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Simple Random SamplingLecture 7
Section 2.5
Robb T. Koether
Hampden-Sydney College
Mon, Jan 30, 2012
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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Review Quiz
Example (Review Quiz)1 Researchers select all the patients that are suffering from
depression at a large hospital for their study.This could be an example of(a) Selection bias.(b) Non-response bias.(c) Response bias.(d) Experimenter bias.
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Review Quiz
Example (Review Quiz)2 Patients agree (or disagree) to be a part of a study to test the
effectiveness of a new treatment.This could be an example of(a) Selection bias.(b) Non-response bias.(c) Response bias.(d) Experimenter bias.
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Review Quiz
Example (Review Quiz)3 Patients join either a treatment group that receives the new
treatment and a control group that does not receive the treatment.This could be an example of(a) Selection bias.(b) Non-response bias.(c) Response bias.(d) Experimenter bias.
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Review Quiz
Example (Review Quiz)4 After two weeks, the researchers check each patient to see how
well they are doing.This could be an example of(a) Selection bias.(b) Non-response bias.(c) Response bias.(d) Experimenter bias.
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Review Quiz Answers
Example (Review Quiz Answers)1. (a) Selection bias.2. (b) Non-response bias.3. (c) Response bias.4. (d) Experimenter bias.
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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Introduction
Suppose I wanted to select a random sample of 4 students fromthis class.
I start in the front left corner and toss a coin.Heads, that student is selected.Tails, he is not selected.
I move on to the next student behind him and use the same rule.I continue in this manner, row by row, until I have 4 students.Is the sample a random sample?Is this a good method?
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Introduction
Suppose I wanted to select a random sample of 4 students fromthis class.I start in the front left corner and toss a coin.
Heads, that student is selected.Tails, he is not selected.
I move on to the next student behind him and use the same rule.I continue in this manner, row by row, until I have 4 students.Is the sample a random sample?Is this a good method?
Robb T. Koether (Hampden-Sydney College) Simple Random Sampling Mon, Jan 30, 2012 9 / 28
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Introduction
Suppose I wanted to select a random sample of 4 students fromthis class.I start in the front left corner and toss a coin.
Heads, that student is selected.Tails, he is not selected.
I move on to the next student behind him and use the same rule.
I continue in this manner, row by row, until I have 4 students.Is the sample a random sample?Is this a good method?
Robb T. Koether (Hampden-Sydney College) Simple Random Sampling Mon, Jan 30, 2012 9 / 28
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Introduction
Suppose I wanted to select a random sample of 4 students fromthis class.I start in the front left corner and toss a coin.
Heads, that student is selected.Tails, he is not selected.
I move on to the next student behind him and use the same rule.I continue in this manner, row by row, until I have 4 students.
Is the sample a random sample?Is this a good method?
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Introduction
Suppose I wanted to select a random sample of 4 students fromthis class.I start in the front left corner and toss a coin.
Heads, that student is selected.Tails, he is not selected.
I move on to the next student behind him and use the same rule.I continue in this manner, row by row, until I have 4 students.Is the sample a random sample?
Is this a good method?
Robb T. Koether (Hampden-Sydney College) Simple Random Sampling Mon, Jan 30, 2012 9 / 28
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Introduction
Suppose I wanted to select a random sample of 4 students fromthis class.I start in the front left corner and toss a coin.
Heads, that student is selected.Tails, he is not selected.
I move on to the next student behind him and use the same rule.I continue in this manner, row by row, until I have 4 students.Is the sample a random sample?Is this a good method?
Robb T. Koether (Hampden-Sydney College) Simple Random Sampling Mon, Jan 30, 2012 9 / 28
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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Simple Random Sample
Definition (Simple random sample)A simple random sample of size n is a random sample that is selectedin such a way that all samples of size n have the same chance of beingselected.
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Simple Random Sample
A consequence of this is that all individuals in the population havethe same chance of being selected for the sample.
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Simple Random Sample
Example (Simple Random Sample)For example, let the population be {Armstrong, Bean, Craycroft,Davis, Freston, Garner}.The possible samples of size 3 are
{A, B, C} {A, C, D} {A, D, G} {B, C, G} {C, D, F}{A, B, D} {A, C, F} {A, F, G} {B, D, F} {C, D, G}{A, B, F} {A, C, G} {B, C, D} {B, D, G} {C, F, G}{A, B, G} {A, D, F} {B, C, F} {B, F, G} {D, F, G}
Choose one of the above samples at random.What is each person’s chance of being in the sample?
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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Simple Random Sample
For larger populations, the previous method is not practical.For example, if N = 100 and n = 6, then there are 1,192,052,400different possible samples.However, it turns out that if we select individuals one at a time,with all individuals equally likely at each step, then all samples areequally likely.Thus, our sample will be a simple random sample.
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Selecting a Simple Random Sample
Select a Sample of Size nGiven a population of size N,
Number the members of the population from 1 to N.Use a random number generator (such as on a calculator) togenerate n random integers from 1 to N.
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Sampling With or Without Replacement
Definition (Sampling with replacement)When we sample with replacement, a selected item may be selectedagain. That is, repetitions are allowed.
Definition (Sampling without replacement)When we sample without replacement, a selected item may not beselected again. That is, repetitions are not allowed.
Sampling may be done with or without replacement.
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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TI-83: Selecting a Sample
TI-83: Selecting a Sample1 Press MATH.2 Use the arrow keys to highlight the PRB menu title.3 Press 5 to select randInt (item #5).4 Enter randInt(1,100). (E.g., if N = 100.)5 Press ENTER. A random number appears.6 Press ENTER repeatedly for more random numbers.
If the sampling is done without replacement, then repetitionsshould be discarded.
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Example
Let the population be the students in this class.Then N = 18.Number the members 1 - 18 in alphabetical order.We will choose a sample of size n = 6.What is each individual’s chance of being in the sample?
PracticeUse randInt(1,18) to select 6 students.
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TI-83: Getting a Set of Random Numbers
To get several random integers at once, possibly with repetitions,use randInt with a third parameter, representing the samplesize.For example, to get 6 random integers from 1 to 18, enterrandInt(1,18,6).However, this may include repetitions.
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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TI-83: Setting the Seed
TI-83: Setting the Seed1 Enter a seed (choose any number whatsoever).2 Press STO. An arrow appears in the display.3 Press MATH, highlight PRB, select rand (item #1).4 Press ENTER. The seed is now set.
In general practice, this is not done.We do it only to “synchronize” our calculators so that we will all getthe same answer.
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Practice
PracticeSet the seed to 157 (an arbitrary choice).Then select a random sample of size 6 from the population of thestudents in this class.
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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Simple Random Sample
Example (Simple Random Sample)Just because a sampling method guarantees that all individuals inthe population have the same chance of being in the sample, itdoes not mean that the sample is a simple random sample.For example, suppose we toss a coin to choose one of thefollowing samples.
{A, B, C} {D, F, G}
What is each person’s chance of being in the sample?What is the probability of getting the sample {A,B,C}?What is the probability of getting the sample {A,B,D}?
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Simple Random Sample
Example (Simple Random Sample)Just because a sampling method guarantees that all individuals inthe population have the same chance of being in the sample, itdoes not mean that the sample is a simple random sample.For example, suppose we toss a coin to choose one of thefollowing samples.
{A, B, C} {D, F, G}What is each person’s chance of being in the sample?
What is the probability of getting the sample {A,B,C}?What is the probability of getting the sample {A,B,D}?
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Simple Random Sample
Example (Simple Random Sample)Just because a sampling method guarantees that all individuals inthe population have the same chance of being in the sample, itdoes not mean that the sample is a simple random sample.For example, suppose we toss a coin to choose one of thefollowing samples.
{A, B, C} {D, F, G}What is each person’s chance of being in the sample?What is the probability of getting the sample {A,B,C}?
What is the probability of getting the sample {A,B,D}?
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Simple Random Sample
Example (Simple Random Sample)Just because a sampling method guarantees that all individuals inthe population have the same chance of being in the sample, itdoes not mean that the sample is a simple random sample.For example, suppose we toss a coin to choose one of thefollowing samples.
{A, B, C} {D, F, G}What is each person’s chance of being in the sample?What is the probability of getting the sample {A,B,C}?What is the probability of getting the sample {A,B,D}?
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Outline
1 Introduction
2 Simple Random Samples
3 Selecting Simple Random SamplesOn the TI-83Setting the Seed
4 Non-Simple Random Samples
5 Assignment
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Assignment
HomeworkRead Section 2.5, pages 98 - 106.Let’s Do It! 2.4.Page 107, exercises 13, 15 - 18.
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