Intro to Business Statistics: Correlation and Regression Correlation Coefficient Pearson r Calculation Interpretation Hypothesis Test Coefficient of Determination r²

What This Is

The correlation coefficient, also known as Pearson's r, measures the strength and direction of a linear relationship between two continuous variables. A retail chain wants to know if there's a relationship between the average temperature in a region and the average daily sales of its stores. By calculating the correlation coefficient, the chain can determine if warmer temperatures lead to higher sales.

Key Formulas & Symbols

• Pearson's r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²]) where xi = individual data point, x̄ = sample mean, yi = individual data point, ȳ = sample mean.
• r² = 1 - [(Σ(xi - x̄)² * Σ(yi - ȳ)²) / (Σ(xi - x̄)² * Σ(yi - ȳ)²)] where r² is the coefficient of determination.
• t = r * √[(n - 2) / (1 - r²)] where t is the test statistic, r is the correlation coefficient, n is the sample size.
• p-value = 2 * P(t > |t|) where p-value is the probability of observing the test statistic (or more extreme) if the null hypothesis is true.
• Null Hypothesis (H₀): ρ = 0 where ρ is the population correlation coefficient.
• Alternative Hypothesis (H₁): ρ ≠ 0 where ρ is the population correlation coefficient.
• Critical Value (t): depends on the degrees of freedom (n - 2) where n is the sample size.
• Coefficient of Determination (r²): measures the proportion of the variance in one variable that is predictable from the other variable.

Step-by-Step Procedure

1. State hypotheses: H₀: ρ = 0 (no correlation) vs. H₁: ρ ≠ 0 (correlation exists).
2. Choose test: Use the t-test for correlation coefficient.
3. Compute test statistic: t = r * √[(n - 2) / (1 - r²)].
4. Find p-value or critical value: Use a t-distribution table or calculator to find the p-value or critical value.
5. Compare to α: Compare the p-value to α (default = 0.05) or compare the test statistic to the critical value.
6. Conclude: If p-value < α or |t| > critical value, reject H₀ and conclude that a significant correlation exists.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that H₀ is true.
• Correction: The p-value is the probability of observing the test statistic (or more extreme) if H₀ is true. It does not directly measure the probability of H₀ being true.
• Mistake: Using the Z-test when the population standard deviation is unknown.
• Correction: Use the t-test when the population standard deviation is unknown.
• Mistake: Failing to check the assumptions of the t-test (normality, independence, and equal variances).
• Correction: Check the assumptions before conducting the t-test.

Quick Practice Problems

1. A marketing firm wants to know if there's a relationship between the number of hours spent watching TV and the number of hours spent browsing the internet. The correlation coefficient is 0.8. What is the p-value?

p-value = 2 * P(t > |t|) = 2 * P(t > 4.2) ≈ 0.0001

1. A quality control team wants to know if there's a relationship between the temperature of a manufacturing process and the quality of the output. The correlation coefficient is 0.5. What is the coefficient of determination?

r² = 1 - [(Σ(xi - x̄)² * Σ(yi - ȳ)²) / (Σ(xi - x̄)² * Σ(yi - ȳ)²)] ≈ 0.25

1. A sales manager wants to know if there's a relationship between the average price of a product and the average sales volume. The correlation coefficient is 0.9. What is the test statistic?

t = r * √[(n - 2) / (1 - r²)] = 0.9 * √[(100 - 2) / (1 - 0.9²)] ≈ 6.3

Last-Minute Cram Sheet

1. Pearson's r measures the strength and direction of a linear relationship.
2. r² measures the proportion of the variance in one variable that is predictable from the other variable.
3. t-test is used to test the correlation coefficient when the population standard deviation is unknown.
4. p-value is the probability of observing the test statistic (or more extreme) if H₀ is true.
5. H₀: ρ = 0 (no correlation) vs. H₁: ρ ≠ 0 (correlation exists).
6. Critical value depends on the degrees of freedom (n - 2).
7. Assumptions of the t-test: normality, independence, and equal variances.
8. ⚠️ p-value is NOT the probability that H₀ is true – it's the probability of observing the data (or more extreme) if H₀ is true.
9. ⚠️ Use the t-test when the population standard deviation is unknown.
10. ⚠️ Check the assumptions before conducting the t-test.