1. (c) 2007 IUPUI SPEA K300 (4392) Outline Correlation and Covariance Bivariate Correlation Coefficient Types of Correlation Correlation Coefficient Formula Correlation Coefficient Computation Short-cut Formula Linear Function (Intercept and Slope)

2. (c) 2007 IUPUI SPEA K300 (4392) Correlation and Covariance It asks how two variables are related When x changes, how does y change? Underlying information is covariance Cov(x,y)=E[(x-xbar)(y-ybar)] Cov(x,y)=Cov(y,x) Cov(x,x)=Var(x), variance is a special type of covariance (covariance of a variable and itself)

3. (c) 2007 IUPUI SPEA K300 (4392) Bivariate Correlation Coefficient (Karl Pearson product moment) correlation coefficient Bivariate correlation coefficient (BCC) for two interval/ratio variables Differentiated from Spearman’s rank correlation coefficient (nonparametric) Differentiated from partial correlation coefficient that controls the impact of other variables No causal relationship imposed. X  Y or Y  X BCC is used for prediction

4. (c) 2007 IUPUI SPEA K300 (4392) Bivariate Correlation Coefficient BCC ranges from -1 to 1 (So does Gamma γ) Covariance component can be negative + means positive relationship; when x increases 1 unit, y increases r unit 0 means no relationship. - means negative relationship; when x increases 1 unit, y decreases r unit. http://noppa5.pc.helsinki.fi/koe/corr/cor7.html

5. (c) 2007 IUPUI SPEA K300 (4392) Positive relationship

6. (c) 2007 IUPUI SPEA K300 (4392) Negative relationship

7. (c) 2007 IUPUI SPEA K300 (4392) No relationship

8. (c) 2007 IUPUI SPEA K300 (4392) Correlation Coefficient Ratio of the covariance component of x and y to the square root of variance components of x and y

9. (c) 2007 IUPUI SPEA K300 (4392) Correlation Coefficient (short-cut) Textbook suggests a short-cut formula below but it is not recommended.

10. (c) 2007 IUPUI SPEA K300 (4392) Illustration: example 10-2, p.526 Noxy(x-xbar)(y-ybar)(x-xbar)^2(y-ybar)^2(x-xbar)(y-ybar) 143128-14.5-8.5210.2572.25123.25 248120-9.5-16.590.25272.25156.75 356135-1.5 2.25 4611433.56.512.2542.2522.75 5671419.54.590.2520.2542.75 67015212.515.5156.25240.25193.75 Sum345819 561.5649.5541.5 Mean57.5137 SSxxSSyySPxy Correlation coefficient0.8967

11. (c) 2007 IUPUI SPEA K300 (4392) Hypothesis Test How reliable is a correlation coefficient? r is a random variable drawn from the sample; ρ is its corresponding parameter H 0 : ρ =0, H a : ρ ≠ 0 TS follows the t distribution with df=n-2 If H 0 is not rejected, r is not reliable regardless of its magnitude (ρ =0)

12. (c) 2007 IUPUI SPEA K300 (4392) Illustration: Example 10-3, p.529 Step 1. H 0 : ρ =0, H a : ρ ≠ 0 Step 2. α=.05, df=4 (=6-2), CV=2.776 Step 3. TS=4.059, r=.897 Step 4. TS>CV, reject H 0 at the.05 level Step 5. ρ ≠ 0

13. (c) 2007 IUPUI SPEA K300 (4392) Linear function A function transforms input into output in its own way Ex: y=square_root(x). Whey you put x (input) into the funciton square_root(), you will get y (output). Linear function consists of a intercept and linear combinations of variables and their slops. Y= a + bX + cX2… Slopes are constant

14. (c) 2007 IUPUI SPEA K300 (4392) Intercept and Slope of a function A linear model: Y = a + b X Dependent variable Y to be explained Independent variable X that explains Y Y-Intercept a: the coordinate of the point at which the line intersects Y axis. Slope b: the change of dependent variable Y per unit change in independent variable X

15. (c) 2007 IUPUI SPEA K300 (4392) Illustration