Factors that impact productivity of an assembly line. A production manager who supervises an assembly operation wants to investigate the effect of the incoming rate (parts per minute), x¹, of components and room temperature, x², on the productivity (number of items produced per minute), y. The component parts approach the worker on a belt and return to the worker if not selected on the first trip past the assembly point. It is thought that an increase in the arrival rate of components has a positive effect on the assembly rate, up to a point, after which increases may annoy the assembler and reduce productivity.

Similarly, it is suspected that lowering the room temperature is beneficial to a point, after which reductions may reduce productivity. The experimenter used the same assembly position for each worker. Thirty-two workers were used for the experiment, two each assigned to the 16 factor level combinations of a 4 × 4 factorial experiment. The data, in parts per minute averaged over a 5-minute period, are shown in the table at the top of p. 710.

(a) Perform an analysis of variance for the data. Display the computed quantities in an ANOVA table.

(b) Write the linear model implied by the analysis of variance.

(c) Do the data provide sufficient evidence to indicate differences among the mean responses for the 16 treatments of the 4 × 4 factorial experiment? Test using α = .05.

(d) Do the data provide sufficient evidence to indicate an interaction between arrival rate x₁ and room temperature x₂ on worker productivity? Test using α = .05.

(e) Find the value of R² that would be obtained if you were to fit the linear model in part b to the data.

(f) Explain why a regression analysis would be a useful addition to the inferential methods used in parts a-e.