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:

y=75.4324+0.0173x1+0.2795x20.1074x30.0556x4y = 75.4324 + 0.0173x_1 + 0.2795x_2 - 0.1074x_3 - 0.0556x_4

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:

  1. What does the R-Square value of 0.8872 imply about the model's performance?
  2. Why might certain predictors (e.g., disposable bags) have high p-values?
  3. How is the Adjusted R-Square different from the R-Square, and why is it important?
  4. What does the Significance F value tell us about the model overall?
  5. 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