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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
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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?
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