Problem:

A university administrator is interested in whether a newbuilding can be planned and built on campus within a four-year timeframe. He considers the process in two phases. Phase I: Phase Iinvolves lobbying the state legislature and governor for permissionand funds, issuing bonds to obtain funds, and obtaining all theappropriate legal documents. Past experience indicates that thetime required to complete phase I is approximately normallydistributed with a mean of 16 months and standard deviation of 4months. If X = phase I time, then X ~ N(? = 16 months, ? = 4months). Phase II: Phase II involves creation of blue prints,obtaining building permits, hiring contractors, and, finally, theactual construction of the building. Past data indicates that thetime required to complete these tasks is approximately normallydistributed with a mean of 18 months and a standard deviation of 12months. If Y = phase II time, then Y ~ N(? = 18 months, ? = 12months). a) A new random variable, T = total time for completingthe entire project, is defined as T = X + Y. What is theprobability distribution of T? (Give both the name of thedistribution and its parameters.) b) Find the probability that thetotal time for the project is less than four years. (In symbols,calculate P(T < 48 months).) c) Find the 95th percentile of thedistribution of T.