Intro to Business Statistics: Correlation and Regression Interpretation of Coefficients

Interpretation of Coefficients

Interpretation of coefficients is a crucial aspect of regression analysis, which helps businesses understand the relationship between variables and make informed decisions. For instance, a retail chain wants to know if the average daily sales exceed $10,000 when the temperature is 75°F, and the price of an item is $20. By analyzing the coefficients, they can determine the impact of temperature and price on sales.

Key Formulas & Symbols

• Coefficient of Determination (R²): Measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s). R² = 1 - (SSE / SST) where SSE = sum of squared errors, SST = total sum of squares.
• Coefficient of Variation (CV): Measures the relative variability of a dataset. CV = (σ / x̄) × 100 where σ = standard deviation, x̄ = mean.
• F-statistic: Used to test the overall significance of a regression model. F = (MSR / MSE) where MSR = mean square regression, MSE = mean square error.
• t-statistic: Used to test the significance of individual coefficients. t = (b - β) / (s / √n) where b = sample coefficient, β = population coefficient, s = standard error, n = sample size.
• p-value: The probability of observing the data (or more extreme) if the null hypothesis is true. p = P(T ≥ |t|) where T = t-distribution.
• R-squared Change (ΔR²): Measures the change in R² when a new variable is added to the model. ΔR² = R²_new - R²_old.
• Adjusted R-squared (R²_adj): Adjusts R² for the number of predictors in the model. R²_adj = 1 - ((n - 1) / (n - k - 1)) × (1 - R²) where n = sample size, k = number of predictors.

Step-by-Step Procedure

1. State hypotheses: Formulate the null and alternative hypotheses. For example, H₀: β = 0 (no relationship between variables) vs. H₁: β ≠ 0 (relationship exists).
2. Choose test: Select the appropriate test statistic (t, F, or Z) based on the type of coefficient being tested and the assumptions of the data.
3. Compute test statistic: Calculate the test statistic using the formula for the chosen test.
4. Find p-value or critical value: Determine the p-value or critical value for the test statistic using a t-distribution table or calculator.
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Make a decision based on the comparison (reject H₀ or fail to reject H₀).

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 provide information about the probability of the null hypothesis being true.
• Mistake: Failing to check the assumptions of the data (normality, linearity, homoscedasticity).
• Correction: Verify that the data meet the assumptions of the test before proceeding.
• Mistake: Ignoring the direction of the coefficient.
• Correction: Consider the direction of the coefficient when interpreting the results.

Quick Practice Problems

1. A marketing firm wants to know if the average daily sales exceed $10,000 when the temperature is 75°F, and the price of an item is $20. The regression equation is Sales = 500 + 2.5 × Temperature + 1.2 × Price. What is the coefficient of determination (R²)?

Answer: R² = 0.8 (80% of the variance in sales is predictable from temperature and price).

1. A company wants to know if the average daily sales exceed $10,000 when the temperature is 75°F, and the price of an item is $20. The regression equation is Sales = 500 + 2.5 × Temperature + 1.2 × Price. What is the t-statistic for the temperature coefficient?

Answer: t = 2.5 / (s / √n) = 2.5 / 0.5 = 5 (assuming s = 0.5 and n = 100).

1. A retail chain wants to know if the average daily sales exceed $10,000 when the temperature is 75°F, and the price of an item is $20. The regression equation is Sales = 500 + 2.5 × Temperature + 1.2 × Price. What is the p-value for the temperature coefficient?

Answer: p = P(T ≥ |t|) = 0.001 (assuming t = 5 and df = 99).

Last-Minute Cram Sheet

1. R² measures the proportion of variance in the dependent variable that is predictable from the independent variable(s).
2. t-statistic is used to test the significance of individual coefficients.
3. p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.
4. F-statistic is used to test the overall significance of a regression model.
5. Adjusted R-squared (R²_adj) adjusts R² for the number of predictors in the model.
6. Coefficient of Variation (CV) measures the relative variability of a dataset.
7. ⚠️ 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.
8. Assumptions of regression analysis: normality, linearity, homoscedasticity, independence.
9. R-squared Change (ΔR²) measures the change in R² when a new variable is added to the model.
10. ⚠️ F-statistic is sensitive to sample size – use caution when interpreting results with small samples.