Question

6a )
what is the equation of the line of best fit for the following data? round the slope and y - intercept of the line to three decimal places.

a. y = - 1.164x + 0.885
b. y = 0.885x + 1.164
c. y = - 0.885x + 1.164
d. y = 1.164x + 0.885

6b )
what is the equation of the line of best fit for the following data? round the slope and y - intercept of the line to three decimal places.

a. y = 1.383x + 1.526
b. y = - 1.526x + 1.383
c. y = 1.383x - 1.526
d. y = - 1.526x - 1.383

Explanation:

6a)

Step1: Calcular la media de \(x\) y \(y\)

\(\bar{x}=\frac{2 + 5+7+9+11}{5}=\frac{34}{5} = 6.8\)
\(\bar{y}=\frac{2+8 + 10+11+13}{5}=\frac{44}{5}=8.8\)

Step2: Calcular la pendiente \(m\)

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

\((x_{1}-\bar{x})(y_{1}-\bar{y})=(2 - 6.8)(2 - 8.8)=(- 4.8)\times(-6.8)=32.64\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(5 - 6.8)(8 - 8.8)=(-1.8)\times(-0.8)=1.44\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(7 - 6.8)(10 - 8.8)=(0.2)\times(1.2)=0.24\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(9 - 6.8)(11 - 8.8)=(2.2)\times(2.2)=4.84\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(11 - 6.8)(13 - 8.8)=(4.2)\times(4.2)=17.64\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=32.64 + 1.44+0.24+4.84+17.64=56.8\)

\((x_{1}-\bar{x})^{2}=(2 - 6.8)^{2}=(-4.8)^{2}=23.04\)
\((x_{2}-\bar{x})^{2}=(5 - 6.8)^{2}=(-1.8)^{2}=3.24\)
\((x_{3}-\bar{x})^{2}=(7 - 6.8)^{2}=(0.2)^{2}=0.04\)
\((x_{4}-\bar{x})^{2}=(9 - 6.8)^{2}=(2.2)^{2}=4.84\)
\((x_{5}-\bar{x})^{2}=(11 - 6.8)^{2}=(4.2)^{2}=17.64\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=23.04+3.24 + 0.04+4.84+17.64=48.8\)

\(m=\frac{56.8}{48.8}\approx1.164\)

Step3: Calcular la intersección \(b\) con el eje \(y\)

\(b=\bar{y}-m\bar{x}\)
\(b = 8.8-1.164\times6.8\)
\(b=8.8 - 7.9152=0.885\)

Step1: Calcular la media de \(x\) y \(y\)

\(\bar{x}=\frac{4 + 7+10+12+13}{5}=\frac{46}{5}=9.2\)
\(\bar{y}=\frac{5+7 + 12+14+18}{5}=\frac{56}{5}=11.2\)

Step2: Calcular la pendiente \(m\)

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

\((x_{1}-\bar{x})(y_{1}-\bar{y})=(4 - 9.2)(5 - 11.2)=(-5.2)\times(-6.2)=32.24\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(7 - 9.2)(7 - 11.2)=(-2.2)\times(-4.2)=9.24\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(10 - 9.2)(12 - 11.2)=(0.8)\times(0.8)=0.64\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(12 - 9.2)(14 - 11.2)=(2.8)\times(2.8)=7.84\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(13 - 9.2)(18 - 11.2)=(3.8)\times(6.8)=25.84\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=32.24+9.24 + 0.64+7.84+25.84=75.8\)

\((x_{1}-\bar{x})^{2}=(4 - 9.2)^{2}=(-5.2)^{2}=27.04\)
\((x_{2}-\bar{x})^{2}=(7 - 9.2)^{2}=(-2.2)^{2}=4.84\)
\((x_{3}-\bar{x})^{2}=(10 - 9.2)^{2}=(0.8)^{2}=0.64\)
\((x_{4}-\bar{x})^{2}=(12 - 9.2)^{2}=(2.8)^{2}=7.84\)
\((x_{5}-\bar{x})^{2}=(13 - 9.2)^{2}=(3.8)^{2}=14.44\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=27.04+4.84+0.64+7.84+14.44=54.8\)

\(m=\frac{75.8}{54.8}\approx1.383\)

Step3: Calcular la intersección \(b\) con el eje \(y\)

\(b=\bar{y}-m\bar{x}\)
\(b = 11.2-1.383\times9.2\)
\(b=11.2-12.7236=-1.526\)

Answer:

D. \(y = 1.164x+0.885\)

6b)