BM 6

ASSOCIATION BETWEEN VARIABLES

Outline

• Introduction
• Correlation analysis
• Scatter plots
• Correlation Coefficient
• Coefficient of determination, r²
• Regression analysis

Objectives

• To be able to draw and interpret scatter diagrams.
• To be able to calculate the correlation coefficient and coefficient of determination.
• To understand and be able to use the least squares method to estimate the regression line.

Introduction

• The lecture introduces two fundamental techniques in statistics:
  • Correlation: A method to measure the association between two variables.
  • Regression: A technique to derive the relationship between two variables, helping to predict dependent variables based on independent variables.
• Example scenarios include:
  • The potential dependence of production costs on the quantity produced.
  • The relationship between product sales and pricing.

Correlation Analysis

• Correlation measures the strength of the association between two variables.
• A change in one variable corresponds to a change in another when they are said to be associated.
• Illustrative examples of potential correlations include:
  • Relationship between production cost and price.
  • Association between advertising efforts and sales revenue.
  • Correlation between the number of deliveries and the time taken for those deliveries.

Class Activity: Determining Dependent and Independent Variables

• Examples include:
  • Study Hours and Exam Grades: Study hours (independent) → Exam grades (dependent)
  • Temperature and Ice Cream Sales: Temperature (independent) → Ice cream sales (dependent)
  • Advertising Budget and Sales Revenue: Budget (independent) → Revenue (dependent)
  • Distance Traveled and Fuel Consumption: Distance (independent) → Fuel (dependent)
  • Employee Training Hours and Productivity: Training hours (independent) → Productivity (dependent)
  • Rainfall Amount and Crop Yield: Rainfall (independent) → Crop yield (dependent)
  • Social Media Engagement and Website Traffic: Engagement (independent) → Traffic (dependent)
  • Education Level and Income: Education (independent) → Income (dependent)

Scatter Diagrams

• A scatter plot visually represents bivariate data, showing potential correlations.
• The independent variable is plotted on the x-axis, while the dependent variable is plotted on the y-axis.
• Analysis of scatter plots can reveal:
  • Patterns indicating the strength and direction of the association.
  • Suitability of data before conducting quantitative analysis.

Degrees of Association/Correlation

• When examining scatter plots, consider:
  • Evidence of a pattern in the plotted points.
  • Types of correlation:
  • Perfect Correlation: All data points lie on a straight line, indicating a precise linear relationship (either positive or negative).
  • Partially Correlated: Some pattern exists but not linear or perfectly associated.
  • Uncorrelated: No discernible relationship between the variables.

Pearson’s Product Moment Correlation Coefficient

• Used to measure the strength of the association between two variables.
• The correlation coefficient (denoted as r) ranges from -1 to +1:
  • A value close to +1 (e.g., 0.9) indicates a strong positive linear relationship.
  • A value close to –1 (e.g., –0.9) indicates a strong negative linear relationship.
  • A value of 0 indicates no linear relationship, though other types of relationships may exist.

Formula for r

• The formula for calculating the Pearson product moment correlation coefficient (r) is given by:
  $r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$

Example 1: Calculating r

• Calculate Pearson's correlation coefficient for the data:
  • Units produced (x): 1, 2, 3, 4, 5, 6
  • Production cost (y): 5.0, 10.5, 15.5, 25.0, 16.0, 22.5

Class Exercise

• Task: Plot a scatter diagram and calculate the Pearson product moment correlation coefficient for the following data:
  • Policy (X) and Overtime hours (Y):
  • 150 → 10
  • 300 → 20
  • 100 → 10
  • 400 → 40
  • 350 → 30
  • 500 → 35

Coefficient of Determination r²

• Before using regression for prediction, evaluate its fit to the data.
• The coefficient of determination (r²) measures how well the independent variable explains the variation in the dependent variable.
• Given by the formula:
  $r^2 = (r)²$
• Calculate r² for the previous examples.

Linear Regression Analysis

• Linear regression defines the relationship between dependent and independent variables using a linear equation.
• The focus is on estimating the line of best fit between two variables using:
  • Least Squares Method: A calculation to minimize the sum of squared differences (errors) between observed and predicted values.

The Linear Regression Model

• The linear regression model is represented by: $y = a + bx$
  • Where:
  • y = dependent variable
  • x = independent variable
  • a = y-intercept (constant)
  • b = slope (gradient)

The Values of a and b

• The formulas to determine the optimal values of a and b that minimize squared errors are:
  $b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$
  $a = \frac{\sum y}{n} - b \frac{\sum x}{n}$

Example: Calculating the Regression Line

• Given data:
  • Units produced (x): 1, 2, 3, 4, 5, 6
  • Production cost (y): 5.0, 10.5, 15.5, 25.0, 16.0, 22.5
• Use the formulas for a and b to calculate the linear regression line.

Class Exercise 2

• Given data regarding output and cost:
  • Output (x) in 000 units and Costs (Y) in P’000:
  • 20 → 82
  • 16 → 70
  • 24 → 90
  • 22 → 85
  • 18 → 73
• Task: Calculate the regression line for the data.

Interpretation of a and b

• After calculating a and b, interpret the meaning of these coefficients in the context of the problem.
• Write the regression equation and plot the regression line on a scatter diagram to visualize the fit in relation to the data points.