1. Correlation Correlation is used to measure strength of the relationship between two variables

2. Scatter Plot Examples y x y x y y x x Strong relationshipsWeak relationships

3. Scatter Plot Examples y x y x No relationship

4. Correlation Coefficient The population correlation coefficient ρ (rho) measures the strength of the association between the variables The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations (continued)

5. Features of ρ  and r Unit free Range between -1 and 1 The closer to -1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker the linear relationship

6. r = +.3r = +1 Examples of Approximate r Values y x y x y x y x y x r = -1 r = -.6r = 0

7. Introduction to Regression Analysis Regression analysis is used to: – Predict the value of a dependent variable based on the value of at least one independent variable – Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain Independent variable: the variable used to explain the dependent variable

8. Correlation Coefficient A number that measures the strength and direction of the linear correlation of a data set is the (linear) correlation coefficient, r. Your graphing calculator will show the correlation coefficient when you calculate linear regression if you select “Diagnostic On” from the Catalog menu.

9. Properties of the Correlation Coefficient, r 1.-1 ≤ r ≤ 1 2.When r > 0, there is a positive linear correlation. 3.When r < 0, there is a negative linear correlation. 4.When |r| ≈ 1, there is a strong linear correlation. 5.When r ≈ 0, there is a weak or no linear correlation.

10. Regression Analysis 1.Enter and plot the data (scatter plot). 2.Find the regression model that fits the problem situation. 3.Analyze the correlation coefficient. 4.Superimpose the graph of the regression model on the scatter plot, and observe the fit. 5.Use the regression model to make the predictions called for in the problem.

11. Example 1 1.What is the correlation coefficient? (round to three decimal places) 2.Describe the correlation. 3.Predict the number of locations in the year 2011. 4.In what year would the number of locations exceed 4000?

12. Example 2 1.What is the correlation coefficient? (round to three decimal places) 2.Describe the correlation. 3.Predict the public school enrollment in the year 2011. 4.In what year would the public school enrollment exceed 100,000?

13. Think, Pair, Share Think about a real world example of someone who would use line of best fit in their work. Turn to your partner and share what you came up with.