Measures the squared diff b/w each data point and the average of the data points - Larger difference b/w data points = larger SST o Divide SST → SSR and SSE ▪ SST = SSR + SSE o Sum of Squares Error (SSE) – measures the variation in the dependent variable that is explained by variables other than the independent variable ▪ SSE = ∑y² - b∑y - m∑xy - Where b = y-intercept - M = slope ▪ In our example → 74.62 o Sum of Squares Regression (SSR) – measures the amount of variation in the dependent variable that is explained by the independent variable
▪ SSR = ∑(predicted value of y at given value of x – average value of the dependent variable from the sample)² ▪ SSR = SST – SSE ▪ In our example → 237.5 ● Calculating the Coefficient of Determination o Coefficient of Determination (R²) – measures the percentage of the total variation of our dependent variable that is explained by our independent variable from a sample ▪ R² = SSR / SST ▪ R² = (r)² ▪ In our example → 0.686 - Conclude that 68.6% of total variation can be explained by the independent variable (hours of study) o The Basics ▪ Value ranges from 0 – 100% ▪ Higher values = more desirable - b/c want to explain most of variation by independent variable - indicate stronger relationship b/w dependent and independent variables ▪ low values may indicate: - using wrong independent variable - need additional independent variables to explain variation in dependent variable ● Conducting a Hypothesis Test to Determine the Significance of the Coefficient of Determination o Population Coefficient of Determination (p²) – measures for an entire population the percentage of the total variation of a dependent variable that is explained by an independent variable ▪ Unknown in our example
▪ Perform hypothesis test to det. if p² is significantly diff from zero based on R² ▪ If reject null hypothesis - Have enough evidence from our sample to conclude that a relationship does exist b/w the 2 variables o Step 1: Hypothesis Statements ▪ H0: p² ≤ 0 ▪ H1: p² > 0 o Step 2: Set level of significance ▪ Set ∞ = 0.05 o Step 3: F-Test Statistic ▪ F = SSR ÷ (SSE / n – 2) ▪ In our example → 8.73 o Step 4: Find the critical F-Score ▪ Identifies the rejection region for the hypothesis test ▪ Follows F-distribution (Table 6, Appendix A) ▪ Degrees of Freedom - D1 = 1 (always one b/c only one independent variable) - D2 = (n – 2) ▪ In our example, ∞ = 0.05, D1 = 1, D2 = 4 - → 7.709 o Step 5: Comparing the F-Test Statistic (F) with the Critical (F-Score) ▪ 8.73 (F) > 7.709 (F-score) → REJECT o Step 6: State your conclusions ▪ Conclude that the coefficient of determination (R²) is greater than zero – appears to be a relationship b/w the two variables o In the REGRESSION OUTPUT – EXCEL ▪ Regression Statistics > R Square - = coefficient of determination, R² ▪ Sum of squares (ANOVA – SS column) - Residual = error sum of squares
- Regression ▪ ANOVA F column = F-score ▪ ANOVA DF shows degrees of freedom for - D1 (regression) - D2 (residual) ▪ ANOVA Significance F = p-value -
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- Fall '12
- Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing