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Downtime of a production process. An operations manager is
Chapter 12, Problem 160SE(choose chapter or problem)
Downtime of a production process. Downtime of a production process. An operations manager is interested in modeling E(y), the expected length of time per month (in hours) that a machine will be shut down for repairs, as a function of the type of machine (001 or 002) and the age of the machine (in years). The manager has proposed the following model:
\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\beta_{3} x_{2}\)
where
\(x_{1}\) = Age of machine
\(x_{2}\) = 1 if machine type 001, 0 if machine type 002
The following data were obtained for n = 20 machine breakdowns.
\(\begin{array}{cccc}
\hline \begin{array}{c}
\text { Downtime } \\
\text { (hours per month) }
\end{array} & \begin{array}{c}
\text { Machine Age, } \\
x_{1} \text { (years) }
\end{array} & \begin{array}{c}
\text { Machine } \\
\text { Type }
\end{array} & x_{2} \\
10 & 1.0 & 001 & 1 \\
20 & 2.0 & 001 & 1 \\
30 & 2.7 & 001 & 1 \\
40 & 4.1 & 001 & 1 \\
9 & 1.2 & 001 & 1 \\
25 & 2.5 & 001 & 1 \\
19 & 1.9 & 001 & 1 \\
41 & 5.0 & 001 & 1 \\
22 & 2.1 & 001 & 1 \\
12 & 1.1 & 001 & 1 \\
10 & 2.0 & 002 & 0 \\
20 & 4.0 & 002 & 0 \\
30 & 5.0 & 002 & 0 \\
44 & 8.0 & 002 & 0 \\
9 & 2.4 & 002 & 0 \\
25 & 5.1 & 002 & 0 \\
20 & 3.5 & 002 & 0 \\
42 & 7.0 & 002 & 0 \\
20 & 4.0 & 002 & 0 \\
13 & 2.1 & 002 & 0 \\
\hline
\end{array}\)
a. Use the data (saved in the file) to estimate the parameters of this model.
b. Do these data provide sufficient evidence to conclude that the second-orcler term \(\left(x_{1}^{2}\right)\) in the model proposed by the operations manager is necessary? Test using \(\alpha=.05\).
c. Test the null hypothesis that \(\beta_{1}=\beta_{2}=0\) using \(\alpha=.10\). Interpret the results of the test in the context of the problem
Questions & Answers
QUESTION:
Downtime of a production process. Downtime of a production process. An operations manager is interested in modeling E(y), the expected length of time per month (in hours) that a machine will be shut down for repairs, as a function of the type of machine (001 or 002) and the age of the machine (in years). The manager has proposed the following model:
\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\beta_{3} x_{2}\)
where
\(x_{1}\) = Age of machine
\(x_{2}\) = 1 if machine type 001, 0 if machine type 002
The following data were obtained for n = 20 machine breakdowns.
\(\begin{array}{cccc}
\hline \begin{array}{c}
\text { Downtime } \\
\text { (hours per month) }
\end{array} & \begin{array}{c}
\text { Machine Age, } \\
x_{1} \text { (years) }
\end{array} & \begin{array}{c}
\text { Machine } \\
\text { Type }
\end{array} & x_{2} \\
10 & 1.0 & 001 & 1 \\
20 & 2.0 & 001 & 1 \\
30 & 2.7 & 001 & 1 \\
40 & 4.1 & 001 & 1 \\
9 & 1.2 & 001 & 1 \\
25 & 2.5 & 001 & 1 \\
19 & 1.9 & 001 & 1 \\
41 & 5.0 & 001 & 1 \\
22 & 2.1 & 001 & 1 \\
12 & 1.1 & 001 & 1 \\
10 & 2.0 & 002 & 0 \\
20 & 4.0 & 002 & 0 \\
30 & 5.0 & 002 & 0 \\
44 & 8.0 & 002 & 0 \\
9 & 2.4 & 002 & 0 \\
25 & 5.1 & 002 & 0 \\
20 & 3.5 & 002 & 0 \\
42 & 7.0 & 002 & 0 \\
20 & 4.0 & 002 & 0 \\
13 & 2.1 & 002 & 0 \\
\hline
\end{array}\)
a. Use the data (saved in the file) to estimate the parameters of this model.
b. Do these data provide sufficient evidence to conclude that the second-orcler term \(\left(x_{1}^{2}\right)\) in the model proposed by the operations manager is necessary? Test using \(\alpha=.05\).
c. Test the null hypothesis that \(\beta_{1}=\beta_{2}=0\) using \(\alpha=.10\). Interpret the results of the test in the context of the problem
ANSWER:Step 1 of 4
Let
\(y = {\rm{Downtime}}\)
\({x_1} = {\rm{Age}}\;{\rm{of}}\;{\rm{machine}}\)
\({x_2} = \left\{ \begin{array}{l}1,\;\;\;{\rm{if}}\;{\rm{machine}}\;{\rm{type}}\;001\\0,\;\;\;{\rm{if}}\;{\rm{machine}}\;{\rm{type}}\;002\end{array} \right\)
Step 2 of 4
(a)
Using the Minitab instructions for the regression analysis, we obtain the parameters of the second-order model is, E(y). The output is shown below.
