Chapter 12 Notes: Association Between I-R Variables & Linear Regression (Part 1)

Chapter 12: Association Between I-R Variables & Linear Regression (Part 1)

Chapter Outline

• Introduction
• Scattergram
• Regression and Prediction
• Find the Regression Line: Y = a + bX; where a is the Y-intercept and b is the slope
• Pearson’s r (correlation)
• Interpreting r^2
• Other Issues in Regression Analysis

Key Concepts

• Scattergram (or a scatter plot): A graph that displays the relationship between two interval-ratio variables.
• Regression Line: Summarizes the linear relationship between two I-R variables X and Y. It predicts a score on Y from a score on X.
• Pearson’s r (correlation): A measure of association for interval-ratio variables.

Example: Regression Analysis

Scenario

• Goal: Determine the relationship between hours of study and exam grade in statistics.
• Method: A random sample of 16 students is drawn from a large statistics class.

Data

The information of the sample is show in the following table:

Scattergram Details

• Axes:
  • The independent variable (X) is arrayed along the horizontal axis.
  • The dependent variable (Y) is arrayed along the vertical axis.
• Data Points:
  • Each dot on a scattergram represents a case.
  • The dot is placed at the intersection of the case’s scores on X and Y.

Scattergram Example

• This scattergram shows the relationship between “hours of study” (X) and “exam grade” (Y) for the 16 students.
• X axis (horizontal) is “hours of study.”
  • Scores range from 1 to 14.
• Y axis (vertical) is “exam grade.”
  • Scores range from 52 to 96.

Scattergram -- Correlation Patterns

• The greater the extent to which dots are clustered around a straight line, the stronger the correlation.
• Positive Correlation: X increases, Y increases.
• Negative Correlation: X increases, Y decreases.

Estimation of Y

• What is the best way to estimate the dependent variable, Y?
  • Mean value?
  • Predictions based on the regression line?

Regression Line

• A single straight line that comes as close as possible to all data points.
• That line minimizes:
  \sum(Y - \hat{Y})^2
• The line passes through the point:
  (\bar{X}, \bar{Y})

Direction of Regression Line

• A positive correlation: regression line rises from left to right.
• A negative correlation: regression line falls from left to right.