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 \(\beta_{1} \approx b_{1}=\boxed{2.174}\).