janelle tracks the number of miles she drives and the number of gallons of gas she has left. what is the…

Explanation:

Step1: Denote miles driven as (x) and gallons in tank as (y). We have data points ((x_1,y_1),(x_2,y_2),\cdots,(x_6,y_6)) where (x = [27,65,83,109,142,175]) and (y=[13,12,11,10,9,8]).

Step2: Calculate the means of (x) and (y).

The mean of (x), (\bar{x}=\frac{27 + 65+83+109+142+175}{6}=\frac{501}{6}=83.5) The mean of (y), (\bar{y}=\frac{13 + 12+11+10+9+8}{6}=\frac{63}{6}=10.5)

Step3: Calculate the numerator and denominator for the correlation - coefficient (r).

The numerator (S_{xy}=\sum_{i = 1}^{6}(x_i-\bar{x})(y_i - \bar{y})) ((27-83.5)(13 - 10.5)=(- 56.5)\times2.5=-141.25) ((65 - 83.5)(12-10.5)=(-18.5)\times1.5=-27.75) ((83 - 83.5)(11 - 10.5)=(-0.5)\times0.5=-0.25) ((109 - 83.5)(10 - 10.5)=(25.5)\times(-0.5)=-12.75) ((142 - 83.5)(9 - 10.5)=(58.5)\times(-1.5)=-87.75) ((175 - 83.5)(8 - 10.5)=(91.5)\times(-2.5)=-228.75) (S_{xy}=-141.25-27.75 - 0.25-12.75-87.75-228.75=-498.5)

The denominator: (S_{xx}=\sum_{i = 1}^{6}(x_i-\bar{x})^2) ((27-83.5)^2=(-56.5)^2 = 3192.25) ((65 - 83.5)^2=(-18.5)^2=342.25) ((83 - 83.5)^2=(-0.5)^2 = 0.25) ((109 - 83.5)^2=(25.5)^2=650.25) ((142 - 83.5)^2=(58.5)^2=3422.25) ((175 - 83.5)^2=(91.5)^2=8372.25) (S_{xx}=3192.25+342.25 + 0.25+650.25+3422.25+8372.25=15979.5)

(S_{yy}=\sum_{i = 1}^{6}(y_i-\bar{y})^2) ((13 - 10.5)^2=(2.5)^2 = 6.25) ((12 - 10.5)^2=(1.5)^2=2.25) ((11 - 10.5)^2=(0.5)^2 = 0.25) ((10 - 10.5)^2=(-0.5)^2=0.25) ((9 - 10.5)^2=(-1.5)^2=2.25) ((8 - 10.5)^2=(-2.5)^2=6.25) (S_{yy}=6.25+2.25 + 0.25+0.25+2.25+6.25=17.5)

(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{-498.5}{\sqrt{15979.5\times17.5}}=\frac{-498.5}{\sqrt{279641.25}}\approx\frac{-498.5}{528.81}\approx - 0.94) Since (|r|\approx0.94) which is close to (1), the strength of the model is strong.

For the linear - regression model (y=mx + b), (m=\frac{S_{xy}}{S_{xx}}=\frac{-498.5}{15979.5}\approx - 0.0312) (b=\bar{y}-m\bar{x}=10.5-(-0.0312)\times83.5=10.5 + 2.6152=13.1152) The linear - regression model is (y=-0.0312x + 13.1152)

Answer:

The correlation coefficient is approximately (-0.94). The strength of the model is strong. The linear - regression model is (y=-0.0312x + 13.1152)