Question

# Forty observations were used to estimate $y=\beta_0+\beta_1 x_1+\beta_2 x_2+\varepsilon$. A portion of the regression results is shown in the accompanying table.\begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline & \text { Coefficients } & \text { Standard Error } & \text { tStat } & p \text {-value } \\ \hline \text { Intercept } & 13.83 & 2.42 & 5.71 & 1.56 \mathrm{E}-06 \\ \hline x_1 & -2.53 & 0.15 & -16.87 & 5.84 \mathrm{E}-19 \\ \hline x_2 & 0.29 & 0.06 & 4.83 & 2.38 \mathrm{E}-05 \\ \hline \end{array} \end{aligned}b. What is the sample regression equation?

Solution

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In this problem is given that $n=40$ observations were used to estimate the multiple linear regression model

$y = \beta_0 + \beta_1x_1 +\beta_2x_2 + \varepsilon\,\,\,(\star)$

where $y$ is the response variable, $x_1$ and $x_2$ are the explanatory variables, $\varepsilon$ is the random error term, and the coefficients $\beta_0$, $\beta_1$ and $\beta_2$ are the unknown parameters.

In general, the sample regression equation for the regression model $(\star)$ has the form:

$\hat{y}=b_0+b_1x_1+b_2x_2\,\,\, (\star\star)$

where coefficients $b_0$, $b_1$ and $b_2$ are numbers estimated from the data, $\hat{y}$ is the predicted value of the response variable $y$, and $x_1$ and $x_2$ are the explanatory variables.

For each explanatory variable $x_j$, $b_j$ represents the corresponding slope coefficient, where $j = 1, 2$.

Using the results in the given table, our task in this part of the problem is to write the sample regression equation.