least squares method
the method of least squares derives a linear regression model by minimizing the sum of squared errors. for a sample with n observations,
the sum of squared errors is $\sum_{i=1}^{n} (y_i - \beta_0 - \beta_1 x_i)^2$.
the sample linear regression function is $\hat{y} = b_0 + b_1 x$, where $\hat{y}$ are the predicted or fitted response values based on the linear
regression model, and the regression parameter estimators, $b_0$ and $b_1$ are the values of the regression parameters, $\beta_0$ and $\beta_1$ that
minimize the sum of squared errors.
the hat notation in $\hat{y}$ is a statistical convention that denotes a sample estimate. $\hat{y} = b_0 + b_1 x$ is the sample linear regression line that
iestimates the population linear regression line $e(y) = \beta_0 + \beta_1 x$.
a linear regression fitted value, $\hat{y}_i = b_0 + b_1 x_i$, is the predicted value of $y$ for the $i$th sample value of $x$ based on the sample linear
regression line.
a linear regression residual, $\epsilon_i = y_i - \hat{y}_i$, is the $i$th estimated regression error based on the sample linear regression line.
for the data above, $b_0 = 2$ and $b_1 = 3$ minimize the sum of squared errors. thus, the sample linear regression line is $\hat{y} = 2 + 3x$. the
fitted value when $x_1 = 0$ is $\hat{y}_1 = 2 + 3(0) = 2$. the corresponding regression residual is $\epsilon_1 = y_1 - \hat{y}_1 = 5 - 2 = 3$
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2.3.3: calculating fitted values and residuals for a sample linear regression line.
use the sample linear regression line $\hat{y} = 2 + 3x$ for the data points below to answer the following questions.
$\begin{array}{|c|c|c|c|c|} \hline x & 0 & 3 & 7 & 10 \\ \hline y & 5 & 5 & 27 & 31 \\ \hline \end{array}$
1) what is the fitted value when $x_2 = 3$?
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2) what is the regression residual when $x_2 = 3$?
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Question

Accepted Answer

# Explanation:
## Step1: Calculate fitted value for $X_2=3$
Substitute $X=3$ into $\hat{Y}=2+3X$:
$\hat{Y}_2 = 2 + 3(3) = 2 + 9 = 11$
## Step2: Calculate residual for $X_2=3$
Use $\varepsilon_i = Y_i - \hat{Y}_i$, substitute $Y_2=5$, $\hat{Y}_2=11$:
$\varepsilon_2 = 5 - 11 = -6$

# Answer:
1) 11
2) -6

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate fitted value for $X_2=3$

Step2: Calculate residual for $X_2=3$

Answer: