11.1 Correlation_Section

Correlation Overview

• Correlation: Measures the strength and direction of the linear relationship between two variables.

• Scatterplot: Visual representation of ordered pairs, with the first coordinate on the horizontal axis and the second on the vertical axis.

• Example: Real estate agent analyzes relationship between house size (sq ft) and selling price ($1000s).

• Associations:

  • Positive association: Large values correlate with large values.

  • Negative association: Large values correlate with small values.

• Correlation Formula:

  • Correlation Coefficient (r) measures relationship strength. Formula: r = (1/(n-1)) * Σ((x - x̄)/s_x) * ((y - ȳ)/s_y)

• Calculator Instructions:

  • Data entry, regression line calculation, and enabling diagnostics for output.

• Interpreting r:

  • Positive r: Positive association, negative r: Negative association, closer to 1/-1 indicates strength.

• Correlation Size Interpretation:

  • .90 to 1.00: Very high positive,

  • .70 to .90: High positive,

  • 0: Negligible,

  • .90 to -1.00: Very high negative.

Example of Applying Correlation Concepts:A researcher is studying the impact of study hours on exam scores among students.

• Scatterplot: The researcher collects data on the number of hours each student studied (horizontal axis) and their corresponding exam scores (vertical axis). A scatterplot is created to visualize the ordered pairs.

• Positive Association: The analysis shows that as study hours increase, exam scores tend to increase as well, indicating a positive association.

• Correlation Coefficient: Using the correlation formula, the researcher calculates the correlation coefficient (r) and finds it to be 0.85. This indicates a strong positive relationship between study hours and exam scores.

• Interpreting r: Since r is close to 1, the researcher concludes that there is a significant correlation, suggesting that more study time generally leads to higher exam scores.