1. Anthony Greene1 Correlation The Association Between Variables

2. Anthony Greene2 When to Use t-test or ANOVA When the independent variable is Categorical

3. Anthony Greene3 When to Use Correlation & Regression When the independent variable is Ratio or Interval

4. Anthony Greene4 ScatterPlots

5. Anthony Greene5 ScatterPlots

6. Times and costs for five word- processing jobs

7. Anthony Greene7 Four data points

8. Anthony Greene8 Direct Relationship: Positive Slope

9. Anthony Greene9 Age and price data for a sample of 11 used cars

10. Anthony Greene10 Scatter diagram for the age and price data of used cars

11. Anthony Greene11 Inverse Relationship: Negative Slope

12. Anthony Greene12 Various degrees of linear correlation (Slide 1 of 3)

13. Anthony Greene13 Various degrees of linear correlation (Slide 3 of 3)

14. Anthony Greene14 Various degrees of linear correlation (Slide 2 of 3)

15. Anthony Greene15 Examples of positive and negative relationships

16. 16 The Simple Idea If the corresponding x and y z-scores are always in agreement, r will be high. If they are sometimes in agreement r will be moderate If they are generally different, r will be near zero If they are in agreement, but in opposite directions, r will be negative

17. Anthony Greene17 The Basic Idea ZxZx ZyZy ZxZx ZyZy ZxZx ZyZy +1.2+1.6+1.2-2.3+1.2-1.6 -1.1-0.7-1.1-0.2-1.1+0.7 +0.8+1.1+0.8+1.6+0.8-1.1 +3.2+2.8+3.2-0.7+3.2-2.8 -2.7-2.3-2.7+1.1-2.7+2.3 +0.1-0.2+0.1+2.8+0.1+0.2

18. Anthony Greene18 We define SS x, SS p and SS y by Notation Used in Regression and Correlation

19. Anthony Greene19 Obtaining the three sums of squares for the used car data using the computational formulas

20. Anthony Greene20 The linear correlation coefficient, r, of n data points is defined by or by the computational formula Linear Correlation Coefficient

21. Anthony Greene21 Linear Correlation Coefficient

22. Anthony Greene22 Coefficient of Determination The coefficient of determination, r 2, is the proportion of variation in the observed values of the response variable that is explained by the regression: The coefficient of the determination always lies between 0 and 1 and is a descriptive measure of of the utility of the regression equation for making predictions. Values of r 2 near 0 indicate that the regression equation is not useful for making predictions, whereas values near 1 indicate that the regression equation is extremely useful for making predictions.

23. Anthony Greene23 t-Distribution for a Correlation Test For samples of size n, the variable has the t-distribution with df = n – 2 if the null hypothesis ρ = 0 ρ or rho is pronounced “row”

24. Anthony Greene24 The t-test for correlation (Slide 1 of 3) With df = n-2 use table B.6

25. Anthony Greene25 The t-test for correlation (Slide 2 of 3)

26. Anthony Greene26 The t-test for correlation (Slide 3 of 3) Step 4 Compute the test statistic r. Table B.6 allows a direct lookup. Alternatively, r has a t distribution and Table B.2 will yield an identical conclusion Step 5 If the value of the test statistic falls in the rejection region, reject the null hypothesis. Step 6 State the conclusion in words

27. Anthony Greene27 Criterion for deciding whether or not to reject the null hypothesis

28. Anthony Greene28 Correlation Matrix abcd a1.00 b0.841.00 c0.680.581.00 d0.120.190.081.00

29. Anthony Greene29 Computer printouts for correlations