91Ó°ÊÓ

Consider the following computer output. $$ \begin{array}{l} \text { The regression equation is } \mathrm{Y}=254+2.77 \times 1-3.58 \times 2 \\ \text { Predictor } \quad \text { Coef } \quad \text { SE Coef } \quad \mathrm{T} \quad \mathrm{P} \\ \text { Constant } \quad 253.810 & 4.781 & ? & ? \\ \text { x1 } & 2.7738 & 0.1846 & 15.02 & ? \\ \text { x2 } & -3.5753 & 0.1526 & ? & ? \\ \mathrm{~S}=5.05756 & \mathrm{R}-\mathrm{Sq}=? & \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=98.4 \% & \\ \text { Analysis of Variance } & & & & \\ \text { Source } & \text { DF } & \mathrm{SS} & \mathrm{MS} & \mathrm{F} \quad \mathrm{P} \\ \text { Regression } & 2 & 22784 & 11392 & ? \quad ? \\ \text { Residual error } & ? & ? & ? & \\ \text { Total } & 14 & 23091 & & \end{array} $$ (a) Fill in the missing quantities. You may use bounds for the \(P\) -values (b) What conclusions can you draw about the significance of regression? (c) What conclusions can you draw about the contributions of the individual regressors to the model?

Step by step solution

01

Calculate Missing Values in Analysis of Variance

To find the missing quantities in the Analysis of Variance table, start by noting that the Total Sum of Squares (SST) is the sum of the Regression Sum of Squares (SSR) and the Residual Sum of Squares (SSE). Given SST = 23091 and SSR = 22784, we use the formula: \[ \text{SST} = \text{SSR} + \text{SSE} \]Thus, \( \text{SSE} = 23091 - 22784 = 307 \).The degrees of freedom (DF) for residual error is the total DF minus the regression DF. Given total DF = 14 and regression DF = 2, the residual DF = 14 - 2 = 12.Thus, Mean Square for Residuals (\( MS_{Residual} \)) = SSE/DF = 307/12 = 25.58.Finally, the F-statistic is given by:\[ F = \frac{MSR}{MSE} = \frac{11392}{25.58} \approx 445.24 \].

02

Calculate T-Statistic for Constant and X2

The T-statistic is calculated as the estimated coefficient divided by its standard error.For the constant: \[ T = \frac{253.81}{4.781} \approx 53.07 \]For \( x2 \): \[ T = \frac{-3.5753}{0.1526} \approx -23.43 \].

03

Calculate R-Squared

R-Squared (or Coefficient of Determination) is calculated as the proportion of the variance in the dependent variable that is predictable from the independent variables. Since the R-Sq(adj) is given as 98.4%, this implies a very high R-Squared, typically calculated as:\[ R^2 = 1 - \frac{SSE}{SST} \approx 1 - \frac{307}{23091} \approx 0.9867 \text{ or } 98.67\% \].

04

Interpret P-Values and Significance

The P-values indicate the probability of observing the results given that the null hypothesis is true. A common significance level is 0.05. The large T-statistic values for the coefficients imply extremely low P-values (i.e., much lower than 0.05), indicating statistically significant results for both the intercept and the regressors.

05

Draw Conclusions Based on Statistics

(a) Filling missing quantities: For the regression table: SSR = 22784, SSE = 307, MSR = 11392, DF for residual = 12, MSE = 25.58, F = 445.24, R-Sq = 98.67%; For P-values, all can be considered highly significant given T-values. (b) The regression model is significant overall, as indicated by the high F-statistic and associated low P-value. (c) Both individual regressors significantly contribute to the model, as indicated by their T-statistics and P-values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Analysis of Variance

Analysis of Variance (ANOVA) is a statistical technique used to determine the significance of one or more factors by comparing means of different samples. In the context of regression, ANOVA helps us analyze whether the independent variables in the model are statistically significant predictors for the dependent variable.
In the exercise, the ANOVA table breaks down the variance into components:

• **Regression**: This variance component captures the explained variability due to the model's independent variables.
• **Residual**: Represents unexplained variability by the model, essentially a measure of error or noise.
• **Total**: The total variability in the dependent variable. It is the sum of regression and residual sums of squares.

The F-statistic is calculated from this table and offers insights into the model's overall significance.

R-Squared

R-Squared, also known as the Coefficient of Determination, quantifies the proportion of the variance in the dependent variable that can be explained by the independent variables in a regression model. This value ranges between 0 and 1, where a higher value indicates a better fit of the model.
In the given exercise, the R-Squared value informs us about the efficacy of the two predictors, \( x_1 \) and \( x_2 \), in explaining the variability of \( Y \). An R-Squared of approximately 98.67% indicates that 98.67% of the variability in \( Y \) is accounted for by the model. This suggests an excellent fit, implying that almost all variability in the dependent variable is explained by the regression model.

T-Statistic

The T-Statistic is used in regression analysis to determine if a particular coefficient is significantly different from zero, which would imply that the associated predictor contributes to the model. In the exercise, the T-Statistic is calculated by dividing each coefficient by its standard error:

• For the constant term, a high T-Statistic of approximately 53.07 reflects a profound impact, usually indicating significance.
• For \( x_1 \), with a T-Statistic of 15.02, it shows that this predictor significantly influences the dependent variable.
• For \( x_2 \), even a negative-coefficient translates to a significant T-Statistic of about -23.43, suggesting its strong but negative contribution.

The bigger the absolute value of the T-Statistic, the more significant the predictor is, contributing substantially to the regression model's accuracy.

F-Statistic

The F-Statistic in regression analysis is a robust measure that tests the overall significance of the model. It specifically examines if at least one of the predictors has a non-zero coefficient, indicating it's relevant to the model.For the exercise, the F-Statistic was calculated to be approximately 445.24. This model's F-Statistic tells us whether the model is better at predicting \( Y \) compared to an empty model, which simply uses the mean of \( Y \) as the prediction. A large F-Statistic, as seen in this instance, implies a significant model, suggesting a strong linear relationship between the combination of independent variables and the dependent variable. This points towards the model having predictive utility, thanks to the contributions of multiple predictors.