Chapter 9. Section Correlation

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1 Chapter 9 Section Correlation Objectives: Introduce linear correlation, independent and dependent variables, and the types of correlation Find a correlation coefficient Test a population correlation coefficient ρ using a table Perform a hypothesis test for a population correlation coefficient ρ Distinguish between correlation and causation Correlation A relationship between two variables. The data can be represented by ordered pairs (x, y) x is the independent (or explanatory) variable y is the dependent (or response) variable A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables. Types of Correlation 1 P a g e

2 Example: Constructing a Scatter Plot An economist want to determine whether there is a linear relationship between a country s gross domestic product (GDP) and carbon dioxide (CO 2 ) emissions. The data are shown in the table. Display the data in a scatter plot and determine whether there appears to be a positive or negative linear correlation or no linear correlation. (Source: World Bank and U.S. Energy Information Administration) Correlation coefficient A measure of the strength and the direction of a linear relationship between two variables. The symbol r represents the sample correlation coefficient. A formula for r is r n xy x y n x x n y y The population correlation coefficient is represented by ρ (rho). The range of the correlation coefficient is -1 to 1. n is the number of data pairs 2 P a g e

3 Linear Correlation Calculating a Correlation Coefficient 3 P a g e

4 Example: Finding the Correlation Coefficient Calculate the correlation coefficient for the gross domestic products and carbon dioxide emissions data. What can you conclude? Using a Table to Test a Population Correlation Coefficient ρ Once the sample correlation coefficient r has been calculated, we need to determine whether there is enough evidence to decide that the population correlation coefficient ρ is significant at a specified level of significance. Use Table 11 in Appendix B. If r is greater than the critical value, there is enough evidence to decide that the correlation coefficient ρ is significant. 4 P a g e

5 Example: Determine whether ρ is significant for five pairs of data (n = 5) at a level of significance of α = If r > 0.959, the correlation is significant. Otherwise, there is not enough evidence to conclude that the correlation is significant. 5 P a g e

6 Example: Using a Table to Test a Population Correlation Coefficient ρ Below is the data for Old Faithful, you used 25 pairs of data to find r Is the correlation coefficient significant? Use α = Hypothesis Testing for a Population Correlation Coefficient ρ A hypothesis test can also be used to determine whether the sample correlation coefficient r provides enough evidence to conclude that the population correlation coefficient ρ is significant at a specified level of significance. A hypothesis test can be one-tailed or two-tailed. Left-tailed test H 0 : ρ 0 (no significant negative correlation) H a : ρ < 0 (significant negative correlation) Right-tailed test H 0 : ρ 0 (no significant positive correlation) H a : ρ > 0 (significant positive correlation) Two-tailed test H 0 : ρ = 0 (no significant correlation) H a : ρ 0 (significant correlation) 6 P a g e

7 The t-test for the Correlation Coefficient Can be used to test whether the correlation between two variables is significant. The test statistic is r The standardized test statistic follows a t-distribution with d.f. = n 2. In this text, only two-tailed hypothesis tests for ρ are considered. Using the t-test for ρ 7 P a g e

8 Example: t-test for a Correlation Coefficient Previously you calculated r (On page 4 on notes). Test the significance of this correlation coefficient. Use α = Correlation and Causation The fact that two variables are strongly correlated does not in itself imply a cause-and-effect relationship between the variables. If there is a significant correlation between two variables, you should consider the following possibilities. 1. Is there a direct cause-and-effect relationship between the variables? Does x cause y? 2. Is there a reverse cause-and-effect relationship between the variables? Does y cause x? 3. Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables? 4. Is it possible that the relationship between two variables may be a coincidence? 8 P a g e

9 Section Linear Regression Objectives: Find the equation of a regression line Predict y-values using a regression equation Regression lines After verifying that the linear correlation between two variables is significant, next we determine the equation of the line that best models the data (regression line). Can be used to predict the value of y for a given value of x. Residual The difference between the observed y-value and the predicted y-value for a given x-value on the line. Regression line (line of best fit) The line for which the sum of the squares of the residuals is a minimum. The equation of a regression line for an independent variable x and a dependent variable y is ŷ = mx + b where m is the slope, b is the y-intercept and is the predicted y-value for a given x value 9 P a g e

10 The Equation of a Regression Line ŷ = mx + b where 2 n xy x y m 2 n x x is the mean of the y-values in the data is the mean of the x-values in the data The regression line always passes through the point x, y Example: Finding the Equation of a Regression Line Find the equation of the regression line for the gross domestic products and carbon dioxide emissions data. 10 P a g e

11 Example: Predicting y-values Using Regression Equations The regression equation for the gross domestic products (in trillions of dollars) and carbon dioxide emissions (in millions of metric tons) data is ŷ = x Use this equation to predict the expected carbon dioxide emissions for the following gross domestic products. (Recall from section 9.1 that x and y have a significant linear correlation.) trillion dollars trillion dollars trillion dollars 11 P a g e

12 Section Measures of Regression and Prediction Intervals Objectives: Interpret the three types of variation about a regression line Find and interpret the coefficient of determination Find and interpret the standard error of the estimate for a regression line Construct and interpret a prediction interval for y Variation About a Regression Line Three types of variation about a regression line Total variation Explained variation Unexplained variation To find the total variation, you must first calculate The total deviation The explained deviation The unexplained deviation Total Deviation = Explained Deviation = Unexplained Deviation = Total variation The sum of the squares of the differences between the y-value of each ordered pair and the mean of y. Total Variation = Explained variation The sum of the squares of the differences between each predicted y-value and the mean of y. Explained Variation = Unexplained variation The sum of the squares of the differences between the y-value of each ordered pair and each corresponding predicted y-value. Unexplained Variation = The sum of the explained and unexplained variation is equal to the total variation. Total variation = Explained variation + Unexplained variation 12 P a g e

13 Coefficient of determination The ratio of the explained variation to the total variation. Denoted by r 2 2 Explained variation r Total variation Example: Coefficient of Determination The correlation coefficient for the gross domestic products and carbon dioxide emissions data as calculated in Section 9.1 is r Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? About the unexplained variation? Standard error of estimate The standard deviation of the observed y i -values about the predicted ŷ-value for a given x i - value. Denoted by s e. s e ( yi yˆ i) n 2 2 n is the number of ordered pairs in the data set The closer the observed y-values are to the predicted y-values, the smaller the standard error of estimate will be. 13 P a g e

14 Example: Standard Error of Estimate The regression equation for the gross domestic products and carbon dioxide emissions data as calculated in section 9.2 is ŷ = x Find the standard error of estimate. 14 P a g e

15 Prediction Intervals Two variables have a bivariate normal distribution if for any fixed value of x, the corresponding values of y are normally distributed and for any fixed values of y, the corresponding x-values are normally distributed. A prediction interval can be constructed for the true value of y. Given a linear regression equation ŷ = mx + b and x 0, a specific value of x, a c-prediction interval for y is ŷ E < y < ŷ + E where E 2 0 x n( x ) tcse 1 n n x ( x) The point estimate is ŷ and the margin of error is E. The probability that the prediction interval contains y is c. Constructing a Prediction Interval for y for a Specific Value of x 15 P a g e

16 Example: Constructing a Prediction Interval Construct a 95% prediction interval for the carbon dioxide emission when the gross domestic product is $3.5 trillion. What can you conclude? Recall, n = 10, ŷ = x , s e = x 15.8, 2 x 32.44, x P a g e