Simple Random Sampling (SRS) (College Board AP® Statistics)
Revision Note
Generating random numbers
What is a random number generator?
A random number generator is a program used to create random numbers
The inputs are
how many random numbers you want
the start and end values that you want to include
The output is a selection of equally-likely random numbers between the start and end values
How do I treat duplicate numbers from random number generators?
A random number generator may give out the same random number twice
These are called duplicates (or repeats)
You need to ignore duplicates if sampling without replacement
Exam Tip
You must mention clearly whether you are including or ignoring duplicates in the exam!
How else can I generate random numbers?
Two-digit random number generators
These give outputs of two-digit numbers from 00 to 99
e.g. if you require numbers between 1 and 70, you need to
ignore 00
ignore any number greater than 70
Random number tables
These may be random digits like 4, 0, 7, 2, 1, ...
Digits are equally likely and independent of each other
Two rows can be used for two-digit numbers (one for each digit)
Or tables like 469066, 155387, 172419, 953505, ...
For two-digit numbers, split as follows: 46, 90, 66, 15, 53, 87, ...
Decimal random number generators on calculators
The output is a random 3-digit decimal between 0 and 1
e.g. 0.541, 0.414, 0.929
For two-digit numbers, use the first two decimal digits etc
Rolling dice (or similar fair probability methods)
Use two fair 10-sided dice (or roll one dice twice) to get numbers between 0 and 99
One for the tens digit and one for the units digit
Draw numbers from a hat (or bag)
Write number on equally-sized pieces of paper and place into a hat and mix well
Randomly select pieces of paper without replacement
Exam Tip
Whilst there are other methods for generating random numbers, it is easy to lose points in the exam when trying to describe them. It is often safer to talk about random number generators instead!
Worked Example
Describe how to find eight distinct random numbers between 1 and 90.
Answer:
Talk about using a random number generator
The word 'distinct' means different, so no repeated numbers are allowed
Use a random number generator to generate random numbers between a start value of 1 and an end value of 90, both inclusive
Ignore any duplicates until eight different random numbers are obtained
Simple random sampling
What is a biased sample?
A sample is biased if the method used to create the sample consistently under-represents or favors certain groups in the population
Biased samples are, on average, unlikely to be representative of the patterns and behaviors of the population
Though it is still possible to select a biased sample that is coincidently representative of the population!
Biased samples can overestimate or underestimate the property of the population that you are studying
What is a simple random sample (SRS)?
A simple random sample (SRS) of size
is one in which every possible sample of size taken from a population has an equal chance of being chosenThis definition refers to all possible samples (not individuals)
Simple random samples are unbiased (fair)
How do I create a simple random sample?
One method for creating a simple random sample of size
from a population of size 
is tofirst label every element in the population a number from 1 to
then generate
random numbers e.g. using a random number generator
then select the
elements from the population that correspond to the random numbers
With this method, every possible sample of size
taken from a population is equally likelyThis means the sample produced is a simple random sample
Exam Tip
When describing how to take simple random samples in the exam, write your answers in context. Replace words like 'element' and 'population' with what they actually are!
Worked Example
A factory has 150 workers.
The manager of a factory wants to sample 20% of the workers.
Describe a process that the manager can use to create a simple random sample.
Answer:
The sample size is 30 because 20% of 150 workers is 30 workers
First label the workers with the numbers from 1 to 150
Then use a random number generator to generate random numbers between 1 and 150 (inclusive)
Ignore any duplicate (repeat) numbers until 30 different random numbers have been generated
Select the 30 workers that correspond to the 30 random numbers
This creates a simple random sample of size 30
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