Question

example 2.3.1: computing the sum of squared errors.
given the data points below, compute the sum of squared errors for the regression equation $y = 7 + 2x$.
\

$$\begin{tabular}{|c|c|c|c|c|} \\hline $x$ & 0 & 3 & 7 & 10\\\\ \\hline $y$ & 5 & 5 & 27 & 31\\\\ \\hline \\end{tabular}$$

solution

for each value of $x$, the regression values from the equation $y = 7 + 2x$ are obtained. the errors are the difference between the values of the data points and the regression values.
\

$$\begin{tabular}{|c|c|c|c|c|} \\hline $x$ & 0 & 3 & 7 & 10\\\\ \\hline $y$ & 5 & 5 & 27 & 31\\\\ \\hline $y = 7 + 2x$ & 7 & 13 & 21 & 27\\\\ \\hline errors & $-2$ & $-8$ & 6 & 4\\\\ \\hline squared errors & 4 & 64 & 36 & 16\\\\ \\hline \\end{tabular}$$

thus, the sum of the squared errors is $4 + 64 + 36 + 16 = 120$.

participation activity | 2.3.2: calculating sum of squared errors for a regression line.

given the data points below, compute the sum of squared errors for the regression equation $y = 2 + 3x$.
\

$$\begin{tabular}{|c|c|c|c|c|} \\hline $x$ & 0 & 3 & 7 & 10\\\\ \\hline $y$ & 5 & 5 & 27 & 31\\\\ \\hline \\end{tabular}$$

1. what are the squared errors for the regression line given by $y = 2 + 3x$? type as a comma - separated list: #, #, #, #

\boxed{}
check show answer

1. what is the sum of squared errors for the regression line give by $y = 2 + 3x$?

\boxed{}
check show answer

Explanation:

Step1: Calculate predicted $\hat{Y}$ values

For each $X$, compute $\hat{Y}=2+3X$:

• $X=0$: $\hat{Y}=2+3(0)=2$
• $X=3$: $\hat{Y}=2+3(3)=11$
• $X=7$: $\hat{Y}=2+3(7)=23$
• $X=10$: $\hat{Y}=2+3(10)=32$

Step2: Compute error values

Error $=Y-\hat{Y}$:

• $X=0$: $5-2=3$
• $X=3$: $5-11=-6$
• $X=7$: $27-23=4$
• $X=10$: $31-32=-1$

Step3: Square each error

Squared error $=(Y-\hat{Y})^2$:

• $3^2=9$
• $(-6)^2=36$
• $4^2=16$
• $(-1)^2=1$

Step4: Sum the squared errors

Sum $=9+36+16+1$

Answer:

1. 9, 36, 16, 1
2. 62