Consider the multiple linear regression model $$ Y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \dots + \beta_{22} x_{22,i} + \epsilon_i, \quad i = 1, 2, \dots, 123, $$ where, for $ j = 0, 1, 2, \dots, 22 $, $ \beta_j $'s are unknown parameters and $ \epsilon_i $'s are independent and identically distributed $ N(0, \sigma^2) $, $ \sigma0 $, random variables. If the sum of squares due to regression is 338.92, the total sum of squares is 522.30 and $ R^2_{\text{adj}} $ denotes the value of adjusted $ R^2 $, then $$ 100 R^2_{\text{adj}} = \underline{\hspace{2cm}} $$ (round off to 2 decimal places).

Question

Consider the multiple linear regression model $$ Y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \dots + \beta_{22} x_{22,i} + \epsilon_i, \quad i = 1, 2, \dots, 123, $$ where, for $ j = 0, 1, 2, \dots, 22 $, $ \beta_j $'s are unknown parameters and $ \epsilon_i $'s are independent and identically distributed $ N(0, \sigma^2) $, $ \sigma>0 $, random variables. If the sum of squares due to regression is 338.92, the total sum of squares is 522.30 and $ R^2_{\text{adj}} $ denotes the value of adjusted $ R^2 $, then $$ 100 R^2_{\text{adj}} = \underline{\hspace{2cm}} $$ (round off to 2 decimal places).

cdquestions Admin · Accepted Answer

Given the sum of squares due to regression and the total sum of squares, we calculate the adjusted $ R^2 $ as follows: $$ R^2_{\text{adj}} = \frac{1 - \frac{SS_{\text{residual}}}{SS_{\text{total}}}}{1 - \frac{1}{n} \cdot \frac{k}{n-1}}, $$ where $ n = 123 $ and $ k = 22 $. After calculating the value, we find: $$ 100 R^2_{\text{adj}} = 57.30. $$ Thus, the value is $ 57.30 $.

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Correct Answer: 57

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