© 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

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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

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Features of theHypothesis Test

• Step 1. H0: 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

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Features of the Hypothesis Test (cont.)

• Step 2. The sampling distribution is the t-distribution with df = n - 2

• When the H0 is true, sample Pearson’s r’s will center around zero

• This test does not require a direct calculation of a standard error

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Features of the Hypothesis Test (cont.)

• Step 4. The test effect is the value of Pearson’s r

• The test statistic is tr

• The p-value is estimated from the t-distribution table, Statistical Table C in Appendix B

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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

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Existence of a Relationship

• Test the H0 that ρ = 0, that there is no relationship between X and Y

• If the H0 is rejected, a relationship exists

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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

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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 r2, not r

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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

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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

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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)

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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