Question

Commercial refrigeration systems. The role of maintenance in energy saving in commercial refrigeration was the topic of an article in the Journal of Quality in Maintenance Engineering (Vol. 18, 2012). The authors provided the following illustration of data relating the efficiency (relative performance) of a refrigeration system to the fraction of total charges for cooling the system required for optimal performance. Based on the data shown in the graph (next page), hypothesize an appropriate model for relative performance (y) as a function of fraction of charge (x). What is the hypothesized sign (positive or negative) of the ${\mathbf{\beta }}_{\mathbf{2}}$parameter in the model?

Short answer

The appropriate model for the scatterplot above is a quadratic model of y on x.

The sign of ${\mathbf{\beta }}_{\mathbf{2}}$will be positive as it can be seen in the graph that the parabola is an upward-sloping curve.

Step-by-step solution

model for the fitted data

The second-order model equation for the fitted data is$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}$

Sign of β2

The sign of ${\mathbf{\beta }}_{\mathbf{2}}$will be positive as it can be seen in the graph that the parabola is an upward-sloping curve.