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

Explanation:

Step1: Denote variables

Let $x$ be the miles - driven and $y$ be the gallons - in - tank. Calculate the necessary sums: Let $n = 6$. $\sum_{i = 1}^{n}x_{i}=27 + 65+83 + 109+142+175=591$ $\sum_{i = 1}^{n}y_{i}=13 + 12+11 + 10+9+8=63$ $\sum_{i = 1}^{n}x_{i}^{2}=27^{2}+65^{2}+83^{2}+109^{2}+142^{2}+175^{2}=27^{2}+4225+6889+11881+20164+30625=74663$ $\sum_{i = 1}^{n}y_{i}^{2}=13^{2}+12^{2}+11^{2}+10^{2}+9^{2}+8^{2}=169+144+121+100+81+64=679$ $\sum_{i = 1}^{n}x_{i}y_{i}=27\times13+65\times12+83\times11+109\times10+142\times9+175\times8=351+780+913+1090+1278+1400=5812$

Step2: Calculate the slope $m$

The formula for the slope $m$ of the regression line is $m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}$ $m=\frac{6\times5812 - 591\times63}{6\times74663-591^{2}}$ $=\frac{34872-37233}{447978 - 349281}=\frac{- 2361}{98697}\approx - 0.024$

Step3: Calculate the y - intercept $b$

The formula for the y - intercept $b$ is $b=\overline{y}-m\overline{x}$, where $\overline{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{591}{6}=98.5$ and $\overline{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{63}{6}=10.5$ $b = 10.5-(-0.024)\times98.5=10.5 + 2.364=12.864$ The linear regression model is $y=-0.024x + 12.864$

Step4: Calculate the correlation coefficient $r$

The formula for the correlation coefficient $r$ is $r=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{\sqrt{(n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2})(n\sum_{i = 1}^{n}y_{i}^{2}-(\sum_{i = 1}^{n}y_{i})^{2})}}$ $r=\frac{6\times5812-591\times63}{\sqrt{(6\times74663 - 591^{2})(6\times679-63^{2})}}$ $=\frac{-2361}{\sqrt{98697\times105}}\approx - 0.73$

Step5: Determine the strength of the model

Since $|r| = 0.73$, and $0.7\leq|r|\lt0.9$, the strength of the model is moderate.

Answer:

Linear regression model: $y=-0.024x + 12.864$ Correlation coefficient: $-0.73$ Strength of the model: Moderate