Homework Review1-in-k Systematic SamplingThe Sample SizeAssignmentAnswers to Even-numbered ProblemsSystematic SamplingLecture 9Section 2.7Robb T. KoetherHampden-Sydney CollegeWed, Sep 7, 2011Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 1 / 25Outline1Homework Review21-in-k Systematic Sampling3The Sample Size4Assignment5Answers to Even-numbered ProblemsRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 2 / 25Outline1Homework Review21-in-k Systematic Sampling3The Sample Size4Assignment5Answers to Even-numbered ProblemsRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 3 / 25Exercise 2.23, p. 116A class consists of 100 students. Suppose that we are interestedin the heights of the people in the class. We could take a stratifiedrandom sample, using gender as the stratification variable.Suppose that it is known that 60 of the 100 students in the classare female.Someone shakes a box containing the 60 female tags, and with ablindfold on, reaches in and draws 10 tags.Then, this same person shakes a box containing the 40 male tags,and with a blindfold on, reaches in and draws 10 tags.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 4 / 25Exercise 2.23, p. 116Now we have two simple random samples, one for the males andone for the females, each of size 10.Suppose that the average height of the males in the sample is 70inches, while the average height for the females in the sample is63 inches.How should these two average heights be combined to producean estimate of the average height of the people in the class?Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 5 / 25SolutionWe should use a weighted average.Weight the female average with 0.60 and the male average with0.40.The estimated average for the class isest. avg. = (0.60)(63) + (0.40)(70)= 37.8 + 28= 65.8.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 6 / 25Outline1Homework Review21-in-k Systematic Sampling3The Sample Size4Assignment5Answers to Even-numbered ProblemsRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 7 / 25Systematic SamplingDefinition (1-in-k Systematic Sampling)1-in-k systematic sampling is a sampling method in which one of thefirst k members of the population is selected at random. Thenbeginning with that member, every kthmember is selected.For example, if k = 10, then one of the first 10 members isselected at random (using randInt(1,10)).Suppose member #6 is selected.Then beginning with member #6, every 10thmember is selected.That is, members 6, 16, 26, 36, and so on.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 8 / 25Systematic Sampling1-in-k Systematic SamplingNumber the members of the population 1 through N.Using randInt(1,k), choose a random starting point in the firstblock of size k. That represents the first member of the sample.From that starting point, put every kthmember in the sample.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 9 / 25ExampleExample (1-in-5 Systematic Sampling)There are 19 students in this class.Let’s choose a 1-in-3 systematic sample from the 19 students inthe class.Use randInt(1,3) to choose the first member. It will be arandom number from 1 to 3.Then choose every 3rdstudent thereafter.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 10 / 25Outline1Homework Review21-in-k Systematic Sampling3The Sample Size4Assignment5Answers to Even-numbered ProblemsRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 11 / 25The Sample SizeIn the last example, notice that 19/3 = 613.So the sample size could end up being either 6 or 7, depending onthe starting point.If the starting point is 1, then we get 7 members:1, 4, 7, 10, 13, 16, 19.If the starting point is 2, then we get 6 members3, 5, 8, 11, 14, 17.Either way, each member has a 1/3 chance of being in thesample, right?Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 12 / 25The Sample SizeCarry out the selection, using a seed of 15.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 13 / 25The Sample SizeThe populationRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 14 / 25The Sample SizeThe “blocks”Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 14 / 25The Sample SizeYThe randomly selected member from the first blockRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 14 / 25The Sample SizeY NY YN N NThe sampleRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 14 / 25The Sample SizeNow carry out the selection again, using a seed of 14.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 15 / 25The Sample SizeThe populationRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 16 / 25The Sample SizeThe “blocks”Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 16 / 25The Sample SizeNThe randomly selected member from the first blockRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 16 / 25The Sample SizeY NN YY NThe sampleRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 16 / 25How to Determine kThe Determination of kDivide the population size N by the desired sample size n.Let k = N/n (rounded off).If you round down, you may get one extra member.If you round up, you may get one less member.It does not matter.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 17 / 25ExampleExample (Systematic Sampling)Suppose we want to select n = 12 members from a population ofsize N = 187.Compute k = 187/12 = 15712≈ 16.Use randInt(1,16) to get a random integer from 1 to 16.Then include that number and every 16th number thereafter in thesample.Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 18 / 25Outline1Homework Review21-in-k Systematic Sampling3The Sample Size4Assignment5Answers to Even-numbered ProblemsRobb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Sep 7, 2011 19 / 25AssignmentHomeworkRead Section 2.7, pages 117 - 121.Let’s Do It! 2.9, 2.10.Page 121, exercises 28, 30 - 32.Chapter 2 review, p. 136, exercises 44, 46, 48 - 50, 51, 53, 55, 56,63.Robb T. Koether
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