For the data set shown below, complete parts (a) through (d) below.
\[
\begin{array}{r|rrrrr}
\mathbf{x} & 3 & 4 & 5 & 7 & 8 \\
\hline \mathbf{y} & 4 & 5 & 8 & 13 & 14
\end{array}
\]
(a) Find the estimates of $\beta_{0}$ and $\beta_{1}$.
$\beta_{0} \approx b_{0}=\square$ (Round to three decimal places as needed.)
$\beta_{1} \approx b_{1}=\square$ (Round to three decimal places as needed.)
Answer
The estimates of \(\beta_{0}\) and \(\beta_{1}\) are approximately -2.942 and 2.174, respectively. So, \(\beta_{0} \approx b_{0}=\boxed{-2.942}\) and \(\beta_{1} \approx b_{1}=\boxed{2.174}\).
Step 1 ：Define the data set with x values as [3, 4, 5, 7, 8] and y values as [4, 5, 8, 13, 14].
Step 2 ：Calculate the mean of x values (x_bar) and y values (y_bar). The mean of x values is 5.4 and the mean of y values is 8.8.
Step 3 ：Calculate the estimate of \(\beta_{1}\) using the formula \(\beta_{1} = \frac{\sum{(x - x_{bar}) * (y - y_{bar})}}{\sum{(x - x_{bar})^2}}\). The calculated value of \(\beta_{1}\) is approximately 2.174.
Step 4 ：Calculate the estimate of \(\beta_{0}\) using the formula \(\beta_{0} = y_{bar} - \beta_{1} * x_{bar}\). The calculated value of \(\beta_{0}\) is approximately -2.942.
Step 5 ：The estimates of \(\beta_{0}\) and \(\beta_{1}\) are approximately -2.942 and 2.174, respectively. So, \(\beta_{0} \approx b_{0}=\boxed{-2.942}\) and \(\beta_{1} \approx b_{1}=\boxed{2.174}\).
