What Is The Estimated Regression Equation Yhat 21873x3 0099x

What Is The Estimated Regression Equationyhat 21873x3 0099x2 0

What Is The Estimated Regression Equationyhat 21873x3 0099x2 0

The primary objective of this analysis is to determine and interpret the estimated regression equation based on the provided coefficients and variables. The regression equation models the relationship between a dependent variable, Y, and three independent variables, X1, X2, and X3, based on the available data. The coefficients in the regression equation quantify the individual contribution of each predictor variable to the dependent variable, holding all other variables constant. This analysis aims to interpret the regression coefficients, assess their significance, and evaluate the potential for variable elimination based on statistical criteria such as p-values and significance levels.

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The estimated regression equation provided appears to be a linear model designed to predict a response variable, Y, based on three predictor variables, X1, X2, and X3. The formal expression of the regression model is typically written as:

Ŷ = β0 + β1X1 + β2X2 + β3X3 + ε

where Ŷ is the predicted value of Y, β0 is the intercept, β1, β2, and β3 are the coefficients for the predictors, and ε is the error term. Based on the options provided, the most representative regression equation is:

Ŷ = -21.873 - 0.099X1 - 0.480X2 + 8.440X3

This model suggests that the predicted value of Y decreases by 0.099 units for each one-unit increase in X1, holding X2 and X3 constant; decreases by 0.480 units for each one-unit increase in X2, holding X1 and X3 constant; and increases by 8.440 units for each one-unit increase in X3, holding X1 and X2 constant. The intercept of -21.873 indicates the predicted value of Y when all predictor variables are zero.

Interpreting the coefficient of X1 involves understanding its effect on the response variable. A one-unit increase in X1 results in an average decrease of 0.099 units in Y, assuming the other variables remain unchanged. This negative relationship suggests that X1 has an inverse effect on Y within the model's context. It's important to recognize that the significance of this coefficient needs to be tested statistically using the t-test for β1. The test statistic for whether β1 is significant has been given as t = 1.17, with an associated F-value of 193, although the precise contexts of these statistics require clarification.

Testing the significance of β1 at a 1% significance level involves comparing the t-statistic to critical values from the t-distribution. Given the t-value of 1.17, the result indicates weak evidence against the null hypothesis (H0: β1=0); therefore, at the 1% level, β1 might not be statistically significant. Conversely, the statement about the given t-statistic of t = 6.6 or -6.6 suggests that in some contexts, the coefficients may be significant, but these are not directly linked to β1 in this context.

Similarly, β2's significance can be evaluated by its t-statistic (which is reported as 6.6 in some parts), indicating a strong effect on Y, and the null hypothesis (H0: β2=0) can be rejected at the 1% significance level. This implies that X2 is an important predictor in the model.

The mean sum of squares regression (MSR), reported as 22,878.5, measures the explained variability in the dependent variable by the model. The high MSR value indicates a substantial proportion of variability is being captured, but it must be contextualized with the mean sum of squares error (MSE) for complete interpretation.

In the context of model refinement, backward elimination involves removing predictors that are not statistically significant, based on their p-values. With an alpha level of 0.05, any variable with a p-value exceeding this threshold should be considered for removal. Given the t-statistic for X1 (1.17), which correlates with a p-value likely exceeding 0.05, we would consider removing X1 from the model in the first step, assuming the p-value confirms this. Conversely, since the t-statistic of 6.6 for X2 indicates significance, X2 would remain in the model.

Overall, the analysis underscores the importance of evaluating coefficient significance, understanding their directional relationships, and applying systematic methods like backward elimination to optimize the regression model. Proper interpretation of coefficients guides decision-making and enhances the model’s predictive accuracy and explanatory power.

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