Find the least squares regression line y = ax + b for the points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \mathbf { K } , \left( x _ { n } , y _ { n } \right)$ by solving the system for a and b. $\begin{array} { r }
n b + \left( \sum _ { i = 1 } ^ { n } x _ { i } \right) a = \left( \sum _ { i = 1 } ^ { n } y _ { i } \right) \\
\left( \sum _ { i = 1 } ^ { n } x _ { i } \right) b + \left( \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } \right) a = \left( \sum _ { i = 1 } ^ { n } x _ { i } y _ { i } \right)
\end{array}$ Points: $( 1,13 ) , ( 3,15 ) , ( 6,18 ) , ( 6,19 )$ space
A)y = 11.81x + 1.11
B)y = -1.09x + 11.81
C)y = -4.30x - 7.24
D)y = 1.11x + 11.81
E)y = 4.01x - 7.24

Question

Find the least squares regression line y = ax + b for the points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \mathbf { K } , \left( x _ { n } , y _ { n } \right)$ by solving the system for a and b. $\begin{array} { r } 
n b + \left( \sum _ { i = 1 } ^ { n } x _ { i } \right) a = \left( \sum _ { i = 1 } ^ { n } y _ { i } \right) \\
\left( \sum _ { i = 1 } ^ { n } x _ { i } \right) b + \left( \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } \right) a = \left( \sum _ { i = 1 } ^ { n } x _ { i } y _ { i } \right)
\end{array}$ Points: $( 1,13 ) , ( 3,15 ) , ( 6,18 ) , ( 6,19 )$ space
A)y = 11.81x + 1.11
B)y = -1.09x + 11.81
C)y = -4.30x - 7.24
D)y = 1.11x + 11.81
E)y = 4.01x - 7.24

Quizplus · Accepted Answer

To find the least squares regression line $y = ax + b$ for the given points, we first calculate the necessary sums: $\sum x_i$, $\sum y_i$, $\sum x_i^2$, and $\sum x_iy_i$, and then solve the system of equations for $a$ and $b$.Given points: $(1, 13), (3, 15), (6, 18), (6, 19)$1. $\sum x_i = 1 + 3 + 6 + 6 = 16$2. $\sum y_i = 13 + 15 + 18 + 19 = 65$3. $\sum x_i^2 = 1^2 + 3^2 + 6^2 + 6^2 = 1 + 9 + 36 + 36 = 82$4. $\sum x_iy_i = (1)(13) + (3)(15) + (6)(18) + (6)(19) = 13 + 45 + 108 + 114 = 280$Now, we have $n = 4$, so we can substitute these values into the system of equations:1. $4b + 16a = 65$2. $16b + 82a = 280$Solving this system for $a$ and $b$, we get:From the first equation, $b = \frac{65 - 16a}{4}$.Substituting $b$ in the second equation gives us a linear equation in $a$, which can be solved to find the values of $a$ and $b$. After solving, we find $a = 1.11$ and $b = 11.81$.Therefore, the least squares regression line is $y = 1.11x + 11.81$.

Find the Least Squares Regression Line Y = Ax (x1,y1),(x2,y2),K,(xn,yn)\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \mathbf { K } , \left( x _ { n } , y _ { n } \right)(x1​,y1​),(x2​,y2​),K,(xn​,yn​)

Find the Least Squares Regression Line Y = Ax $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \mathbf { K } , \left( x _ { n } , y _ { n } \right)$