Question

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 400 metric tons of lemon imports. is the prediction worthwhile? use a significance level of 0.05.
lemon imports 225 262 354 485 546
crash fatality rate 16 15.7 15.5 15.4 14.9

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.)

Explanation:

Step1: Calculate necessary sums

First, we need to find the sums of \( x \) (lemon imports), \( y \) (crash fatality rate), \( xy \), and \( x^2 \).
Given \( x \): 225, 262, 354, 485, 546
\( y \): 16, 15.7, 15.5, 15.4, 14.9

\( \sum x = 225 + 262 + 354 + 485 + 546 = 1872 \)
\( \sum y = 16 + 15.7 + 15.5 + 15.4 + 14.9 = 77.5 \)
\( \sum xy = (225×16) + (262×15.7) + (354×15.5) + (485×15.4) + (546×14.9) \)
\( = 3600 + 4113.4 + 5487 + 7469 + 8135.4 = 28804.8 \)
\( \sum x^2 = 225^2 + 262^2 + 354^2 + 485^2 + 546^2 \)
\( = 50625 + 68644 + 125316 + 235225 + 298116 = 777926 \)

Step2: Calculate slope (\( b \)) and y-intercept (\( a \))

The formula for the slope \( b \) is:
\( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 5 \) (number of data points)

\( b = \frac{5×28804.8 - 1872×77.5}{5×777926 - 1872^2} \)
First, calculate numerator: \( 5×28804.8 = 144024 \), \( 1872×77.5 = 145080 \), so numerator \( = 144024 - 145080 = -1056 \)
Denominator: \( 5×777926 = 3889630 \), \( 1872^2 = 3504384 \), so denominator \( = 3889630 - 3504384 = 385246 \)
\( b = \frac{-1056}{385246} ≈ -0.002741 \) (rounded to four decimal places)

The formula for the y-intercept \( a \) is:
\( a = \bar{y} - b\bar{x} \)
where \( \bar{x} = \frac{\sum x}{n} = \frac{1872}{5} = 374.4 \)
\( \bar{y} = \frac{\sum y}{n} = \frac{77.5}{5} = 15.5 \)

\( a = 15.5 - (-0.002741)×374.4 \)
\( = 15.5 + 1.026 \) (approximate, since \( 0.002741×374.4 ≈ 1.026 \))
\( ≈ 16.526 \) (rounded to three decimal places)

Answer:

\( \hat{y} = 16.526 + (-0.0027)x \) (Note: The slope was calculated as approximately -0.002741, rounded to four decimal places is -0.0027, and y-intercept 16.526)