systematic sampling - lecture 9 section 2people.hsc.edu/faculty-staff/robbk/math121/lectures/spring...

Systematic SamplingLecture 9

Section 2.7

Robb T. Koether

Hampden-Sydney College

Wed, Jan 27, 2010

Robb T. Koether (Hampden-Sydney College) Systematic Sampling Wed, Jan 27, 2010 1 / 22

Outline

1 Homework Review

2 1-in-k Systematic Sampling

3 The Sample Size

4 Assignment

5 Answers to Even-numbered Problems


Outline

1 Homework Review


3 The Sample Size

4 Assignment



Exercise 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.


Exercise 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?


SolutionWe should use a weighted average.Weight the female average with 0.60 and the male average with0.40.The estimated average for the class is

est. avg. = (0.60)(63) + (0.40)(70)

= 37.8 + 28= 65.8.


Outline

1 Homework Review


3 The Sample Size

4 Assignment



Systematic Sampling

Definition (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 k th member is selected.

For example, if k = 10, then one of the first 10 members isselected at random. Suppose member #6 is selected. Thenbeginning with member #6, every 10th member is selected, i.e.,members 6, 16, 26, 36, and so on.


Systematic Sampling

1-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 k th member in the sample.


Example

Example (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 3rd student thereafter.


Outline

1 Homework Review


3 The Sample Size

4 Assignment



The Sample Size

In 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 members

3, 5, 8, 11, 14, 17.

Either way, each member has a 1/3 chance of being in thesample, right?


The Sample Size

Row 1 Row 2

Row 3


The Sample Size

Row 1 Row 2

Row 3

Y


The Sample Size

Row 1 Row 2

Row 3

Y

N Y N

N

N

Y


How to Determine k

The 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.


Example

Example (Systematic Sampling)Suppose we want to select n = 12 members from a population ofsize N = 187.Compute k = 187/12 = 15 7

12 ≈ 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.


Outline

1 Homework Review


3 The Sample Size

4 Assignment



Assignment

HomeworkRead 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.


Outline

1 Homework Review


3 The Sample Size

4 Assignment



Answers to Even-numbered Problems

Page 121, Problems 28, 30, 322.28 (a) 3.

(b) 4.2.30 (a) Stratified.

(b) No.(c) Selection.

2.32 (a) 114 .

(b) 40.



Page 136, Problems 44, 46, 48, 50, 562.44 (c) statistic is to a sample.2.46 (a) The population is the set of all adult U.S. residents.

The sample is the set of 1500 people who wereselected. n = 1500.

(b) The population is the set of all 1-gallon milk cartonsin today’s shipment. The sample is the set of 5cartons that were selected. n = 5.

(c) The population is the set of all 740 members of thelocal women’s business association. The sample isthe set of 38 questionnaires that were returned.n = 38.



Page 136, Problems 44, 46, 48, 50, 562.48 (a) (i) selection bias.

(b) (ii) nonresponse bias.(c) (iii) response bias.

2.50 (a) Simple random sample, or convenience sampling(see p. 90).

(b) The set of all people who called the help line duringthe six-week period, or the set of all drug users.

(c) No. The headline seems to apply the results of thesurvey to all workers, but the sample did not comefrom the population of all workers.



Page 136, Problems 44, 46, 48, 50, 562.56 (a) Stratified sampling.

(b) 75416 .

(c) 76%.(d) (i) statistic.


systematic sampling - lecture 9 section 2people.hsc.edu/faculty-staff/robbk/math121/lectures/spring...

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