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Statistics 431 Home Work # 10 (Professor S. Hedayat) In this home work we shall concentrate on an important family of sampling plans known as systematic sampling plans which we covered in the class. Please prepare your answer in an organized and clean format. Remember that you should exhibits all your data and results with the same (accuracy) decimal points. Sampling Project: A software company has 20 software engineers who work 5 days a week and only 5 hours per day. The hours are 9 to 12 in the morning and 2 to 4 in the afternoon. The manager is aware that the total productivities changes from hours to hours and from day to day. With the help of a survey statistician she wants to select 5 “randomly” selected hours and measure the productivity for the selected hour by the 20 software engineers and then estimate the total productivities. The survey statistician recommends that these 5 hours should cover every day of the week and every working hour of the day. The survey statistician prepares a 5 by 5 array with rows indicating the days of the week and columns the working hours of a day. Then he recommends that for Monday one of the 5 hours be selected with probability of 1/5 and then the remaining 4 hours will be the corresponding hours either on the main/left or broken diagonal associated with the random selected hour on Monday. Upon randomization of hours on Monday the working hour of 2-3 PM was selected. 1- Draw a 5 by 5 array and identify the selected 5 hours. 2- Specify the support and the probability over the support for the sampling plan 1 3- Compute the first-order and the second-order inclusion probabilities for this sampling plan. 4- Use the notation Pij for the total productivity for the 20 engineers for the day i and the hour j and then provide an unbiased estimate of the total productivities P.. for these 20 engineers in a given week. 5- If possible provide an unbiased estimate of the variance of your estimate in 4. If this cannot be done then can you impose some reasonable and minimum conditions of the problem so that you can provide this unbiased estimate of the variance? 2