearity: There may be some relationship between x and y even when there is no linear correlation.

10. Basic Concepts of Regression

11. Regression The regression equation expresses a relationship between x (called the explanatory variable, predictor variable or independent variable), and y (called the response variable or dependent variable). ^ The typical equation of a straight liney = mx + b is expressed in the formy = b0 + b1x, where b0 is the y-intercept and b1 is the slope. ^

12. Definitions Regression Equation ^ y= b0 + b1x Given a collection of paired data, the regression equation algebraically describes the relationship between the two variables. • Regression Line • The graph of the regression equation is called the regression line (or line of best fit, or leastsquares line).

13. Notation for Regression Equation y-intercept of regression equation Slope of regression equation Equation of the regression line 0b0 1b1 y = 0 + 1xy = b0 + b1x ^ Population Parameter Sample Statistic

14. Requirements 1. The sample of paired (x, y) data is a random sample of quantitative data. 2. Visual examination of the scatterplot shows that the points approximate a straight-line pattern. 3. Any outliers must be removed if they are known to be errors. Consider the effects of any outliers that are not known errors.

15. Formulas for b0 and b1 calculators or computers can compute these values (slope) (y-intercept)

16. Example: Refer to the sample data given in Table below. Use technology to find the equation of the regression line in which the explanatory variable (or x variable) is the cost of a slice of pizza and the response variable (or y variable) is the corresponding cost of a subway fare.

17. Example: Requirements are satisfied: simple random sample; scatterplot approximates a straight line; no outliers Here are results from four different technologies technologies

18. All of these technologies show that the regression equation can be expressed asy = 0.0346 +0.945x, where y is the predicted cost of a subway fare and x is the cost of a slice of pizza. ^ ^ Example:

19. Example: Graph the regression equation(from the preceding Example) on the scatterplot of the pizza/subway fare data and examine the graph to subjectively determine how well the regression line fits the data.

20. Example:

21. For a pair of sample x and y values, the residual is the difference between the observed sample value of y and the y-value that is predicted by using the regression equation. That is, Definition ^ residual = observed y – predicted y = y – y

22. Residuals

23. A straight line satisfies the least-squares property if the sum of the squaresof the residuals is the smallest sum possible. Definitions

24. A residual plot is a scatterplot of the (x, y) values after each of they-coordinate values has been replaced by the residual value y – y (where y denotes the predicted value of y). That is, a residual plot is a graph of the points (x, y – y). Definitions ^ ^ ^

25. Residual Plot Analysis When analyzing a residual plot, look for a pattern in the way the points are configured, and use these criteria: The residual plot should not have an obvious pattern that is not a straight-line pattern. The residual plot should not become thicker (or thinner) when viewed from left to right.

26. Residuals Plot - Pizza/Subway

27. Residual Plots

28. Residual Plots

29. Residual Plots

30. Definition Coefficient of determination is the amount of the variation in y that is explained by the regression line. explained variation. r2= total variation The value of r2 is the proportion of the variation in y that is explained by the linear relationship between x and y.

31. ( ) å 2 x å = SS ( x ) “sum of squ ares for x” 2 = - x n ( ) å 2 y å = SS ( y ) “sum of squ ares for y” 2 = - y n å å x y å = SS ( xy ) “sum of squ ares for xy” = - xy n Alternate Formula for r

32. Example • Example: The table below presents the weight (in thousands of pounds) x and the gasoline mileage (miles per gallon) y for ten different automobiles. Find the linear correlation coefficient:

33. - SS ( xy ) 42 . 79 = = = - r 0. 47 SS ( x ) SS ( y ) ( 7 . 449 )( 1116 . 9 ) Completing the Calculation for r

34. The equation is determined by: b0: y-intercept b1: slope • Values that satisfy the least squares criterion: The Line of Best Fit Equation

35. Example • Example: A recent article measured the job satisfaction of subjects with a 14-question survey. The data below represents the job satisfaction scores, y, and the salaries, x, for a sample of similar individuals: 1) Draw a scatter diagram for this data 2) Find the equation of the line of best fit

36. ^ y SS ( xy ) 118 . 75 = = = b 0. 5174 1 SS ( x ) 229 . 5 ( ) å å - × y b x - 133 (0. 5174 )( 234 ) 1 = = = b 1 . 4902 0 n 8 = + 1 . 49 0. 517 x Solution 1) Equation o f the line of best f it: Line of Best Fit

37. Job Satisfaction Survey Solution 2) 22 21 20 19 18 Job Satisfaction 17 16 15 14 13 12 21 23 25 27 29 31 33 35 37 Salary Scatter Diagram