Intro to Business Statistics: Statistical Software and Applications Running Regression Analysis

What This Is

Running regression analysis is a statistical method used to model the relationship between a dependent variable (y) and one or more independent variables (x). This technique helps businesses understand how changes in independent variables affect the dependent variable, enabling informed decisions. For instance, a retail chain wants to know if average daily sales exceed $10,000 when the number of promotions and advertising expenses are considered.

Key Formulas & Symbols

• Regression Equation: y = β₀ + β₁x + ε, where y = dependent variable, x = independent variable, β₀ = intercept, β₁ = slope, and ε = error term.
• Coefficient of Determination (R²): R² = 1 - (SSE / SST), where SSE = sum of squared errors, SST = total sum of squares.
• F-Statistic: F = (MSR / MSE), where MSR = mean square regression, MSE = mean square error.
• T-Statistic: t = (β₁ - β₁₀) / (s / √(1/n + 1/n)), where β₁ = estimated slope, β₁₀ = hypothesized slope, s = standard error of the slope, n = sample size.
• Standard Error of the Slope (s): s = √(MSE / ∑(xi - x̄)²), where MSE = mean square error, xi = individual data points, x̄ = sample mean.
• Degrees of Freedom (df): df = n - 2, where n = sample size.
• Critical F-Value: F(α, df1, df2), where α = significance level, df1 = numerator degrees of freedom, df2 = denominator degrees of freedom.
• Critical T-Value: t(α, df), where α = significance level, df = degrees of freedom.

Step-by-Step Procedure

1. State Hypotheses: Formulate null and alternative hypotheses (e.g., H₀: β₁ = 0 vs. H₁: β₁ ≠ 0).
2. Choose Test: Select the appropriate test statistic (e.g., F-test for overall significance or t-test for individual coefficients).
3. Compute Test Statistic: Calculate the F-statistic or t-statistic using the given formulas.
4. Find p-Value or Critical Value: Determine the p-value associated with the test statistic or find the critical value from a distribution table.
5. Compare to α: Compare the p-value or critical value to the chosen significance level (α = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H₀ if p-value < α).

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true. It does not directly indicate the probability of the null hypothesis being true.
• Mistake: Failing to check for multicollinearity among independent variables.
• Correction: Multicollinearity can lead to unstable estimates and incorrect conclusions. Check for high correlations among independent variables and consider using techniques like variance inflation factor (VIF) or principal component analysis (PCA).
• Mistake: Ignoring the assumption of normality in the residuals.
• Correction: Non-normal residuals can lead to incorrect conclusions. Check for normality using plots or tests like the Shapiro-Wilk test and consider transformations or alternative models if necessary.

Quick Practice Problems

1. A marketing manager wants to know if the number of social media followers affects sales. The regression equation is y = 100 + 2x, where y = sales and x = number of social media followers. What is the slope of the regression line?

Answer: 2. The slope represents the change in sales for a one-unit change in the number of social media followers.

1. A company wants to determine if the price of a product affects demand. The regression equation is y = 1000 - 5x, where y = demand and x = price. What is the p-value associated with the slope coefficient?

Answer: Not provided (requires actual data and calculations). However, the p-value would indicate the probability of observing the data (or more extreme) if the null hypothesis that the slope is 0 is true.

1. A quality control manager wants to know if the number of defects per unit affects the cost of production. The regression equation is y = 100 + 3x, where y = cost and x = number of defects per unit. What is the coefficient of determination (R²)?

Answer: Not provided (requires actual data and calculations). However, R² would indicate the proportion of variance in the cost that is explained by the number of defects per unit.

Last-Minute Cram Sheet

1. F-Statistic: F = (MSR / MSE), where MSR = mean square regression, MSE = mean square error.
2. T-Statistic: t = (β₁ - β₁₀) / (s / √(1/n + 1/n)), where β₁ = estimated slope, β₁₀ = hypothesized slope, s = standard error of the slope, n = sample size.
3. Critical F-Value: F(α, df1, df2), where α = significance level, df1 = numerator degrees of freedom, df2 = denominator degrees of freedom.
4. Critical T-Value: t(α, df), where α = significance level, df = degrees of freedom.
5. Coefficient of Determination (R²): R² = 1 - (SSE / SST), where SSE = sum of squared errors, SST = total sum of squares.
6. Standard Error of the Slope (s): s = √(MSE / ∑(xi - x̄)²), where MSE = mean square error, xi = individual data points, x̄ = sample mean.
7. Degrees of Freedom (df): df = n - 2, where n = sample size.
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. Assumption of Normality: Check for normality using plots or tests like the Shapiro-Wilk test.
10. ⚠️ Multicollinearity can lead to unstable estimates and incorrect conclusions. Check for high correlations among independent variables.