Question

Based on the following regression output, what proportion the total variation in Y is explained by X? Regression Statistics Multiple R 0.917214 R Square 0.841282 Adjusted R Square 0.821442 Standard Error 9.385572 Observations 10 ANOVA di SS MS Significance F 1 Regression 3735.3060 3735.30600 42.40379 0.000186 Residual 8 704.7117 88.08896 9 Total 4440.0170 Coefficients Standard Error t Stat P-value Lower 95% Intercept 31.623780 10.442970 3.028236 0.016353 7.542233 X Variable 1.131661 0.173786 6.511819 0.000186 0.730910 o a. 0.917214 o b.9.385572 oc. 0.841282 o d. 0.821442

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Answer 1

1. The answer is (c) 0.841282 which is the value of R square. R square explains the proportion of variability in the dependent variable (Y) by the independent variable (X). Option (a) is wrong because it is multiple R which denotes the correlation between actual and predicted values of dependent variable. Option (b) is out of question since it denotes standard error and not variability. Option (d) is adjusted R square and it accounts for variability based on actual independent variables. Hence it is incorrect.

2. The correct answer is (b) X1 and X3 represent similar functions. So multi collinearity exists and hence R square value is same. However, adjusted R2 takes into account the number of independent variables that actually affect the dependent variablea and since variation caused by X1 is included in X3, the adjusted R2 value is lower since 2 variables are used here. When only X3 is used, the adjsuted R2 value is higher. Option (d) is wrong since standard error is not related to R2. Option (a) is wrong becayse increase in ESS leads to increased R2. Option (c) fails to explain the similarity of R2 value between 2 models (X3 alone, X1 and X3). Hence, it is incorrect.

3. R2 measures the percentage of variability in dependent variable Y explained by the model involving independent variables. Hence, (a) is the correct answer. Option (b) is wrong since R square does not account for unxplained variability. Option (c) is wrong since R square=ESS/TSS not RSS/ESS. Model Sophistication is not indicated by R square. Hence,(d) is also incorrect.