Unit 3: Simple Random Samples

by Sylvie B.

Simple Random Sample

A sample of size n drawn from a population of size N where every possible sample of size n has equal chance of being selected.

Example

There are 5 patients in a clinical trial who have a particular condition: Patient A, B, C, D, and E. The researcher wants to select a random sample of 2 patients to receive the new treatment and compare their outcomes to the others.

1. {Patient A, Patient B}

2. {Patient A, Patient C}

3. {Patient A, Patient D}

4. {Patient A, Patient E}

5. {Patient B, Patient C}

6. {Patient B, Patient D}

7. {Patient B, Patient E}

8. {Patient C, Patient D}

9. {Patient C, Patient E}

10. {Patient D, Patient E}

Five over two possible samples —> 10 samples

Each person also has an equal chance of getting picked (4/10).

However, what makes this a SRS is that each of the ten samples is equally as likely to get picked (1/10 chance).

Drawing Simple Random Samples: Example

Names from a hat

This is the simplest SRS.
It puts no constraints on who can be selected together or the likelihood of any individual being picked.

Drawing Simple Random Samples: Random Digits Table

Random Digits Table: a long string of digits (digits being the numbers 0 to 9 inclusive) which has the following properties:

Each entry is equally likely to be any of the digits 0 to 9
Each entry is independent of the previous entry
Each string of numbers is independent of any other string of numbers

5 steps of how to use a random digits table to select a sample of size n:

Label every unit (element) in your population from 1 to n.
Randomly select a location in the table to start at.
With 1-10 units, look at the digits one at a time (where 0 corresponds to 10). With 11-100 units, look at the digits two at a time (01 corresponds to 1, 02 corresponds to 2,. . . and 00 corresponds to 100, etc.)
For each grouping of numbers ask:
- Is this number already in my sample?
- Does it correspond to a number outside of my sample range?
- If the answer to both questions is no, put the corresponding unit in your sample.
- If the answer to either is yes, discard the number and move to the next number.
Continue until you have picked n units.

Example 1: Lotto 6/49

In Lotto 6/49, 6 numbers are selected from the numbers 1-49. We will use the following line from a random digits table to pick 6 winning numbers. We will label the numbers 01,02,. . . ,09,10,11,. . . ,49. The place in the table is already picked and is the following:

69051 64817 09517 84534 06489 87201 97245

69, 51, 78 —> not in population

Sample: 05, 16, 48, 17, 09, 45

Example 2: Teambuilding

Suppose we have 8 people who must be divided in to two teams of four players. We can select the teams as follows: Label them: 1. Finlay, 2. Barry, 3. Cathy, 4. Neil, 5. Terry, 6. Ken, 7. Mike, 8. Marlene.

Now draw an SRS of size 4 using the following line of a random digits table:

66925 55658 39100 78458 11206 19876 87151 31260

6, 5 —> already selected

9 —> not in population

Group 1: 6, 2, 5, 8

Group 2: 1, 3, 4, 7 (remaining individuals)

Equally Likely Sampling Units vs Equally Likely Groups

Just because every person has an equal chance of being selected, that does not mean every group of size n has the same chance of being selected.

Example:

Let’s revisit that group of patients: Patient A to E. Suppose we labeled randomly pick one of these patients. Then that person and the next person on the list would be the sample.

Patient A, Patient B
Patient B, Patient C
Patient C, Patient D
Patient D, Patient E
Patient E, Patient A

Every person appears in two of the groups so each person has probability 2/5 of being selected, i.e., each person is equally likely to be selected.

A/B : Probability = 1/5
A/C : Probability = 0/5
A/D : Probability = 0/5
A/E : Probability = 1/5
B/C : Probability = 1/5
B/D : Probability = 0/5
B/E : Probability = 0/5
C/D : Probability = 1/5
C/E : Probability = 0/5
D/E : Probability = 1/5

Here not all groups of size 2 are equally as likely to be selected.

Facit:

Equal probability groups are included ⇒ Equal probability individuals are included
Equal probability individuals are included ⇏ Equal probability groups are included.

Estimation with Simple Random Samples: Expected Values

An expected value can be thought of as a weighted average of outcomes based on the probability each outcome occurs.
The expected value of a random variable Y is given by