Correlation and Regression

Introduction

This chapter introduces the concepts of correlation and regression in statistics, focusing on how to measure and interpret the relationship between two quantitative variables. The primary tool for assessing linear relationships is the linear correlation coefficient (r), and regression analysis is used to model and predict outcomes.

10-1 Correlation

Key Concepts

• Correlation exists when the values of one variable are associated with the values of another variable.

• Linear correlation occurs when the plotted points of paired data approximate a straight line.

• Only linear relationships are considered in this context; nonlinear relationships require different methods.

Interpreting Scatterplots

Scatterplots are graphical representations of paired data. They help visualize the type and strength of the relationship between two variables.

• Positive linear correlation: As x increases, y increases (points trend upward).

• Negative linear correlation: As x increases, y decreases (points trend downward).

• No correlation: Points are scattered randomly, showing no apparent pattern.

• Nonlinear relationship: Points follow a curved pattern, not a straight line.

Measuring the Strength of Linear Correlation with r

The linear correlation coefficient (r) quantifies the strength and direction of a linear relationship between two variables.

• r > 0: Positive correlation

• r < 0: Negative correlation

• r ≈ 0: No linear correlation

• |r| close to 1: Strong linear correlation

Definition: Linear Correlation Coefficient r

The linear correlation coefficient r measures the strength of the linear correlation between paired quantitative x and y values in a sample. It is also known as the Pearson product moment correlation coefficient.

Calculating and Interpreting the Linear Correlation Coefficient r

• Objective: Determine whether there is a linear correlation between two variables.

Notation

• n: Number of pairs of sample data

• Σ: Summation symbol (sum of indicated items)

• Σx: Sum of all x values

• Σx²: Sum of squares of x values

• (Σx)²: Square of the sum of x values

• Σxy: Sum of products of paired x and y values

• r: Linear correlation coefficient for sample data

• ρ: Linear correlation coefficient for a population

Requirements for Using r

• The sample of paired (x, y) data is a simple random sample of quantitative data.

• Visual examination of the scatterplot must confirm that the points approximate a straight-line pattern.

• Outliers must be removed if known to be errors; otherwise, calculate r with and without outliers to assess their impact.

• Formal requirement: The pairs of (x, y) data must have a bivariate normal distribution. Additional info: This ensures the validity of inferential procedures involving r.

Formulas for Calculating r

• Formula 10-1 (Calculation):

• Formula 10-2 (Understanding):

where and are the z-scores for the sample values of x and y, respectively.

Rounding r

• Round r to three decimal places for comparison with critical values.

Interpreting r

• Using P-value: If P-value ≤ α (significance level), there is evidence of linear correlation. If P-value > α, there is not.

• Using Critical Values (Table A-6): If |r| ≥ critical value, conclude linear correlation; if |r| < critical value, conclude no linear correlation.

Properties of the Linear Correlation Coefficient r

• The value of r is always between -1 and 1 inclusive:

• Changing the scale of either variable does not affect r.

• Interchanging x and y does not affect r.

• r measures only linear relationships, not nonlinear ones.

• r is sensitive to outliers; a single outlier can dramatically affect its value.

Example: Finding r Using Technology

Given paired data for Powerball jackpots and ticket sales:

Using statistical software, r is automatically calculated as 0.947 (rounded).

Example: Finding r Using Formula 10-1

Calculation:

Example: Finding r Using Formula 10-2

Each sample value is replaced by its z-score:

Then,

Summary Table: Types of Scatterplot Relationships

Conclusion

The linear correlation coefficient r is a fundamental statistic for quantifying the strength and direction of a linear relationship between two quantitative variables. Proper calculation, interpretation, and understanding of its properties are essential for valid statistical analysis. Always check requirements and use appropriate methods for calculation and interpretation.