35 an insurance agent is looking at records to determine if there is a relationship between a drivers age…

Explanation:

Step1: Calculate necessary sums

Let (n = 12) (number of data - points). Calculate (\sum x), (\sum y), (\sum x^{2}), (\sum y^{2}), (\sum xy). (\sum x=17 + 18+21+25+30+35+40+45+50+55+60+65 = 461) (\sum y=49 + 49+48+38+31+33+24+25+16+10+5+6 = 344) (\sum x^{2}=17^{2}+18^{2}+21^{2}+25^{2}+30^{2}+35^{2}+40^{2}+45^{2}+50^{2}+55^{2}+60^{2}+65^{2}) (=289 + 324+441+625+900+1225+1600+2025+2500+3025+3600+4225 = 20809) (\sum y^{2}=49^{2}+49^{2}+48^{2}+38^{2}+31^{2}+33^{2}+24^{2}+25^{2}+16^{2}+10^{2}+5^{2}+6^{2}) (=2401+2401+2304+1444+961+1089+576+625+256+100+25+36 = 12118) (\sum xy=(17\times49)+(18\times49)+(21\times48)+(25\times38)+(30\times31)+(35\times33)+(40\times24)+(45\times25)+(50\times16)+(55\times10)+(60\times5)+(65\times6)) (=833+882+1008+950+930+1155+960+1125+800+550+300+390 = 9983)

Step2: Calculate the slope (m)

The formula for the slope (m) of the regression - line is (m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}) (m=\frac{12\times9983 - 461\times344}{12\times20809-461^{2}}) (=\frac{119796-158584}{249708 - 212521}=\frac{-38788}{37187}\approx - 1.04)

Step3: Calculate the y - intercept (b)

The formula for the y - intercept (b) is (b=\overline{y}-m\overline{x}), where (\overline{x}=\frac{\sum x}{n}=\frac{461}{12}\approx38.42) and (\overline{y}=\frac{\sum y}{n}=\frac{344}{12}\approx28.67) (b = 28.67-(-1.04)\times38.42) (=28.67 + 1.04\times38.42=28.67+39.96 = 68.63)

The linear regression equation is (y=-1.04x + 68.63)

Step4: Calculate the correlation coefficient (r)

The formula for the correlation coefficient (r) is (r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}}) (r=\frac{12\times9983 - 461\times344}{\sqrt{(12\times20809 - 461^{2})(12\times12118-344^{2})}}) (=\frac{-38788}{\sqrt{37187\times12118\times12 - 344^{2}\times37187}}) (=\frac{-38788}{\sqrt{37187\times(145416 - 118336)}}) (=\frac{-38788}{\sqrt{37187\times27080}}) (=\frac{-38788}{\sqrt{1007033960}}) (=\frac{-38788}{31734.05}\approx - 0.97)

The correlation coefficient (r\approx - 0.97). A correlation coefficient close to (-1) indicates a strong negative linear relationship. In the context of this problem, it means that as the age of the driver increases, the percentage of accidents caused by speeding decreases significantly.

Answer:

The linear regression equation is (y=-1.04x + 68.63). The correlation coefficient is (r\approx - 0.97), which indicates a strong negative linear relationship between the driver's age and the percentage of accidents caused by speeding.