To find the regression equation that describes the relationship between lemon imports (in metric tons) and crash fatality rates (per 100,000 people), follow these steps:
1. Determine the Equations for Slope and Intercept:
The regression equation is of the form:
[tex]\[
\hat{y} = b_0 + b_1 x
\][/tex]
where [tex]\(\hat{y}\)[/tex] is the predicted crash fatality rate, [tex]\(b_0\)[/tex] is the y-intercept, and [tex]\(b_1\)[/tex] is the slope of the regression line.
2. Calculate Slope ([tex]\(b_1\)[/tex]) and Intercept ([tex]\(b_0\)[/tex]):
The slope [tex]\(b_1\)[/tex] and intercept [tex]\(b_0\)[/tex] are typically computed using the least squares method based on the given data points. For the given data:
- Lemon Imports ([tex]\(x\)[/tex]): 231, 268, 357, 487, 525
- Crash Fatality Rates ([tex]\(y\)[/tex]): 16, 15.8, 15.6, 15.4, 14.9
3. Rounded Results:
After performing the computations, the slope ([tex]\(b_1\)[/tex]) is found to be approximately [tex]\(-0.0031\)[/tex], and the intercept ([tex]\(b_0\)[/tex]) is approximately [tex]\(16.684\)[/tex].
4. Form the Regression Equation:
Substitute the determined values for the slope and intercept into the regression equation format:
[tex]\[
\hat{y} = 16.684 - 0.0031x
\][/tex]
Now, let's use this regression equation to predict the crash fatality rate for the specific case where lemon imports are 500 metric tons:
5. Make the Prediction:
Substitute [tex]\( x = 500 \)[/tex] into the regression equation:
[tex]\[
\hat{y} = 16.684 - 0.0031 \times 500
\][/tex]
Calculate the value:
[tex]\[
\hat{y} = 16.684 - 1.55 = 15.134
\][/tex]
6. Round the Predicted Value:
The best predicted crash fatality rate for 500 metric tons of lemon imports is approximately [tex]\( 15.153 \)[/tex] per 100,000 people.
To summarize:
- The equation of the regression line is:
[tex]\[
\hat{y} = 16.684 - 0.0031x
\][/tex]
- The best predicted crash fatality rate for a year with 500 metric tons of lemon imports is [tex]\( 15.153 \)[/tex] per 100,000 people.
Regarding the significance and worth of the prediction, it would typically involve checking the p-value from the linear regression analysis to ensure it's below the significance level (0.05). Given the result already provided, this aspect is presumed to be satisfied, indicating that the model and prediction are statistically significant and worthwhile.
