Math Problem Statement
start have gotten no simpler. Congress has decided to grant small businesses federal funding based on how well a sample of 25 households do in keeping CO2 levels down. To predict these CO2 levels (y), we investigate four variables over the course of a year:
(i) the average number of hours/month appliances are left in standby mode instead of shut off (x1); (ii) the number of times residents use disposable shopping bags instead of reusable ones (x2); (iii) the number of times residents carpooled to work for the year (x3); and, (iv) for the year, total pounds recycled instead of thrown in regular trash (x4).
The output below is from a regression model whose aim is to predict CO2 levels:
Regression Statistics
Multiple R 0.94192204
R Square 0.88721714
Adjusted R Square 0.86466057
Standard Error 10.8899991
Observations 25
ANOVA
df SS MS F Significance F Regression 4 18658.31838 4664.58 39.33298 3.28733E-09 Residual 20 2371.841623 118.5921
Total 24 21030.16
Coefficients Standard Error t Stat P-value Lower 95% Intercept 75.4324495 9.048340478 8.336606 6.13E-08 56.55794204 Hrs in standby 0.01732133 0.0114359 1.514646 0.145507 -0.006533534 Disposable bags 0.27953303 0.219842622 1.271514 0.218134 -0.179050647 Times Carpooled -0.1074275 0.030876793 -3.47923 0.002366 -0.171835341 Pounds Recycled -0.0555547 0.021787858 -2.5498 0.019085 -0.10100335 The multiple regression equation is: Question 1 options:
y L
9 . 05 + 0 . 01 x 1 + 0 . 22 x 2
0 . 03 x 3
0 . 02 x 4 + 9 . 05
y L
0 . 89 x 1 + 0 . 86 x 2 + 10 . 89 x 3
25 x 4 + 0 . 94
y L
0 . 02 x 1 + 0 . 28 x 2
0 . 11 x 3
0 . 06 x 4 + 75 . 43
y L
1 . 51 x 1 + 1 . 27 x 2
3 . 48 x 3
2 . 55 x 4 + 8 . 34
correct regression equation not listed Previous Page Next Page
Solution
The correct multiple regression equation should be determined based on the coefficients provided in the output. From the Coefficients table:
- Intercept: 75.4324
- Hrs in standby (x₁): 0.0173
- Disposable bags (x₂): 0.2795
- Times Carpooled (x₃): -0.1074
- Pounds Recycled (x₄): -0.0556
Using these coefficients, the regression equation is:
From the provided options, this equation corresponds to:
y = 0.02x₁ + 0.28x₂ - 0.11x₃ - 0.06x₄ + 75.43
Answer:
y = 0.02x₁ + 0.28x₂ - 0.11x₃ - 0.06x₄ + 75.43
Would you like a detailed explanation of how this regression equation is derived?
Here are some follow-up questions for better understanding:
- What does the R-Square value of 0.8872 imply about the model's performance?
- Why might certain predictors (e.g., disposable bags) have high p-values?
- How is the Adjusted R-Square different from the R-Square, and why is it important?
- What does the Significance F value tell us about the model overall?
- Why are some variables, such as carpooling, negatively correlated with CO2 levels?
Tip: When interpreting regression models, always check both the significance of individual predictors and the overall model significance.
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Math Problem Analysis
Mathematical Concepts
Multiple Regression
Statistical Analysis
Linear Models
Formulas
y = Intercept + (Coefficient_x1 * x1) + (Coefficient_x2 * x2) + (Coefficient_x3 * x3) + (Coefficient_x4 * x4)
Theorems
Least Squares Estimation
Hypothesis Testing in Regression
Suitable Grade Level
Grades 11-12 (Advanced High School), College Level
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