# 1 (1 a) Show that the following functions are probability density functions for some value of k and determine k. Then determine the mean and variance of X. (a) f(x)= kx2 for 0 0 and 100 < x<(100 + k) (1 b) For each of the density functions in Problem 1 (a) , perform the following using Excel (Show the underlying computation in form of spreadsheet along with graphs) (a) Graph the density function and mark the location of the mean on the graph. (b) Find the cumulative distribution function. (c) Graph the cumulative distribution function. PROBLEM # 2 The CCDF P (T>t ) 1F (t) of a cutting tool life(time to failure T ) is given by the alpha distribution P (T>t ) 1F ( t )=1 1 c 1 t ( ( )) Where t> 0 , is in hours a) Plot F ( t )= 1 c 1 t ( ( )) against t/c varying from t +0 (zero not included) to t/c =5 with an increment of t=0.05 for =0.05, .1, 0.2,0 .3 , 0.4 .Please Note alpha distribution is essentially 1 c a normal distribution with z= 1 t ( ) .Not only show the % desired curves , but also show the computation behind those curves by providing the Excel Spreadsheet. b) What is the Median Time to Failure, t0.5 ? c) How frequently should the tool be changed if failures are to be held to no more than p %? d) Mode of T, (most probable life which is the value of t where df (t) =0 dt PROBLEM # 3 Two operators perform the same machining operation. Their supervisor wants to estimate the difference in the mean machining times between them. No assumption can be made as to whether the variabilities of machining time are the same for both operators. It can be assumed, however, that the distribution of machining time is normal for each operator. A random sample of 10 from the first operator gives an average machining time of 4.2 min with a standard deviation of 0.5 min. a random sample of 6 from the second operator yields an average machining time of 5.1 min with a standard deviation 0.8 min. Find a 95% confidence interval for the difference in the mean machining times between the two operators. PROBLEM # 4 The Occupational Safety and Health Administration (OSHA) mandates certain regulations that have to be adopted by corporations. Prior to the implementation of the OSHA program, a company found that for a sample of 40 randomly selected months, the mean employee time lost due to job-related accident was 45 h. After implementation of the OSHA program, for a random sample of 45 months, the mean employee time lost due to job-related accidents was 39 h. It can be assumed that the variability of time lost due to accidents is about the same before and after implementation of the OSHA program (with a standard deviation being 3.5 h)