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

Y: outcome

p(Y=y): probability of outcome

When every outcome is equally likely, the expected value is just the average of all outcomes.

Estimation with Simple Random Samples: Variance

The population variance or variance of a random variable Y is given by

How far away is the outcome from the mean --> standard deviation—> squared standard deviation = variance

Estimate of the population mean

The estimator of the population mean μ

Estimate of the population variance

Estimated variance of ȳ

with s being the sample standard deviation

Finite Population Correction

Bound on the Error of Estimation

2: both sides of mean

square root of estimated variance —> standard deviation

Estimator of the Population Total τ

Note:

Estimated Variance of τ

Sample Size Required to Estimate µ With a Bound on the Error of Estimation B

where

Sample Size Required to Estimate τ with a bound on error B

Estimating σ^2

When unavailable, we can typically approximate the population variance σ^2 in two ways:

s^2

(1/4)^2 of the range of the dataset

(1/4 of the range of the dataset for standard deviation)

Sample Size Warning

Remember to always round up sample size!

If we round down sample size, we will not meet the error bound requirements.

Estimator of population proportion p

Estimated variance of p̂

Bound on the error of estimation

Sample size required to estimate p with a bound on the error of estimation B

Note: For p we will need to estimate it with a known reliable approximation or by p = 0.5 with no prior information.

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