Chapter 10r Linear Regression Revisited

Correlation A numerical measure of the direction and strength of a linear association. –Like standard deviation was a numerical measure of spread.

Correlation Coefficient - Facts The correlation coefficient is denoted by the letter r. –Safe to assume r is always correlation in this class. The sign of the correlation coefficient give the direction of the association. –Positive is positive and negative is negative.

Correlation Coefficient - Facts The correlation coefficient is always between -1 and +1. –A low correlation is closer to zero and strong closer to either -1 or +1. Ex. r = 0.21 or (weak), r = or 0.98(strong). –If correlation is equal to exactly -1 or +1 then the data points all fall on an exact straight line.

Correlation Coefficient - Facts Correlation coefficient has no units. –The correlation is just that the correlation. Learn it on its own scale, not as a percentage. Correlation doesn’t change if center or scale of original data is changed. –Depends only on the z-score.

What is STRONG/WEAK? Again a judgment call. Rule of thumb: –0 to +/- 0.5 Weak –+/- 0.5 to +/ Moderate –+/- 0.8 to +/- 1.0 Strong

Computing Correlation Use your technology to help you find this number. –Calculator

Hypothesis Testing for ρ (rho) Before we do a linear regression we can conclude whether or not there is a significant linear relationship between the variables or if r is due to chance. In order to do this we use a t Test for the correlation coefficient –Ho: ρ = 0 No correlation between x and y variables –Ha: ρ ≠ 0 Significant correlation between the variables

Example - Correlation HT Data was obtained in a study on the number of hours that nine people exercise each week and the amount of milk (in ounces) each person consumes each week. Test the significance of the correlation coefficient at α = 0.01.

Example - Correlation HT Weekly Exercise Hours (X)Amount of Milk Consumed (Y)

Example - Correlation HT Step 1 –Ho: ρ = 0 –Ha: ρ ≠ 0 Step 2 –α = 0.01 Step 3 (note d.f.) –t (n – 2) = t (9 – 2 = 7) = t (7)

Example - Correlation HT Step 4 –Enter the lists into your calculator. –STAT -> TESTS -> LinRegTTest Make sure the right lists are there for X and Y Check appropriate Ha (should be not equal) Calculate –Report the r value = –t (7) = –P-value = 0.864

Example - Correlation HT Step 5 –0.864 > 0.01 –DO NOT REJECT Ho Step 6 –There is not significant evidence to suggest a correlation between the variables. This means that you would probably not do the linear regression analysis on these variables.