find the regression equation, letting the first variable be the predictor (x) variable. using the listed…

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.\nlemon imports 225 262 354 485 546 \ncrash fatality rate 16 15.7 15.5 15.4 14.9 \n\nfind the equation of the regression line.\nŷ = □ + (□)x\n(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 400 metric tons of lemon imports. is the prediction worthwhile? use a significance level of 0.05.\nlemon imports 225 262 354 485 546 \ncrash fatality rate 16 15.7 15.5 15.4 14.9 \n\nfind the equation of the regression line.\nŷ = □ + (□)x\n(round the y-intercept to three decimal places as needed. round the slope to four decimal places as needed.)

Answer

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)