State the multiple regression equation. ]+ OX + (O×2 Round to two decimal places as needed.)

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### Table: Regression Analysis Results

This table presents the results of a regression analysis, showcasing the relationships between different variables. The table includes columns for the variable, coefficients, standard error, T statistic, and p-value.

- **Variable**
  - **Intercept**
    - Coefficient: 11.144293
    - Standard Error: 7.196818
    - T Statistic: 1.55
    - p-Value: 0.1248
  - **Revenue**
    - Coefficient: 0.6787574
    - Standard Error: 0.321135
    - T Statistic: 2.11
    - p-Value: 0.0372
  - **Efficiency**
    - Coefficient: 0.8446421
    - Standard Error: 0.077995
    - T Statistic: 10.83
    - p-Value: <0.0001

### Explanation

- **Coefficients**: Indicate the change in the dependent variable for a one-unit change in the independent variable, holding other variables constant.
- **Standard Error**: Measures the accuracy of the coefficients' estimates.
- **T Statistic**: Used to determine if a coefficient is significantly different from zero.
- **p-Value**: Assesses the significance of the results. A lower p-value (< 0.05) suggests strong evidence against the null hypothesis, indicating that the variable is a significant predictor. 

In this analysis:
- The Revenue variable's coefficient is 0.6787574 with a p-value of 0.0372, suggesting a statistically significant positive effect.
- The Efficiency variable's coefficient is 0.8446421 with a highly significant p-value of <0.0001. 

This table helps in understanding the impact each independent variable has on the outcome of the model.
Transcribed Image Text:### Table: Regression Analysis Results This table presents the results of a regression analysis, showcasing the relationships between different variables. The table includes columns for the variable, coefficients, standard error, T statistic, and p-value. - **Variable** - **Intercept** - Coefficient: 11.144293 - Standard Error: 7.196818 - T Statistic: 1.55 - p-Value: 0.1248 - **Revenue** - Coefficient: 0.6787574 - Standard Error: 0.321135 - T Statistic: 2.11 - p-Value: 0.0372 - **Efficiency** - Coefficient: 0.8446421 - Standard Error: 0.077995 - T Statistic: 10.83 - p-Value: <0.0001 ### Explanation - **Coefficients**: Indicate the change in the dependent variable for a one-unit change in the independent variable, holding other variables constant. - **Standard Error**: Measures the accuracy of the coefficients' estimates. - **T Statistic**: Used to determine if a coefficient is significantly different from zero. - **p-Value**: Assesses the significance of the results. A lower p-value (< 0.05) suggests strong evidence against the null hypothesis, indicating that the variable is a significant predictor. In this analysis: - The Revenue variable's coefficient is 0.6787574 with a p-value of 0.0372, suggesting a statistically significant positive effect. - The Efficiency variable's coefficient is 0.8446421 with a highly significant p-value of <0.0001. This table helps in understanding the impact each independent variable has on the outcome of the model.
A nonprofit analyst seeks to determine which variables should be used to predict nonprofit charitable commitment, a nonprofit organization's commitment to its charitable purpose. Two independent variables under consideration are Revenue, a measurement of total revenue, in billions of dollars, as a measure of nonprofit size \(X_1\), and Efficiency, a measurement of the percent of private donations remaining after fundraising expenses as a measure of nonprofit fundraising efficiency \(X_2\). The dependent variable \(Y\) is Commitment, a measurement of the percent of total expenses that are allocated directly to charitable services. Both variables \(X_2\) and \(Y\) are encoded as percents. Data are collected from a random sample of 98 nonprofit organizations, with the results provided in the accompanying table. Complete parts (a) through (c) below.

a. State the multiple regression equation.

\[
\hat{Y}_i = \boxed{} + \left( \boxed{} \right) X_{1i} + \left( \boxed{} \right) X_{2i}
\]

(Round to two decimal places as needed.)
Transcribed Image Text:A nonprofit analyst seeks to determine which variables should be used to predict nonprofit charitable commitment, a nonprofit organization's commitment to its charitable purpose. Two independent variables under consideration are Revenue, a measurement of total revenue, in billions of dollars, as a measure of nonprofit size \(X_1\), and Efficiency, a measurement of the percent of private donations remaining after fundraising expenses as a measure of nonprofit fundraising efficiency \(X_2\). The dependent variable \(Y\) is Commitment, a measurement of the percent of total expenses that are allocated directly to charitable services. Both variables \(X_2\) and \(Y\) are encoded as percents. Data are collected from a random sample of 98 nonprofit organizations, with the results provided in the accompanying table. Complete parts (a) through (c) below. a. State the multiple regression equation. \[ \hat{Y}_i = \boxed{} + \left( \boxed{} \right) X_{1i} + \left( \boxed{} \right) X_{2i} \] (Round to two decimal places as needed.)
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