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 500 metric tons of lemon imports. is the prediction worthwhile? use a significance level of 0.05.\nlemon imports 226 269 358 494 535\ncrash fatality rate 16 15.9 15.6 15.4 15.1\nfind the equation of the regression line.\n\\(\\hat{y}=\\square+(\\square)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 500 metric tons of lemon imports. is the prediction worthwhile? use a significance level of 0.05.\nlemon imports 226 269 358 494 535\ncrash fatality rate 16 15.9 15.6 15.4 15.1\nfind the equation of the regression line.\n\\(\\hat{y}=\\square+(\\square)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 the means of x and y

Let $x$ be lemon imports and $y$ be crash - fatality rate. $n = 5$ $\bar{x}=\frac{226 + 269+358+494+535}{5}=\frac{1882}{5}=376.4$ $\bar{y}=\frac{16 + 15.9+15.6+15.4+15.1}{5}=\frac{78}{5}=15.6$

Step2: Calculate the numerator and denominator for the slope $b_1$

$\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(226 - 376.4)(16 - 15.6)+(269 - 376.4)(15.9 - 15.6)+(358 - 376.4)(15.6 - 15.6)+(494 - 376.4)(15.4 - 15.6)+(535 - 376.4)(15.1 - 15.6)$ $=(- 150.4)\times0.4+(-107.4)\times0.3+( - 18.4)\times0+(117.6)\times(-0.2)+(158.6)\times(-0.5)$ $=-60.16-32.22 + 0-23.52-79.3$ $=-195.2$ $\sum_{i = 1}^{n}(x_i-\bar{x})^2=(226 - 376.4)^2+(269 - 376.4)^2+(358 - 376.4)^2+(494 - 376.4)^2+(535 - 376.4)^2$ $=(-150.4)^2+(-107.4)^2+(-18.4)^2+(117.6)^2+(158.6)^2$ $=22620.16 + 11534.76+338.56+13829.76+25153.96$ $=73477.2$ $b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}=\frac{-195.2}{73477.2}\approx - 0.0027$

Step3: Calculate the y - intercept $b_0$

$b_0=\bar{y}-b_1\bar{x}=15.6-(-0.0027)\times376.4$ $=15.6 + 1.01628\approx16.616$

Answer:

$\hat{y}=16.616+( - 0.0027)x$