Notes: Unit 2 - Exploring Two-Variable Data

Two Categorical Variables

• Qualitative data: Often involves two categorical variables which may or may not be dependent.
• Two-way Table/Contingency Table: Used to display such data.

Example 2.1: The Cuteness Factor

• Study with 250 volunteers observing different categories (baby animals, adult animals, tasty foods).
• Row Variable: Pictures viewed.
• Column Variable: Level of focus.
• Table Total: Sum of all cell values.
• Marginal Frequencies: Totals for each row and column, used to form proportions/percentages.

Two Quantitative Variables

• Bivariate Quantitative Data Sets: Concerned with relationships between two numerical variables.
• Scatterplot: Provides visual representation of the potential relationship.
• Correlation Coefficient: Measures strength of linear relationship.

Example 2.2: Comic Books

• Scatterplot comparing speed and strength of comic characters.
• Positive Association: Larger values of one variable correlate with larger values of another.
• Negative Association: Larger values of one correlate with smaller values of another.

Correlation

• Only measures linear relationships.
• Designated by r: Formula uses means and standard deviations.
• Unit-Free: Unaffected by switching x and y.
• Range: -1 to +1.
• Coefficient of Determination (r^2): Ratio of variance in predicted values to observed values.

Example 2.3: Football Statistics

• Correlation between Total Points and Yards Gained: r = 0.84, r² = 0.7056.

Least Squares Regression

• Best-fitting Line: Minimizes the sum of squares of vertical differences between observed and predicted values.
• Passes through means of X and Y.

Example 2.4: Teen Survey

• Regression for relationship between number of friends and evening Facebook checks.
• Slope interpretation: Each friend leads to 0.5492 more checks.

Residuals

• Difference: Between observed and predicted values.
• Positive Residual: Model underestimated.
• Negative Residual: Model overestimated.

Outliers, Influential Points, Leverage

• Outliers: Points with large discrepancies from the pattern.
• Influential Scores: Removal changes regression line significantly.
• High Leverage: x-values far from the mean.

Example 2.5: GPA vs. TV Time

• Identification of outliers based on regression context.

More on Regression

• Implications of the regression equation in terms of correlation.

Example 2.8 & 2.9

• Calculations using attendance and popcorn sales data for predictions.

Transformations to Achieve Linearity

• Transformation: Logarithmic transformations can reveal linear relationships.

Example 2.10

• Population data suggests a linear model, but nonlinear might be stronger.

This summary captures key concepts and examples from Unit 2, providing a comprehensive guide to understanding two-variable data analysis.