Math Problem Statement

Carbon foot print: Carbon dioxide CO2 is produced by burning fossil fuels such as oil and natural gas, and has been connected to global warming. The following table presents the average amounts (in metric tons) of CO2 emissions for certain years per person in the United States and per person in the rest of the world.

Year Non-U.S. U.S. 1991 3.6 19 1992 3.5 18.8 1995 3.3 19.5 1997 3.4 20 1998 3.3 19.5 2000 3.3 20 2001 3.3 19.6

Compute the least-squares regression line for predicting U.S. emissions from non-U.S. emissions. Round the slope and y-intercept to at least four decimal places. Regression line equation:

Solution

To compute the least-squares regression line for predicting U.S. emissions (yy) from non-U.S. emissions (xx), we will use the formula for the regression line:

y=mx+by = mx + b

Where:

  • mm is the slope of the line, given by:

m=n∑xy−(∑x)(∑y)n∑x2−(∑x)2m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}

  • bb is the y-intercept, given by:

b=∑y−m∑xnb = \frac{\sum y - m\sum x}{n}

Here:

  • xx represents non-U.S. emissions,
  • yy represents U.S. emissions,
  • nn is the number of data points.

Let's go step by step:

Data:

\hline \text{Year} & \text{Non-U.S. emissions (x)} & \text{U.S. emissions (y)} \\ \hline 1991 & 3.6 & 19 \\ 1992 & 3.5 & 18.8 \\ 1995 & 3.3 & 19.5 \\ 1997 & 3.4 & 20 \\ 1998 & 3.3 & 19.5 \\ 2000 & 3.3 & 20 \\ 2001 & 3.3 & 19.6 \\ \hline \end{array}$$ ### Step 1: Calculate the necessary sums - $$\sum x$$ (sum of non-U.S. emissions), - $$\sum y$$ (sum of U.S. emissions), - $$\sum x^2$$ (sum of squares of non-U.S. emissions), - $$\sum xy$$ (sum of the product of non-U.S. and U.S. emissions). Let's calculate these sums: $$\sum x = 3.6 + 3.5 + 3.3 + 3.4 + 3.3 + 3.3 + 3.3 = 23.8$$ $$\sum y = 19 + 18.8 + 19.5 + 20 + 19.5 + 20 + 19.6 = 136.4$$ $$\sum x^2 = 3.6^2 + 3.5^2 + 3.3^2 + 3.4^2 + 3.3^2 + 3.3^2 + 3.3^2 = 12.96 + 12.25 + 10.89 + 11.56 + 10.89 + 10.89 + 10.89 = 79.53$$ $$\sum xy = (3.6 \times 19) + (3.5 \times 18.8) + (3.3 \times 19.5) + (3.4 \times 20) + (3.3 \times 19.5) + (3.3 \times 20) + (3.3 \times 19.6) = 68.4 + 65.8 + 64.35 + 68 + 64.35 + 66 + 64.68 = 461.58$$ ### Step 2: Calculate the slope $$m$$ and the y-intercept $$b$$ Now, we can plug the sums into the formulas: - $$n = 7$$ (number of data points), - $$\sum x = 23.8$$, - $$\sum y = 136.4$$, - $$\sum x^2 = 79.53$$, - $$\sum xy = 461.58$$. #### Slope $$m$$: $$m = \frac{7 \times 461.58 - (23.8 \times 136.4)}{7 \times 79.53 - (23.8)^2}$$ $$m = \frac{3231.06 - 3247.52}{556.71 - 566.44} = \frac{-16.46}{-9.73} \approx 1.6925$$ #### Y-intercept $$b$$: $$b = \frac{136.4 - 1.6925 \times 23.8}{7} = \frac{136.4 - 40.35}{7} = \frac{96.05}{7} \approx 13.7214$$ ### Final Regression Line: The regression line equation is: $$y = 1.6925x + 13.7214$$ --- Let me know if you'd like to go over any of the steps in more detail! Here are some related questions you might find interesting: 1. How would the regression equation change if more years of data were included? 2. How can the correlation coefficient be calculated to assess the fit of this regression line? 3. What does the slope of this regression line signify in terms of emissions? 4. How might this regression model change if it included data from other countries? 5. How does multivariable regression differ when multiple predictors are used to predict CO2 emissions? **Tip:** Always double-check your sums and intermediate calculations when working with regression to ensure accuracy!

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Least-Squares Method
Statistics

Formulas

y = mx + b
m = (nΣxy - (Σx)(Σy)) / (nΣx^2 - (Σx)^2)
b = (Σy - mΣx) / n

Theorems

Least Squares Regression Theorem

Suitable Grade Level

Grades 11-12

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