Questions: Find the regression equation, letting the first variable be the predictor (x) variable. Using the listed lemon/crash data, where lemon imports are in metric tons and the fatality rates are per 100,000 people, find the best predicted crash fatality rate for a year in which there are 475 metric tons of lemon imports. Is the prediction worthwhile? Use a significance level of 0.05. Lemon Imports: 230, 267, 354, 476, 515 Crash Fatality Rate: 15.9, 15.7, 15.4, 15.3, 15 Find the equation of the regression line. ŷ = + (x) (Round the y-intercept to three decimal places as needed. Round the slope to four decimal places as needed.)

Find the regression equation, letting the first variable be the predictor (x) variable. Using the listed lemon/crash data, where lemon imports are in metric tons and the fatality rates are per 100,000 people, find the best predicted crash fatality rate for a year in which there are 475 metric tons of lemon imports. Is the prediction worthwhile? Use a significance level of 0.05.

Lemon Imports: 230, 267, 354, 476, 515
Crash Fatality Rate: 15.9, 15.7, 15.4, 15.3, 15

Find the equation of the regression line.

ŷ = + (x)

(Round the y-intercept to three decimal places as needed. Round the slope to four decimal places as needed.)

Transcript text: Find the regression equation, letting the first variable be the predictor ( $x$ ) variable. Using the listed lemon/crash data, where lemon imports are in metric tons and the fatality rates are per 100,000 people, find the best predicted crash fatality rate for a year in which there are 475 metric tons of lemon imports. Is the prediction worthwhile? Use a significance level of 0.05. \begin{tabular}{llllll} \hline Lemon Imports & 230 & 267 & 354 & 476 & 515 \\ Crash Fatality Rate & 15.9 & 15.7 & 15.4 & 15.3 & 15 \\ \hline \end{tabular} Find the equation of the regression line. \[ \hat{y}= \] $\square$ \[ +(\square) \mathrm{x} \] (Round the $y$-intercept to three decimal places as needed. Round the slope to four decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Means

The means of the lemon imports ($ \bar{x} $) and crash fatality rates ($ \bar{y} $) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 368.4 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 15.46 \]

Step 2: Calculate the Correlation Coefficient

The correlation coefficient ($ r $) is found to be:

\[ r = -0.9622 \]

Step 3: Calculate the Slope ($ \beta $)

The numerator for the slope ($ \beta $) is calculated as:

\[ \text{Numerator for } \beta = \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 28308.3 - 5 \cdot 368.4 \cdot 15.46 = -169.02 \]

The denominator for the slope is:

\[ \text{Denominator for } \beta = \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 741306 - 5 \cdot 368.4^2 = 62713.2 \]

Thus, the slope ($ \beta $) is:

\[ \beta = \frac{-169.02}{62713.2} = -0.0027 \]

Step 4: Calculate the Intercept ($ \alpha $)

The intercept ($ \alpha $) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 15.46 - (-0.0027) \cdot 368.4 = 16.4529 \]

Step 5: Write the Regression Equation

The equation of the regression line is:

\[ \hat{y} = 16.4529 + (-0.0027)x \]

Step 6: Predict the Crash Fatality Rate

To predict the crash fatality rate for 475 metric tons of lemon imports, we substitute $ x = 475 $ into the regression equation:

\[ \hat{y} = 16.4529 + (-0.0027) \cdot 475 = 15.1704 \]

Step 7: Evaluate the Prediction

The correlation coefficient ($ r = -0.9622 $) indicates a strong negative linear relationship between lemon imports and crash fatality rates. Since the absolute value of the correlation coefficient is greater than 0.05, the prediction is considered worthwhile.