Simple Random Sampling (SRS) (College Board AP® Statistics)

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

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

Examiner Tips and Tricks

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

Examiner Tips and Tricks

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 n is one in which every possible sample of size n taken from a population has an equal chance of being chosen

    • This 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 n from a population of size N is to

    • first label every element in the population a number from 1 to N

    • then generate n random numbers

      • e.g. using a random number generator

    • then select the n elements from the population that correspond to the n random numbers

  • With this method, every possible sample of size n taken from a population is equally likely

    • This means the sample produced is a simple random sample

Examiner Tips and Tricks

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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Mark Curtis

Author: Mark Curtis

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.