© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 15: Correlation and Regression Part 2: Hypothesis Testing and Aspects of a Relationship

© 2008 McGraw-Hill Higher Education When to Test a Hypothesis Using Correlation and Regression 1)There is one representative sample from a single population 2)There are two interval/ratio variables 3)There are no restrictions on sample size, but generally, the larger the n, the better 4)A scatterplot of the coordinates of the two variables fits a linear pattern

© 2008 McGraw-Hill Higher Education Test Preparation Before proceeding with the hypothesis test, check the scatterplot for a linear pattern Calculate the Pearson’s r correlation coefficient and the regression coefficient, b Compute the means of X and Y and use them and b to compute a Specify the regression equation, insert values of X, solve for Ý, and plot the line on the scatterplot Provide a conceptual diagram

© 2008 McGraw-Hill Higher Education Features of the Hypothesis Test Step 1. H 0 : ρ = 0 That is, there is no relationship between X and Y The Greek letter rho (ρ) is the correlation coefficient obtained if Pearson’s correlation coefficient were computed for the population A ρ of zero asserts that there is no correlation in the population and that the regression line has no slope

© 2008 McGraw-Hill Higher Education Features of the Hypothesis Test (cont.) Step 2. The sampling distribution is the t- distribution with df = n - 2 When the H 0 is true, sample Pearson’s r’s will center around zero This test does not require a direct calculation of a standard error

© 2008 McGraw-Hill Higher Education Features of the Hypothesis Test (cont.) Step 4. The test effect is the value of Pearson’s r The test statistic is t r The p-value is estimated from the t- distribution table, Statistical Table C in Appendix B

© 2008 McGraw-Hill Higher Education Four Aspects of a Relationship With correlation and regression analysis, because both variables are of interval/ratio level, the analysis is mathematically rich All four aspects of a relationship apply

© 2008 McGraw-Hill Higher Education Existence of a Relationship Test the H 0 that ρ = 0, that there is no relationship between X and Y If the H 0 is rejected, a relationship exists

© 2008 McGraw-Hill Higher Education Direction of a Relationship Direction is indicated by the sign of r and b, and by observing the slope of the pattern of coordinates in a scatterplot A positive relationship is revealed with an upward slope, and r and b will be positive A negative relationship is revealed with a downward slope, and r and b will be negative

© 2008 McGraw-Hill Higher Education Strength of a Relationship Strength is determined by the proportion of the total variation in Y explained by X This proportion is quickly obtained by squaring Pearson’s r correlation coefficient Focus on r 2, not r

© 2008 McGraw-Hill Higher Education Nature of a Relationship 1)Interpret the regression coefficient, b, the slope of the regression line. State the effect on Y of a one-unit change in X 2)Provide best estimates using the regression line equation. Insert chosen values of X, compute Ý ’s and interpret them in everyday language

© 2008 McGraw-Hill Higher Education Careful Interpretation of Findings A correlation applies to a population, not to an individual E.g., predictions of Y for a value of X provide the best estimate of the mean of Y for all subjects with that X-score

© 2008 McGraw-Hill Higher Education Careful Interpretation of Findings (cont.) A statistical relationship may exist but not mean much. Be wary of statistically significant but small Pearson’s r’s Distinguish statistical significance (i.e., the existence of a relationship) from practical significance (i.e., the strength of the relationship)

© 2008 McGraw-Hill Higher Education Statistical Follies: Spurious Correlation A spurious correlation is one that is conceptually false, nonsensical, or theoretically meaningless E.g., for the period of the 1990s, there is a positive correlation between the amount of carbon dioxide released into the atmosphere and the level of the Dow Jones stock index