Example Coefficient of Determination For the case of our example the proportion

# Example coefficient of determination for the case of

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in y is explained. Example: Coefficient of Determination For the case of our example, the proportion of the difference in y (weight) that is explained by x (height) can be found as follows: R 2 = S xy 2 S xx ×S yy = r 2 For the height and weight data r= 0.956 so: R 2 = ( 0.956 ) 2 = 0.914 This means that 91.4% of the variation in weight is explained by the variations in their heights.
Multiple Linear Regression A linear regression model with more than one independent variable: ^ Y = a + b 1 X 1 + b 2 X 2 + + b n X n + e - Regression coefficient bi shows the change on the dependent variable Y when the associated independent variable Xi is increased by one measurement unit while the remaining independent variables are kept constant. - Example: An employee’s salary has found to depend on his/her age and years of employment according to the model: o Salary= 10,312.45 + 675.21Age + 1083.23Employment + e Regression Coefficients - Check independent variables significance - Test the null hypothesis with a two tailed test - Choose a level of confidence % (Type 1 error) typically 5%, - If p-value 0.05 then accept the null hypothesis and hence the independent variable is not significant and should be removed from the model. - If p-value 0.05 then reject the null hypothesis and hence the independent variable is significant.
Goodness of fit and Adjusted R 2 - Coefficient of determination is affected by the number of independent variables as well as sample size.
- Adding independent variables to the model will result to a better R 2 even if the new variable is insignificant.

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