Intro to Marketing Research: Correlation and Regression Pearson Correlation Coefficient r Calculation Interpretation Hypothesis Testing Scatterplots

What It Is

The Pearson Correlation Coefficient (r) is a statistical measure used to assess the linear relationship between two continuous variables. It ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation), with 0 indicating no correlation. A famous example is the study by Galton (1886) on the relationship between the height of parents and their children, which showed a strong positive correlation (r = 0.63). This matters for marketing decision-making as it helps identify the strength and direction of relationships between variables, enabling marketers to make informed decisions about product development, pricing, and target audience selection.

Key Terms & Concepts

• Pearson Correlation Coefficient (r): a statistical measure of the linear relationship between two continuous variables.
  • Formula: r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])
  • Example: A study on the relationship between the price of a product and its sales volume.
• Correlation vs. Causation: correlation does not imply causation.
  • Example: A study on the relationship between the number of hours watched TV and the number of hours spent exercising, which shows a negative correlation (r = -0.5), but does not imply that watching TV causes people to exercise less.
• Scatterplot: a graphical representation of the relationship between two variables.
  • Example: A scatterplot showing the relationship between the price of a product and its sales volume.
• Coefficient of Determination (R²): a measure of the proportion of variance in one variable that is explained by the other variable.
  • Formula: R² = r²
  • Example: A study on the relationship between the price of a product and its sales volume, which shows an R² value of 0.7, indicating that 70% of the variance in sales volume is explained by the price of the product.
• Hypothesis Testing: a statistical method used to test a hypothesis about the relationship between two variables.
  • Example: A study on the relationship between the price of a product and its sales volume, which tests the hypothesis that there is a positive correlation between the two variables.
• Null Hypothesis (H0): a statement of no effect or no difference.
  • Example: H0: There is no correlation between the price of a product and its sales volume.
• Alternative Hypothesis (H1): a statement of an effect or a difference.
  • Example: H1: There is a positive correlation between the price of a product and its sales volume.
• Type I Error: a false positive result, where a null hypothesis is rejected when it is true.
  • Example: A study on the relationship between the price of a product and its sales volume, which rejects the null hypothesis of no correlation when it is true.
• Type II Error: a false negative result, where a null hypothesis is not rejected when it is false.
  • Example: A study on the relationship between the price of a product and its sales volume, which fails to reject the null hypothesis of no correlation when it is false.
• Significance Level (α): the maximum probability of a Type I error.
  • Example: α = 0.05, which means that there is a 5% chance of rejecting the null hypothesis when it is true.
• Critical Region: the region of the distribution where the null hypothesis is rejected.
  • Example: A critical region of z > 1.96, which means that the null hypothesis is rejected if the z-score is greater than 1.96.
• Regression Equation: a mathematical equation that describes the relationship between two variables.
  • Formula: y = a + bx
  • Example: A regression equation that describes the relationship between the price of a product and its sales volume.
• Coefficient of Variation (CV): a measure of the relative variability of a variable.
  • Formula: CV = (σ / x̄) * 100
  • Example: A study on the relationship between the price of a product and its sales volume, which shows a CV value of 20%, indicating that the sales volume is 20% more variable than the price of the product.

Common Misunderstandings

• Misunderstanding: The Pearson Correlation Coefficient (r) measures the strength of the relationship between two variables.
• Correction: The Pearson Correlation Coefficient (r) measures the linear relationship between two variables, but does not indicate the strength of the relationship. The strength of the relationship is indicated by the coefficient of determination (R²).
• Misunderstanding: A scatterplot is a graphical representation of the relationship between two variables.
• Correction: A scatterplot is a graphical representation of the relationship between two variables, but it does not indicate the strength or direction of the relationship. The Pearson Correlation Coefficient (r) is used to indicate the strength and direction of the relationship.
• Misunderstanding: Hypothesis testing is used to test a hypothesis about the relationship between two variables.
• Correction: Hypothesis testing is used to test a hypothesis about the population parameter, but it is not used to test a hypothesis about the relationship between two variables. The Pearson Correlation Coefficient (r) is used to test the hypothesis about the relationship between two variables.

Quick Application / Identification

Scenario: A marketing manager wants to know if there is a relationship between the price of a product and its sales volume. She collects data on the price and sales volume of the product over a period of time and plots a scatterplot. What does the scatterplot indicate?

Answer: The scatterplot indicates the relationship between the price of the product and its sales volume, but it does not indicate the strength or direction of the relationship. The Pearson Correlation Coefficient (r) is used to indicate the strength and direction of the relationship.

Last‑Minute Revision

• The Pearson Correlation Coefficient (r) ranges from -1 to 1.
• The coefficient of determination (R²) measures the proportion of variance in one variable that is explained by the other variable.
• A scatterplot is a graphical representation of the relationship between two variables.
• Hypothesis testing is used to test a hypothesis about the population parameter.
• The null hypothesis (H0) is a statement of no effect or no difference.
• The alternative hypothesis (H1) is a statement of an effect or a difference.
• Type I error is a false positive result, where a null hypothesis is rejected when it is true.
• Type II error is a false negative result, where a null hypothesis is not rejected when it is false.
• The significance level (α) is the maximum probability of a Type I error.
• The critical region is the region of the distribution where the null hypothesis is rejected.
• The regression equation is a mathematical equation that describes the relationship between two variables.
• The coefficient of variation (CV) is a measure of the relative variability of a variable.
• ⚠️ A Type I error occurs when the null hypothesis is rejected when it is true.
• ⚠️ A Type II error occurs when the null hypothesis is not rejected when it is false.
• ⚠️ The significance level (α) is usually set at 0.05.
• ⚠️ The critical region is usually determined by the significance level (α).
• ⚠️ The regression equation is usually linear.