Your company has for a long time used Imperial Deliveries for freight. Lately, however, a promising new competitor has surfaced: Centurium Falcon Freight. You decide to test the rate of delivery errors to compare the two. Let \(\pi_{I D}\) be the proportion of erroneous deliveries at Imperial Deliveries, and let \(\pi_{C F F}\) be the rate for Centurium Falcon Freight.

Your internal routines are built around delivery by Imperial Deliveries, making a change of freight company cost a bit. However, at the same time there is a fair bit to save if the new alternative is an improvement. A cost analysis using utility functions indicates that you should test the hypothesis \(H_{1}: \pi_{C F F}<\pi_{I D}\) with significance \(\alpha=0.15\).

You run 200 trial deliveries with both companies. Imperial Deliveries make four delivery errors, whereas Centurium Falcon Freight make one error. Investigate, with prior \(\beta_{(1,1)}\), and decide between the competing hypotheses.