Compute MSR and MSE. The MSR, mean square due to regression is calculated as follows where SSR is the sum of squares due to regression and p is the number of independent variables in the estimated regression equation. MSR – SSR The estimated regression equation was given to be y = 28.1270 +0.5803×0.497%,. This equation has two independent variables, so The value of SSR was given to be SSR – 6,228.375. Use these to find the value of MSR, rounding the result to three decimal places. SSR MSR- 6,228.375
The Correct Answer and Explanation is:
To compute MSR (Mean Square due to Regression), we use the formula: MSR=SSRp\text{MSR} = \frac{\text{SSR}}{p}
Where:
- SSR is the Sum of Squares due to Regression.
- p is the number of independent variables.
Given:
- SSR = 6,228.375
- The estimated regression equation is: y=28.1270+0.5803×1+0.497x2y = 28.1270 + 0.5803x_1 + 0.497x_2 So, there are two independent variables: x1x_1 and x2x_2, hence p=2p = 2.
Step-by-step Calculation:
MSR=6,228.3752=3,114.1875\text{MSR} = \frac{6,228.375}{2} = 3,114.1875
Final Answer (rounded to three decimal places):
MSR=3,114.188\boxed{\text{MSR} = 3,114.188}
Explanation
In regression analysis, the Mean Square due to Regression (MSR) is a key measure used to evaluate how well the regression model explains the variability in the dependent variable (y). It represents the average variability in the response variable that can be attributed to the model’s independent variables.
The MSR is calculated by dividing the Sum of Squares due to Regression (SSR) by the number of independent variables (p) in the model. The SSR quantifies the amount of variation explained by the regression model. It is a part of the total variation in the data (Total Sum of Squares, SST), and its complement is the Sum of Squares due to Error (SSE), which measures the variation not explained by the model.
In this case, the regression model is: y=28.1270+0.5803×1+0.497x2y = 28.1270 + 0.5803x_1 + 0.497x_2
This equation shows there are two independent variables, x1x_1 and x2x_2. With an SSR value of 6,228.375, we apply the formula for MSR: MSR=SSRp=6,228.3752=3,114.1875\text{MSR} = \frac{\text{SSR}}{p} = \frac{6,228.375}{2} = 3,114.1875
Rounding this to three decimal places, we get 3,114.188.
MSR plays a crucial role in the F-test for overall model significance in regression analysis. It is compared with the Mean Square Error (MSE) to compute the F-statistic. A higher MSR relative to MSE indicates a better-fitting model, suggesting that the independent variables collectively contribute significantly to predicting the outcome.
