9.1 Correlation Key Concepts: –Scatter Plots –Correlation –Sample Correlation Coefficient, r –Hypothesis Testing for the Population Correlation Coefficient, ρ

9.1 Correlation What exactly do we mean by correlation? –If two variables are correlated, it means a relationship exists between them. –Examples of correlated variables: Job Satisfaction and Job Attendance Number of Cows per Square Mile and Crime Rate Height and Weight High School GPA and College GPA Square Footage and Price (of a house)

9.1 Correlation Two questions we need to answer: 1.Does a linear (or straight line) correlation exist between the two variables? 2.If the variables appear linearly correlated, how strong is the correlation? –We can answer (1) using a scatter plot The independent (explanatory) variable is x The dependent (response) variable is y –Example: How well does High School GPA, x, “explain” College GPA, y? –See section 2.2 for a review of scatter plots

9.1 Correlation Once the scatter plot is complete, we should be able to see if a linear relationship exists between the two variables. –See p. 470 for what we mean by Negative Linear Correlation, Positive Linear Correlation, No Correlation, and Nonlinear Correlation. Next, we need a way to quantify or measure the strength of the linear relationship between the two variables.

9.1 Correlation The Correlation Coefficient measures the strength and the direction of the linear relationship between two variables. The sample correlation coefficient, r, is defined as: where n is the number of pairs of data

9.1 Correlation Things we need to know about the sample correlation coefficient, r : –r will always lie between -1 and 1, inclusive: -1 ≤ r ≤ 1 –If r = -1, we say there is a perfect negative linear correlation between the two variables. –If r = 1, there is a perfect positive linear correlation between the two variables. –The strength of the linear relationship between the variables is determined by r ’s proximity to 1 or -1. In other words, the closer r is to 1 or -1, the stronger the linear relationship. The closer r is to 0, the weaker the linear relationship. Practice: #22 p. 482 (Age and Vocabulary)

9.1 Correlation Once we have the sample linear correlation coefficient, r, we can use it in a t-Test to make an inference about the population linear correlation coefficient, ρ (Greek letter “rho”). –Why bother? Remember we found r using a limited set of data. What about the rest of the population? Do we have enough evidence from the sample data to claim that a significant linear correlation exists between our two variables? –Example: If we have analyzed the High School GPA and College GPA of 25 students, is there enough evidence to claim that a significant linear correlation exists between the High School GPA and College GPA of all students?

9.1 Correlation t-Test for the Population Correlation Coefficient –We will use the two-tailed version of this test: H 0 : ρ = 0 (no significant correlation exists) H a : ρ ≠ 0 (a significant correlation exists) –The test statistic is r and the standardized test statistic is given by: Note: t follows a t-distribution with n – 2 degrees of freedom

9.1 Correlation Practice using the t-Test: #32 p. 484 (Braking Distances: Wet Surface)