Wagner Printers performs all types of printing, including custom work, such as advertising displays, and standard work, such as business cards. Market prices exist for standard work, and Wagner Printers must match or better these prices to get the business. The key issue is whether the existing market price covers the cost associated with doing the work. On the other hand, most of the custom work must be priced individually. Because all custom work is done on a job-order basis, Wagner routinely keeps track of all the direct labor and direct materials costs associated with each job. However, the overhead for each job must be estimated. The overhead is applied to each job using a predetermined (normalized) rate based on estimated overhead and labor hours. Once the cost of the prospective job is determined, the sales manager develops a bid that reflects both the existing market conditions and the estimated price of completing the job.

In the past, the normalized rate for overhead has been computed by using the historical average of overhead per direct labor hour. Wagner has become increasingly concerned about this practice for two reasons. First, it hasn't produced accurate forecasts of overhead in the past. Second, technology has changed the printing process, so that the labor content of jobs has been decreasing, and the normalized rate of overhead per direct labor hour has steadily been increasing. The file C11_03.xlsx shows the overhead data that Wagner has collected for its shop for the past 52 weeks. The average weekly overhead for the last 52 weeks is $54,208, and the average weekly number of labor hours worked is 716. Therefore, the normalized rate for overhead that will be used in the upcoming week is about $76 (=54208/716) per direct labor hour.

Wagner is now preparing a bid for an important order that may involve a considerable amount of repeat business. The estimated requirements for this project are 15 labor hours, 8 machine hours, $150 direct labor cost, and $750 direct material cost. Using the existing approach to cost estimation, Wagner has estimated the cost for this job as $2040(= 150 + 750 + (76 x 15)). Given the existing data, what cost would you estimate for this job?

2. Spectral Versus Singular Value Decomposition: Consider a linear operator A with matrix representation 1 3 A _ [7 _1] (a) Compute the spectral decomposition and singular 1mlue decomposition of A. (1:) Sketch ANSI}, the image of the unit circle under .A. (c) Compare your answers to parts (a) and {b} and draw a conclusion. Consider a Markov chain {Xn : n = 0, 1, 2, ...} with state space {1, 2, 3} and one-step transition probability matrix O NIH NIH P = O 0 O (a) Mark O or X: ( ) The Markov chain is irreducible. ( ) The Markov chain is aperiodic. ( ) The Markov chain is transient. ( ) The Markov chain is recurrent. ( ) The Markov chain is null recurrent. ( ) The Markov chain is ergodic. (b) Calculate P(X5 = 1/X2 = 1). (c) Find limn + P(Xn = 1/X2 = 1).6.10 Find the method of moments estimators of the parameters 6 and 62 in the beta distribution with the probability density function: r(@, + 02) f (x) = -' (1 - x) for 0 Ex s 1 r(or(62) based on a random sample X1. X2. .... X,. (Hint: Equate the mean and variance of the beta distribution, the formulas for which are given in Section 2.8.4, to the corresponding sample quantities, , and (2 - ;, respectively, and solve.)we we variance o'. Problem # 4 Suppose that the proportion of defectives shipped by a vendor, which varies somewhat form shipment to shipment, may be looked upon as a random variable having the beta distribution with a = 1 and B = 4. (a) Find the mean of this beta distribution, namely, the average proportion of defectives in a shipmer from this vendor. b) Find the probability that a shipment from this vendor will contain at most half defectives