Question

Carbon dioxide (COv2) 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 COv2 emissions for certain years per person in the United States and per person in the rest of the world.
year- nonus vs us
1990 3.6 / 19.2
1991 3.6/19.0
1992 3.5/18.8
1994 3.3/19.8
1995 3.3/19.5
1997 3.4/20.0
1998 3.3/19.5
1999 3.2/19.6
2004 3.7/19.6
2005 3.8/19.5

(a) Compute the least-squares regression line for predicting U.S. emissions from non-U.S. emissions.
Round the slope and y-intercept values to four decimal places.
Answer
No the answer is 21.5166-0.5956x

(b) If the non-U.S. emissions differ by 0.2 from one year to the next, by how much would you predict the U.S. emissions to differ? Round your answer to four decimal places.
Answer
(c) Predict the U.S. emissions for a year when the non-U.S. emissions level is 3.4.
Round the answer to two decimal places.
Answered by GPT-4o mini
To compute the least-squares regression line for predicting U.S. emissions from non-U.S. emissions, we need to calculate the slope \( b \) and the y-intercept \( a \) of the regression line given by the formula:

\[ y = a + bx \]

Where:
- \( y \) is the U.S. emissions
- \( x \) is the non-U.S. emissions

The formulas for the slope \( b \) and intercept \( a \) are:

1. **Slope \( b \)**:

\[
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

2. **Intercept \( a \)**:

\[
a = \frac{\sum y - b(\sum x)}{n}
\]

### Step-by-Step Calculation

First, let's compile our data:

| Year | Non-U.S. Emissions (x) | U.S. Emissions (y) |
|------|-------------------------|--------------------|
| 1990 | 3.6 | 19.2 |
| 1991 | 3.6 | 19.0 |
| 1992 | 3.5 | 18.8 |
| 1994 | 3.3 | 19.8 |
| 1995 | 3.3 | 19.5 |
| 1997 | 3.4 | 20.0 |
| 1998 | 3.3 | 19.5 |
| 1999 | 3.2 | 19.6 |
| 2004 | 3.7 | 19.6 |
| 2005 | 3.8 | 19.5 |

Now we calculate the necessary sums:

- \( n = 10 \) (number of data points)
- \( \sum x = 3.6 + 3.6 + 3.5 + 3.3 + 3.3 + 3.4 + 3.3 + 3.2 + 3.7 + 3.8 = 34.3 \)
- \( \sum y = 19.2 + 19.0 + 18.8 + 19.8 + 19.5 + 20.0 + 19.5 + 19.6 + 19.6 + 19.5 = 197.5 \)
- \( \sum xy = (3.6 \cdot 19.2) + (3.6 \cdot 19.0) + (3.5 \cdot 18.8) + (3.3 \cdot 19.8) + (3.3 \cdot 19.5) + (3.4 \cdot 20.0) + (3.3 \cdot 19.5) + (3.2 \cdot 19.6) + (3.7 \cdot 19.6) + (3.8 \cdot 19.5) \)
- \( = 69.12 + 68.64 + 65.8 + 65.34 + 64.35 + 68.0 + 64.35 + 62.72 + 72.52 + 74.1 = 691.22 \)
- \( \sum x^2 = (3.6^2) + (3.6^2) + (3.5^2) + (3.3^2) + (3.3^2) + (3.4^2) + (3.3^2) + (3.2^2) + (3.7^2) + (3.8^2) \)
- \( = 12.96 + 12.96 + 12.25 + 10.89 + 10.89 + 11.56 + 10.89 + 10.24 + 13.69 + 14.44 = 134.67 \)

Now, substituting these values into the slope formula:

\[
b = \frac{10(691.22) - (34.3)(197.5)}{10(134.67) - (34.3)^2}
\]

Calculating the components:

\[
b = \frac{6912.2 - 6785.75}{1346.7 - 1177.69} = \frac{126.45}{169.01} \approx 0.7472
\]

Now substituting \( b \) back to find \( a \):

\[
a = \frac{197.5 - 0.7472 \cdot 34.3 }{10}
\]

Calculating:

\[
a = \frac{197.5 - 25.66376}{10} = \frac{171.83624}{10} \approx 17.1836
\]

### Final Regression Equation

Thus, the least-squares regression line for predicting U.S. emissions from non-U.S. emissions is:

\[
\hat{y} = 17.1836 + 0.7472x
\]

**Rounded values**:
- Slope \( b \approx 0.7472 \)
- Intercept \( a \approx 17.1836 \)
Answered by GPT-4o mini
Based on the relation provided, where you have determined the regression line to be:

\[
\hat{y} = 21.5166 - 0.5956x
\]

In this case, to find out how much the U.S. emissions are predicted to differ when the non-U.S. emissions differ by 0.2, we can use the slope of the regression line.

The slope \( b = -0.5956 \) indicates that for every 1 unit increase in non-U.S. emissions, the U.S. emissions will decrease by 0.5956 units. Therefore, if the non-U.S. emissions increase by 0.2, the change in U.S. emissions can be calculated as follows:

### Calculation of Change in U.S. Emissions:

\[
\Delta y = b \times \Delta x
\]

Where:
- \(\Delta y\) is the change in U.S. emissions.
- \(b = -0.5956\) (the slope).
- \(\Delta x = 0.2\) (the change in non-U.S. emissions).

Substituting the values in:

\[
\Delta y = -0.5956 \times 0.2
\]
\[
\Delta y = -0.11912
\]

### Conclusion:

Thus, if the non-U.S. emissions differ by 0.2 from one year to the next, the predicted U.S. emissions would differ by approximately:

\[
\Delta y \approx -0.1191 \text{ (rounded to four decimal places)}
\]

### Final Answer:

The predicted U.S. emissions would differ by **-0.1191 metric tons** per person.
Answered by GPT-4o mini
To predict the U.S. emissions when the non-U.S. emissions level is \( x = 3.4 \), we can use the regression equation that you provided:

\[
\hat{y} = 21.5166 - 0.5956x
\]

Substituting \( x = 3.4 \) into the equation:

\[
\hat{y} = 21.5166 - 0.5956(3.4)
\]

Calculating the value:

1. Calculate \( 0.5956 \times 3.4 \):

\[
0.5956 \times 3.4 = 2.02004
\]

2. Now substitute this back into the equation:

\[
\hat{y} = 21.5166 - 2.02004
\]
\[
\hat{y} = 19.49656
\]

### Final Calculation:

Rounding \( 19.49656 \) to two decimal places:

\[
\hat{y} \approx 19.50
\]

### Conclusion:

The predicted U.S. emissions for a year when the non-U.S. emissions level is 3.4 is **19.50 metric tons** per person.