The statistical assignment that I can’t do because I don’t understand.I hope the tutor can help me, assist me to complete it together and do it independently without plagiarism.
FOUN046: Mathematics for Science
Statistical Assignment: Sampling
(10% towards the final grade).
This assignment is a literature and website based assignment. You have eight references available to you to answer your questions. These references are available for use available in the FOUN046 Teams folder under files, Statistics assignment folder. No other research is needed.
The four resources in the book are a selection of material from the following references.
Power Point notes from a website: Why Sample?
Modular Mathematics for GCSE—chapter 19: Sampling, Surveys and Questionnaires
Statistics AEB—chapter 2: Data Collection
Advanced Level Statistics — chapter 9: Sampling and estimation
Another four resources from the internet are also provided. The reference is on the document.
Your Student Book 2 may also be useful in helping you answer the questions.
Reference each question.
Assignment resources:
For the book, use the titles as given above when referencing each question in your assignment.
For the other resources, use the reference in the document as given.
Using Other References from the internet: You may use other references to help you answer the questions. Please reference these resources correctly (as taught in FOUN001) for each question. Be careful about the time you spend researching other resources and make sure they are reliable academic sources (as taught in FOUN001).
You are required in to hand in this assignment on or before Tuesday 19 January 2021 tutorial time (4 pm NZ time). If you are an on campus student by giving it to your teacher in class, or by email, for marking and comments. If you are an online student by email for marking and comments.
Emails must be sent to Una’s email address. Use your university email when sending the assignment. Emails from personal addresses with attachments may not arrive as they will be treated as junk mail by the university system. It is your responsibility to ensure that your emailed assignment arrives to the right place and on time.
Do not use One Drive to send your assignment.
Your assignment must be a Word document. Put your name and class in your file name on your Word document. Keep a draft copy somewhere safe.
Your declaration must be completed and handed in with your assignment. Assignments will not be marked unless a completed declaration is handed in.
If you have any queries, please ask, or message, me before you start. Good luck!
Please Read very carefully before you start your assignment.
Declaration of Work – Academic Integrity
Refer to the Student Assessment Guide book for full details.
Student Name (full name, including “English” name)
ID Number (from your ID card)
Paper
FOUN046 Maths for Science
Assignment Name
Sampling in Statistics
Teacher
Una Byrne/George Morris
If you have any doubts or questions, please see your teacher before you hand in your work.
STUDENT DECLARATION: you must sign this declaration of academic integrity as part of your assignment.
I confirm and promise
· that this assignment is entirely my own work and is original work
· that I have not committed plagiarism when completing the attached piece of work
· that source materials have been fully referenced
· that this assignment, or substantial parts of it, have not been submitted for assessment
in any other programme, any other semester or paper
· that I have not copied from another student or allowed my work to be copied
· that I have not submitted the same or a very similar assignment to that of another student
· that I have not worked on an assignment with other students
· that I have not assisted someone else to commit academic misconduct by writing their assignment or by sharing files
· that I have not made up data or information
· that I have not committed any form of academic misconduct
· that I have read and am aware of academic integrity and the academic misconduct details in my Student Assessment Guide.
Do not forget to complete this declaration. Assignments will not be marked without it.
Name/ Signature
Date Class:
3
2
FOUN046 Mathematics for Science Assignment Term 3 2020 Total marks: 60
You must answer the questions in your own words to demonstrate your understanding of the statistical ideas. Answers must be presented in paragraphs, or sentences, as prescribed in Academic English I and make use of relevant statistical language.
Also each question must be clearly referenced. (6 marks)
1. Define briefly (at most two sentences) what is meant by each of the following three terms in Statistics:
a) a sample b) a sample design c) secondary data. (9 marks)
2. a) Define briefly (at most three sentences) what is meant by bias in Statistics.
b) List two factors that might introduce statistical bias into a sample.
c) Give a statistical example of when using a biased sample could be an advantage in
scientific research. (8 marks)
3. a) List five different factors you must be careful about when designing a survey
questionnaire.
b) Briefly describe (at most two sentences) two factors from your list in a). (9 marks)
4. In this question you are going to create a sample from a database. Follow the instructions below to help you create your sample.
a) Select a large set of data, which has a science, or health related, theme. Provide a reference for the data. If it is not easily available online, print or scan the set of data and attach it to the assignment.
We recommend
http://www.gapminder.org/data/ as a good source to begin your search for a suitable theme and data. Choose a science, or health related, theme that is of interest to you. State why you have chosen this set of data. Check with your teacher if you are unsure about your choice. (2 marks)
systematic sampling; random sampling using a calculator; stratified sampling; cluster sampling
b) Choose two suitable statistical sampling methods from the box above that could be used to create your sample. Write a detailed description of each. Your description must include all of the steps you need to do to choose a sample from a population.
(8 marks)
c) Describe in detail how you would create a sample from your chosen set of data. Your answer must include the following relevant details
i. your choice of one of your two methods described in part b) to create your sample and state why you have chosen to use this method. (2 marks)
ii. the size of the sample you wish to create. Justify, statistically, why you have chosen that size. (2 marks)
iii. all the steps you need to calculate to create your sample. Each step should include a description and statistical calculations. (8 marks)
iv. your final sample. This should be detailed and presented in table form or as a list. You may wish to use Excel. (6 marks)
Cookies & Privacy Feedback
Systematic random sampling
Systematic random sampling is a type of probability sampling technique [see our article Probability sampling if you do not
know what probability sampling is]. With the systematic random sample, there is an equal chance (probability) of selecting
each unit from within the population when creating the sample. The systematic sample is a variation on the simple random
sample. Rather than referring to random number tables to select the cases that will be included in your sample, you select
units directly from the sample frame [see our article, Sampling: The basics, if you are unsure about the terms unit, sample,
sampling frame and population]. This article explains (a) what systematic random sampling is, (b) how to create a
systematic random sample, and (c) the advantages and disadvantages (limitations) of systematic random sampling.
Systematic random sampling explained
Creating a systematic random sample
Advantages and disadvantages (limitations) of systematic random sampling
Systematic random sampling explained
Imagine that a researcher wants to understand more about the career goals of students at the University of Bath. Let’s say
that the university has roughly 10,000 students. These 10,000 students are our population (N). In order to select a sample
(n) of students from this population of 10,000 students, we could choose to use a systematic random sample.
With systematic random sampling, there would an equal chance (probability) that each of the 10,000 students could be
selected for inclusion in our sample. Each of the 10,000 students is known as a unit, a case or an object (these terms are
sometimes used interchangeably; we use the word unit). If our desired sample size was around 400 students, each of these
students would subsequently be sent a questionnaire to complete (imagining we choose to collect our data using a
questionnaire).
Creating a simple random sample
To create a systemic random sample, there are seven steps: (a) defining the population; (b) choosing your sample size; (c)
listing the population; (d) assigning numbers to cases; (e) calculating the sampling fraction; (f) selecting the first unit; and
(g) selecting your sample.
GETTING STARTED QUANTITATIVE DISSERTATIONS FUNDAMENTALS
Quantitative Dissertations Dissertation Essentials Research Strategy Data Analysis
http://dissertation.laerd.com/privacy.php#cookies
http://dissertation.laerd.com/feedback.php
http://dissertation.laerd.com/probability-sampling-explained.php
http://dissertation.laerd.com/simple-random-sampling-an-overview.php
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/getting-started.php
http://dissertation.laerd.com/systematic-random-sampling.php
http://dissertation.laerd.com/fundamentals.php
http://dissertation.laerd.com/systematic-random-sampling.php
http://dissertation.laerd.com/data-analysis.php
STEP ONE: Define the population
STEP TWO: Choose your sample size
STEP THREE: List the population
STEP FOUR: Assign numbers to cases
STEP FIVE: Calculate the sampling fraction
STEP SIX: Select the first unit
STEP SEVEN: Select your sample
STEP ONE
Define the population
In our example, the population is the 10,000 students at the University of Bath. The population is expressed as N. Since we
are interested in all of these university students, we can say that our sampling frame is all 10,000 students. If we were only
interested in female university students, for example, we would exclude all males in creating our sampling frame, which
would be much less than 10,000.
STEP TWO
Choose your sample size
Let’s imagine that we choose a sample size of 100 students. The sample is expressed as n. This number was chosen
because it reflects the limit of our budget and the time we have to distribute our questionnaire to students. However, we
could have also determined the sample size we needed using a sample size calculation, which is a particularly useful
statistical tool. This may have suggested that we needed a larger sample size; perhaps as many as 400 students.
STEP THREE
List the population
To select a sample of 100 students, we need to identify all 10,000 students at the University of Bath. If you were actually
carrying out this research, you would most likely have had to receive permission from Student Records (or another
department in the university) to view a list of all students studying at the university. You can read about this later in the
article under Disadvantages (limitations) of systematic random sampling.
STEP FOUR
Assign numbers to cases
We now need to assign a consecutive number from 1 to N, next to each of the students. In our case, this would mean
assigning a consecutive number from 1 to 10,000 (i.e. N = 10,000; your population of students at the university).
STEP FIVE
Calculate the sampling fraction
Assuming we have chosen a sample size of 100 students, we now need to work out the sampling fraction, which is simply
the sample size selected (expressed as n) divided by the population size (N). In this case:
The sampling fraction tells us that we need to select 1 student in every 100 students from the population of 10,000 students
at the university. After doing this 100 times, we will have our sample of 100 students. However, first we need to select the
first unit (i.e., the first student), which starts the process of creating our sample.
STEP SIX
Select the first unit
Since we need to select 1 student in every 100 students, first we use a random number table to select the first student.
Imagine the first number in the random number table was 0009, we would ignore the first three digits and focus on the last
digit, 9, since this number fits between 0 and 100. As such, our first student would be the 9th on our list of 10,000 students.
STEP SEVEN
Select your sample
Now that we know the first unit, namely the 9th student on the list, we can select the other 99 students to make up our
sample of 100 students. Since we need to select 1 student in every 100 students from the list, we use the 9th student as the
starting point and then select every 100th student from this point. As such, we select the 109th student on the list, the 209th
student, the 309th student, and so forth.
Advantages and disadvantages (limitations) of systematic random sampling
The advantages and disadvantages (limitations) of systematic random sampling are explained below. Many of these are
similar to other types of probability sampling technique, but with some exceptions. Whilst systematic random sampling is
one of the “gold standards” of sampling techniques, it presents many challenges for students conducting dissertation
research at the undergraduate and master’s level.
Advantages of systematic random sampling
The aim of the systemic random sample is to reduce the potential for human bias in the selection of cases to be
included in the sample. As a result, the systemic random sample provides us with a sample that is highly
representative of the population being studied, assuming that there is limited missing data.
Since the units selected for inclusion within the sample are chosen using probabilistic methods, systemic random
sampling allows us to make statistical conclusions from the data collected that will be considered to be valid.
Relative to the simple random sample, the selection of units using a systematic procedure can be viewed as superior
because it improves the potential for the units to be more evenly spread over the population.
Disadvantages (limitations) of systematic random sampling
A systematic random sample can only be carried out if a complete list of the population is available.
If the list of the population has some kind of standardised arrangement (order/pattern), systematic sampling could pick
out similar cases rather than completely random ones. For example, when Student Records put together the list of the
10,000 students (our example), the list may have been ordered so that each record moved from a male to female
student (i.e., record #1 was a male student, record #2 a female student, record #3 a male student again, and so
forth). This may have been intentional or unintentional. Either way, if we select the 9th student in every hundred from
the list (as per our example; i.e., the 9th, 109th, 209th student, and so forth), we will always select a male student
(i.e., all odd numbers in the list are male students, whilst all even numbers are female students). This will lead to a
very biased sample. In reality, such a bias in the list should be easily seen and corrected. However, sometimes such a
standardised arrangement (order/pattern) may not be obvious or visible, resulting in sampling bias.
Attaining a complete list of the population can be difficult for a number of reasons:
Even if a list is readily available, it may be challenging to gain access to that list. The list may be protected by
privacy policies or require a length process to attain permissions.
There may be no single list detailing the population you are interested in. As a result, it may be difficult and time
consuming to bring together numerous sublists to create a final list from which you want to select your sample.
As an undergraduate and master?s level dissertation student, you may simply not have sufficient time to do this.
Many lists will not be in the public domain and their purchase may be expensive; at least in terms of the research
funds of a typical undergraduate or master’s level dissertation student.
In terms of human populations (as opposed to other types of populations; see the article: Sampling: The basics),
some of these populations will be expensive and time consuming to contact, even where a list is available.
Assuming that your list has all the contact details of potential participants in the first instance, managing the
different ways (postal, telephone, email) that may be required to contact your sample may be challenging, not
forgetting the fact that your sample may also be geographical scattered.
In the case of human populations, to avoid potential bias in your sample, you will also need to try and ensure that an
adequate proportion of your sample takes part in the research. This may require recontacting nonrespondents, can be
very time consuming, or reaching out to new respondents.
Contact Us Cookies & Privacy Copyright Terms & Conditions © 2012 Lund Research Ltd
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/contact-us.php
http://dissertation.laerd.com/privacy.php
http://dissertation.laerd.com/copyright.php
http://dissertation.laerd.com/t-and-c.php
Please Read very carefully before you start your assignment.
Declaration of Work – Academic Misconduct
Refer to the Student Assessment Guide book for full details.
Student Name (full name, including “English” name) ID Number (from your ID card)
Paper
FOUN046 Maths for Science
Assignment Name
Sampling in Statistics
Teacher
If you have any doubts or questions please see your teacher before you hand in your work.
STUDENT DECLARATION: you must sign this statement as part of your assignment.
I confirm and promise
that this assignment is entirely my own work and is original work
that I have not committed plagiarism when completing the attached piece of work
that source materials have been fully referenced
that this assignment, or substantial parts of it, have not been submitted for assessment
in any other programme or paper
that I have not committed any form of academic misconduct
that I have read and am aware of the academic misconduct details in my Student
Assessment Guide.
Signature(s)
Date
Cookies & Privacy Feedback
Stratified random sampling
Stratified random sampling is a type of probability sampling technique [see our article Probability sampling if you do not
know what probability sampling is]. Unlike the simple random sample and the systematic random sample, sometimes we
are interested in particular strata (meaning groups) within the population (e.g., males vs. females; houses vs. apartments,
etc.) [see our article, Sampling: The basics, if you are unsure about the terms unit, sample, strata and population]. With
the stratified random sample, there is an equal chance (probability) of selecting each unit from within a particular stratum
(group) of the population when creating the sample. This article explains (a) what stratified random sampling is, (b) how to
create a stratified random sample, and (c) the advantages and disadvantages (limitations) of stratified random sampling.
Stratified random sampling explained
Creating a stratified random sample
Advantages and disadvantages (limitations) of stratified random sampling
Stratified random sampling explained
Imagine that a researcher wants to understand more about the career goals of students at the University of Bath. Let’s say
that the university has roughly 10,000 students. These 10,000 students are our population (N). In order to select a sample
(n) of students from this population of 10,000 students, we could choose to use a simple random sample or a systematic
random sample. However, sometimes we are interested in particular strata (groups) within the population. Therefore, the
stratified random sample involves dividing the population into two or more strata (groups). These strata are expressed as
H.
For example, imagine we were interested in comparing the differences in career goals between male and female students at
the University of Bath. If this was the case, we would want to ensure that the sample we selected had a proportional
number of male and female students. This is known as proportionate stratification (as opposed to disproportionate
stratification, where the sample size of each of the stratum is not proportionate to the population size of the same stratum).
With stratified random sampling, there would an equal chance (probability) that each female or male student could be
selected for inclusion in each stratum of our sample. However, in line with proportionate stratification, the total number of
male and female students included in our sampling frame would only be equal if 5,000 students from the university were
male and the other 5,000 students were female. Since this is unlikely to be the case, the number of units that should be
selected for each stratum (i.e., the number of male and female students selected) will vary. We explain how this is
achieved in the next section: Creating a stratified random sample.
GETTING STARTED QUANTITATIVE DISSERTATIONS FUNDAMENTALS
Quantitative Dissertations Dissertation Essentials Research Strategy Data Analysis
http://dissertation.laerd.com/privacy.php#cookies
http://dissertation.laerd.com/feedback.php
http://dissertation.laerd.com/probability-sampling.php
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/systematic-random-sampling.php
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/systematic-random-sampling.php
http://dissertation.laerd.com/getting-started.php
http://dissertation.laerd.com/stratified-random-sampling.php
http://dissertation.laerd.com/fundamentals.php
http://dissertation.laerd.com/stratified-random-sampling.php
http://dissertation.laerd.com/data-analysis.php
Creating a stratified random sample
To create a stratified random sample, there are seven steps: (a) defining the population; (b) choosing the relevant
stratification; (c) listing the population; (d) listing the population according to the chosen stratification; (e) choosing your
sample size; (f) calculating a proportionate stratification; and (g) using a simple random or systematic sample to select
your sample.
STEP ONE: Define the population
STEP TWO: Choose the relevant stratification
STEP THREE: List the population
STEP FOUR: List the population according to the chosen stratification
STEP FIVE: Choose your sample size
STEP SIX: Calculate a proportionate stratification
STEP SEVEN: Use a simple random or systematic sample to select your sample
STEP ONE
Define the population
In our example, the population is the 10,000 students at the University of Bath. The population is expressed as N. Since we
are interested in all of these university students, we can say that our sampling frame is all 10,000 students. If we were only
interested in female university students, for example, we would exclude all males in creating our sampling frame, which
would be much less than 10,000.
STEP TWO
Choose the relevant stratification
If we wanted to look at the differences in male and female students, this would mean choosing gender as the stratification,
but it could similarly involve choosing students from different subjects (e.g., social sciences, medicine, engineering,
education, etc.), year groups, or some other variable(s). For the purposes of this example, we will use gender
(male/female) as our strata.
STEP THREE
List the population
We need to identify all 10,000 students at the University of Bath. If you were actually carrying out this research, you would
most likely have had to receive permission from Student Records (or another department in the university) to view a list of
all students studying at the university. You can read about this later in the article under Disadvantages (limitations) of
stratified random sampling.
STEP FOUR
List the population according to the chosen stratification
As with the simple random sampling and systematic random sampling techniques, we need to assign a consecutive number
from 1 to NK to each of the students in each stratum. As a result, we would end up with two lists, one detailing all male
students and one detailing all female students.
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/systematic-random-sampling.php
STEP FIVE
Choose your sample size
Let’s imagine that we choose a sample size of 100 students. The sample is expressed as n. This number was chosen
because it reflects the limit of our budget and the time we have to distribute our questionnaire to students. However, we
could have also determined the sample size we needed using a sample size calculation, which is a particularly useful
statistical tool. This may have suggested that we needed a larger sample size; perhaps as many as 400 students.
STEP SIX
Calculate a proportionate stratification
Imagine that of the 10,000 students, 60% of these are female and 40% male. We need to ensure that the number of units
selected for the sample from each stratum is proportionate to the number of males and females in the population. To
achieve this, we first divide the desired sample size (n) by the proportion of units in each stratum. Therefore, to calculate
the number of female students required in our sample, we divide 100 by 0.60 (i.e., 0.60 = 60% of the population of
students at the university), which gives us a total of 60 female students. If we do the same for male students, we get 40
students (i.e., 40% of students are male, where 100 ÷ 0.40 = 40). This means that we need to select 60 female students
and 40 male students for our sample of 100 students.
STEP SEVEN
Use a simple random or systematic sample to select your sample
Now that we have chosen to sample 40 male and 60 female students, we still need to select these students from our two
lists of male and female students (see STEP FOUR above). We do this using either simple random sampling or systematic
random sampling [click on the links to see what to do next].
Advantages and disadvantages (limitations) of stratified random sampling
The advantages and disadvantages (limitations) of stratified random sampling are explained below. Many of these are
similar to other types of probability sampling technique, but with some exceptions. Whilst stratified random sampling is one
of the ‘gold standards’ of sampling techniques, it presents many challenges for students conducting dissertation research at
the undergraduate and master’s level.
Advantages of stratified random sampling
The aim of the stratified random sample is to reduce the potential for human bias in the selection of cases to be
included in the sample. As a result, the stratified random sample provides us with a sample that is highly
representative of the population being studied, assuming that there is limited missing data.
Since the units selected for inclusion within the sample are chosen using probabilistic methods, stratified random
sampling allows us to make statistical conclusions from the data collected that will be considered to be valid.
Relative to the simple random sample, the selection of units using a stratified procedure can be viewed as superior
because it improves the potential for the units to be more evenly spread over the population. Furthermore, where the
samples are the same size, a stratified random sample can provide greater precision than a simple random sample.
Because of the greater precision of a stratified random sample compared with a simple random sample, it may be
possible to use a smaller sample, which saves time and money.
The stratified random sample also improves the representation of particular strata (groups) within the population, as
well as ensuring that these strata are not overrepresented. Together, this helps the researcher to compare strata, as
well as make more valid inferences from the sample to the population.
Disadvantages (limitations) of stratified random sampling
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/systematic-random-sampling.php
A stratified random sample can only be carried out if a complete list of the population is available.
It must also be possible for the list of the population to be clearly delineated into each stratum; that is, each unit from
the population must only belong to one stratum. In our example, this would be fairly simple, since our strata are male
and female students. Clearly, a student could only be classified as either male or female. No student could fit into both
categories (ignoring transgender issues).
Furthermore, imagine extending the sampling requirements such that we were also interested in how career goals
changed depending on whether a student was an undergraduate or graduate. Since the strata must be mutually
exclusive and collectively exclusive, this means that we would need to sample four strata from the population:
undergraduate males, undergraduate females, graduate males, and graduate females. This will increase overall sample
size required for the research, which can increase costs and time to carry out the research.
Attaining a complete list of the population can be difficult for a number of reasons:
Even if a list is readily available, it may be challenging to gain access to that list. The list may be protected by
privacy policies or require a length process to attain permissions.
There may be no single list detailing the population you are interested in. As a result, it may be difficult and time
consuming to bring together numerous sublists to create a final list from which you want to select your sample.
As an undergraduate and master’s level dissertation student, you may simply not have sufficient time to do this.
Indeed, it will be more complex and time consuming to prepare this list compared with simple random sampling
and systematic random sampling.
Many lists will not be in the public domain and their purchase may be expensive; at least in terms of the research
funds of a typical undergraduate or master’s level dissertation student.
In terms of human populations (as opposed to other types of populations; see the article: Sampling: The basics),
some of these populations will be expensive and time consuming to contact, even where a list is available.
Assuming that your list has all the contact details of potential participants in the first instance, managing the
different ways (postal, telephone, email) that may be required to contact your sample may be challenging, not
forgetting the fact that your sample may also be geographical scattered.
In the case of human populations, to avoid potential bias in your sample, you will also need to try and ensure that an
adequate proportion of your sample takes part in the research. This may require recontacting nonrespondents, can be
very time consuming, or reaching out to new respondents.
Contact Us Cookies & Privacy Copyright Terms & Conditions © 2012 Lund Research Ltd
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/systematic-random-sampling.php
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/contact-us.php
http://dissertation.laerd.com/privacy.php
http://dissertation.laerd.com/copyright.php
http://dissertation.laerd.com/t-and-c.php
Cookies & Privacy Feedback
Simple random sampling
Simple random sampling is a type of probability sampling technique [see our article, Probability sampling, if you do not
know what probability sampling is]. With the simple random sample, there is an equal chance (probability) of selecting each
unit from the population being studied when creating your sample [see our article, Sampling: The basics, if you are unsure
about the terms unit, sample and population]. This article (a) explains what simple random sampling is, (b) how to create a
simple random sample, and (c) the advantages and disadvantages of simple random sampling.
Simple random sampling explained
Creating a simple random sample
Advantages and disadvantages of simple random sampling
Simple random sampling explained
Imagine that a researcher wants to understand more about the career goals of students at a single university. Let’s say that
the university has roughly 10,000 students. These 10,000 students are our population (N). Each of the 10,000 students is
known as a unit (although sometimes other terms are used to describe a unit; see Sampling: The basics). In order to select
a sample (n) of students from this population of 10,000 students, we could choose to use a simple random sample.
With simple random sampling, there would an equal chance (probability) that each of the 10,000 students could be selected
for inclusion in our sample. If our desired sample size was around 200 students, each of these students would subsequently
be sent a questionnaire to complete (imagining we choose to collect our data using a questionnaire).
Creating a simple random sample
To create a simple random sample, there are six steps: (a) defining the population; (b) choosing your sample size; (c)
listing the population; (d) assigning numbers to the units; (e) finding random numbers; and (f) selecting your sample.
STEP ONE: Define the population
STEP TWO: Choose your sample size
STEP THREE: List the population
GETTING STARTED QUANTITATIVE DISSERTATIONS FUNDAMENTALS
Quantitative Dissertations Dissertation Essentials Research Strategy Data Analysis
http://dissertation.laerd.com/privacy.php#cookies
http://dissertation.laerd.com/feedback.php
http://dissertation.laerd.com/probability-sampling.php
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/getting-started.php
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/fundamentals.php
http://dissertation.laerd.com/simple-random-sampling.php
http://dissertation.laerd.com/data-analysis.php
STEP FOUR: Assign numbers to the units
STEP FIVE: Find random numbers
STEP SIX: Select your sample
STEP ONE
Define the population
In our example, the population is the 10,000 students at the single university. The population is expressed as N. Since we
are interested in all of these university students, we can say that our sampling frame is all 10,000 students. If we were only
interested in female university students, for example, we would exclude all males in creating our sampling frame, which
would be much less than 10,000 students.
STEP TWO
Choose your sample size
Let’s imagine that we choose a sample size of 200 students. The sample is expressed as n. This number was chosen
because it reflects the limit of our budget and the time we have to distribute our questionnaire to students. However, we
could have also determined the sample size we needed using a sample size calculation, which is a particularly useful
statistical tool. This may have suggested that we needed a larger sample size; perhaps as many as 400 students.
STEP THREE
List the population
To select a sample of 200 students, we need to identify all 10,000 students at the university. If you were actually carrying
out this research, you would most likely have had to receive permission from Student Records (or another department in
the university) to view a list of all students studying at the university. You can read about this later in the article under
Disadvantages of simple random sampling.
STEP FOUR
Assign numbers to the units
We now need to assign a consecutive number from 1 to N, next to each of the students. In our case, this would mean
assigning a consecutive number from 1 to 10,000 (i.e., N = 10,000; the population of students at the university).
STEP FIVE
Find random numbers
Next, we need a list of random numbers before we can select the sample of 200 students from the total list of 10,000
students. These random numbers can either be found using random number tables or a computer program that generates
these numbers for you.
STEP SIX
Select your sample
Finally, we select which of the 10,000 students will be invited to take part in the research. In this case, this would mean
selecting 200 random numbers from the random number table. Imagine the first three numbers from the random number
table were:
0011 (the 11th student from the numbered list of 10,000 students)
9292 (the 9,292nd student from the list)
2001 (the 2,001st student from the list)
We would select the 11th, 9,292nd and 2,001st students from our list to be part of the sample. We keep doing this until we
have all 200 students that we want in our sample.
Advantages and disadvantages of simple random sampling
The advantages and disadvantages of simple random sampling are explained below. Many of these are similar to other
types of probability sampling technique, but with some exceptions. Whilst simple random sampling is one of the ‘gold
standards’ of sampling techniques, it presents many challenges for students conducting dissertation research at the
undergraduate and master’s level.
Advantages of simple random sampling
The aim of the simple random sample is to reduce the potential for human bias in the selection of cases to be included
in the sample. As a result, the simple random sample provides us with a sample that is highly representative of the
population being studied, assuming that there is limited missing data.
Since the units selected for inclusion in the sample are chosen using probabilistic methods, simple random sampling
allows us to make generalisations (i.e., statistical inferences) from the sample to the population. This is a major
advantage because such generalisations are more likely to be considered to have external validity.
Disadvantages of simple random sampling
A simple random sample can only be carried out if the list of the population is available and complete.
Attaining a complete list of the population can be difficult for a number of reasons:
Even if a list is readily available, it may be challenging to gain access to that list. The list may be protected by
privacy policies or require a lengthy process to attain permissions.
There may be no single list detailing the population you are interested in. As a result, it may be difficult and time
consuming to bring together numerous sublists to create a final list from which you want to select your sample.
As an undergraduate and master?s level dissertation student, you may simply not have sufficient time to do this.
Many lists will not be in the public domain and their purchase may be expensive; at least in terms of the research
funds of a typical undergraduate or master’s level dissertation student.
In terms of human populations (as opposed to other types of populations; see the article: Sampling: The basics),
some of these populations will be expensive and time consuming to contact, even where a list is available.
Assuming that your list has all the contact details of potential participants in the first instance, managing the
different ways (e.g., postal, telephone, email) that may be required to contact your sample may be challenging,
not forgetting the fact that your sample may also be geographical scattered.
In the case of human populations, to avoid potential bias in your sample, you will also need to try and ensure that an
adequate proportion of your sample takes part in the research. This may require recontacting nonrespondents, can be
very time consuming, or reaching out to new respondents.
If you are an undergraduate or master’s level dissertation student considering using simple random sampling, you may also
want to read more about how to put together your sampling strategy [see the section: Sampling Strategy].
Contact Us Cookies & Privacy Copyright Terms & Conditions © 2012 Lund Research Ltd
http://dissertation.laerd.com/sampling-the-basics.php
http://dissertation.laerd.com/sampling-strategy.php
http://dissertation.laerd.com/contact-us.php
http://dissertation.laerd.com/privacy.php
http://dissertation.laerd.com/copyright.php
http://dissertation.laerd.com/t-and-c.php
Student Name:_____________________________
FOUN046 Mathematics
for Science
Student Book 2
Foundation Year
~ 2 ~
Copyright 2016
University of Otago Foundation Year
UOLCFY
PO Box 56
Dunedin
New Zealand
~ 3 ~
Contents
Information for Students ………………………………………………………………………………………………………… 4
Overview of Learning Outcomes: …………………………………………………………………………………………….. 8
Week 6 Repeated Lecture: Straight Lines and Exponential Graphs ………………………………………… 10
Week 8 Lecture: Probability ………………………………………………………………………………………………… 20
Week 9 Lecture: Statistics: Types of data and Measures of Statistics. …………………………………………. 27
Week 10 Lecture: Normal Distribution ……………………………………………………………………………………. 37
Week 11 Lecture: Estimation …………………………………………………………………………………………………. 47
Week 12 Lecture: Confidence Intervals. ………………………………………………………………………………….. 50
Week 7 Tutorial 2: Straight Lines ……………………………………………………………………………………………. 53
Week 7 Tutorial 3: Exponential Graphs………………………………………………………………………………… 60
Week 8 Tutorial 1: Growth and Decay Models. ………………………………………………………………………… 62
Week 8 Tutorial 2: Probability, definition, union and intersection of events. ………………………………. 67
Week 8 Tutorial 3: Complement of Events, Odds of Events. ………………………………………………………. 70
Week 9 Tutorial 1: Successive Events and Decision Diagrams. …………………………………………………… 73
Week 9 Tutorial 2: Two Way Tables in Probability ……………………………………………………………………. 79
Week 9 Tutorial 3: Measures of Central Tendency …………………………………………………………………… 86
Week 10 Tutorial 1: Measures of Spread, range and interquartile range. ……………………………………. 92
Week 10 Tutorial 2: Measures of Spread, standard deviation. …………………………………………………… 97
Week 10 Tutorial 3: Normal Distribution Tables …………………………………………………………………….. 104
Week 11 Tutorial 1: Normal Distribution Applications …………………………………………………………….. 107
Week 11 Tutorial 2: Estimation ……………………………………………………………………………………………. 110
Week 11 Tutorial 3: Working in groups: Statistical Graphs ………………………………………………………. 114
Week 12 Tutorial 1: Confidence Intervals ………………………………………………………………………………. 133
~ 4 ~
Information for Students
Paper Outcomes
Mathematical problem solving is an important technique for the solution of problems
in any area of society. This paper aims to develop the ideas and concepts from
Mathematics in such a way that the student develops their problem solving techniques
in Mathematics and can apply the processes to Science. This paper will give the
student a sound knowledge of mathematical concepts and prepare them for the
demands of 100-level Science, Mathematical and Statistical papers at the University
of Otago.
Learning Outcomes
By the end of the paper, successful students will be able to:
1. explore the use of formulae, relationships, equations, expressions and
statistical techniques in a variety of contexts
2. use number, algebra, probability, statistics and trigonometry in different
situations and interpret their results
3. develop their mathematical skills in number, algebra, trigonometry,
probability, statistics and some curve sketching
4. gain and demonstrate an understanding and appreciation of problem solving
techniques in a variety of contexts.
Paper Outline
Book 2
Topic 5: Graphs for Science
Topic 6: Probability
Topic 7: Statistics
Paper Materials
Two FOUN046 Student Books are supplied by Foundation Year. They contain
lecture notes and tutorial exercises.
Students will also get extra handouts and notes in tutorials.
Students are encouraged to use a range of mathematical books from the university
libraries and online maths resources to help them develop and explore their
mathematical studies more.
The weekly programme for week 7 – 12 is on the next page. The programme for
Week 1 -7 is in Book 1.
~ 5 ~
Assessment
The final grade for FOUN046 Mathematics for Science will be based on one internal
test (20%), one written assignment (10%) and a final examination worth 70%.
Type Learning Outcomes Weighting
Test 1 1-4: topics 1 , 2 ,3 and 4 20%
Assignment
(written)
1-4: Statistical Sampling 10%
Final Examination
(written)
1-4: topics 5, 6 and 7. 70%
Total 100%
Week Lecture (1 hour) Tutorial (1 hour) Tutorial (1 hour) Tutorial (1 hour)
1
2
3
4
5
6 Topic 5:
Graphing for
Science
7 Mid Term tests Mid term tests
Maths Test (20%)
St lines:properties, sketch
and St line models
Sketch exponential
graphs
8 Topic 6:
Probability
Growth and Decay
models
Union, intersection of
events
Complement of events,
Odds of events
Assignment: Sampling
issued and details
9 Topic 7: Statistics:
Types of Data
Measures of
statistics
Successive Events,
Decision Diagrams
Assignment queries
Two Way Tables
Assignment queries
Measures of Central
Tendency & Spread 1
Assignment queries
10 Normal
Distribution
Measures of Central
Tendency & Spread 2
Assignment queries
Measures of Central
Tendency & Spread 3
Assignment queries
Normal Distribution
tables
Assignment queries
11 Estimation
Assignment:
Sampling due
(10%)
Normal Distribution
Applications
Estimation Working in Groups:
Statistical Graphs:
drawing and reading of
graphs.
12 Confidence
Intervals
Confidence Intervals
Revision and catch up Revision and catch up
~ 6 ~
Assignment: Statistical Sampling: Week 8 – 11: Students will be issued with a
problem solving exercise covering the topic Sampling. They are expected to complete
the assignment and communicate their mathematical ideas using the correct
mathematical language and produce their work to a high standard. Students will have
about two weeks to complete the assignment.
This assignment must be handed in by 11 am on the due date, which will be a
Monday.
Final External Examination Duration: Two hours
Included: all work covered in topics 5, 6 and 7.
The examination will consist of four compulsory questions.
Student Expectations
LECTURES
Students will have one lecture a week. These lectures cover one whole topic or
part of a topic in detail. Some topics are only taught during the tutorials particularly
in the first few weeks. See weekly programme for details.
Students should bring the student book to all lectures and tutorials.
Students should take notes during the lectures. These notes will help students with
their own study.
Students should study the lecture notes before and after the lecture. These notes
should be reviewed before the tutorial for that lecture.
After a lecture, the student may need to complete any accompanying exercises as part
of their own study. Queries about the exercises can be brought to consultation.
Students should complete any exercises specified at the lecture before the tutorial on
that topic. The tutorial may make use of the ideas contained in these exercises.
Students who encounter any problems or difficulties should discuss these with the
teacher during the follow up tutorial sessions or at consultation time.
Question Type Marks allocation
1 10 parts, short answer, 4 marks each 40
2 – 4 3 problem solving context questions,
worth 20 marks each.
60
Total 100
~ 7 ~
TUTORIALS
Students will have three tutorials a week.
A tutorial may review lecture material by practicing relevant tutorial exercises.
A tutorial may be self-contained. It may consist of reviewing, or learning, concepts,
practicing them and solving problems which include these concepts.
Tutorial time will also provide students with time to ask questions, to consolidate
mathematical concepts learnt and to work as part of a group.
Students should bring to each tutorial:
student book with completed lecture notes
attempted exercises
exercise book, A4 refill paper and graph paper if needed
calculator.
During tutorials, students should:
clarify any concepts that they find difficult
seek help from the teacher for any individual problems arising from the
tutorials
if appropriate, actively work as part of a group.
After tutorials, students should:
complete any set homework
do any questions in exercises that were not completed in tutorial time.
Assessment Procedures
For all assessment procedures, please refer to the Student Assessment Guide
booklet given to you at Orientation. This is also posted on Blackboard.
~ 8 ~
Overview of Learning Outcomes:
Topic 5: GRAPHING IN SCIENCE
At the end of this topic a student should be able to:
plot and describe points on a graph
find the gradient of a line
describe the gradient of a line in terms of its direction and rate of change
sketch lines using the gradient and y-intercept method
sketch lines using the intercepts method
describe the features of a graph in words
sketch simultaneous linear equations
solve graphically word problems involving lines
describe the features of an exponential graph
sketch exponential graphs
use exponents and logarithms in growth model problems
use exponents and logarithms in decay model problems
graph exponential curves and use them to solve problems.
Topic 6: PROBABILITY
At the end of this topic, a student should be able to:
define probability
calculate probabilities
use and understand correct probability notations
find the probability of the union, intersection and complement of events
use Venn diagrams to find all the possible outcomes and calculate the probabilities of events
calculate the odds for an event
use decision diagrams to find all the possible outcomes and calculate the probabilities of
events with replacements and without replacements
use two way tables to find all the possible outcomes and calculate the probabilities of events
apply probability theory to a range of problems.
~ 9 ~
Topic 7: STATISTICS
At the end of this topic, a student should be able to:
identify and classify quantities into categorical data and numerical data, discrete and
continuous data
organize and interpret categorical and discrete data using statistical graphs such as bar graphs,
pie charts and line graphs
organize, draw and interpret continuous data using statistical graphs such as histograms,
frequency polygons and cumulative frequency curves
for a set of data calculate and interpret measures of central tendency such as mean, mode and
median
for a set of data calculate and interpret measures of dispersion, range, interquartile range,
percentiles and standard deviation
define and describe a range of sampling techniques
for a set of data, generate a statistical sample
interpret the spread of data in a normal distribution curve
calculate z-scores in a standard normal distribution
explain the significance of the z-score value
describe a normal distribution
interpret the spread of data in a normal distribution curve
calculate data values described as a normal distribution
calculate probabilities using the standard normal distribution tables
use estimation to compare samples and populations
use confidence intervals to describe samples or populations.
~ 10 ~
Week 6 Repeated Lecture: Straight Lines and Exponential Graphs
This lecture is also in book 1. Transfer your notes to this book so that you can use them in
tutorials.
Graphs of Linear Equations
0 0
−1
−2
−3
−4
−3
2
3
1 2 3 4 −1
1
−2 −4
4
• A(4,2)
(−3, 4) B•
• C (1, −3)
-coordinate y-coordinate
y
origin
The points on a graph are called co-ordinates.
Graphing Linear Equations
A straight line consists of infinitely many points. However, we need to plot only two of them
to sketch a graph of the line.
Two methods:
(i) When equations are given in the gradient-intercept form
C
y = m + c y-intercept
gradient(slope)
y
•
A graph gives a visual picture of the relationship between two variables.
~ 11 ~
Example: Sketch the line with equation y =3 + 2 using the gradient y-intercept method.
Example: Sketch the line with equation y +
3
4
= –1 using the gradient y-intercept method.
(ii) The axes intercepts method.
Example:
Use the axis intercepts method to sketch the following lines
(a) + 2y = 8 (b) 2 – y – 6 = 0.
y
Put 0x to find the y axis -intercept
Put 0y to find the axis -intercept
c
x
~ 12 ~
Simultaneous linear equations
Example: + 2y = 4 equation 1
7 – 5y = 9 equation 2
Simultaneous equations will be solved graphically.
Graphical solution of simultaneous equations
Example: Solve, graphically, the equations 2 + y = 10 and y = 3 − 5
Graphs of Exponential Equations
Exponential equations can be found all around us.
Examples:
Amoebae reproduce by dividing after a certain time.
Radioactive substances have “half lives”.
Double time or compounding models.
Population models.
Simultaneous equations are two equations containing two unknowns.
When we solve the above equations, we are trying to find the values of the unknown
quantities, which are true for each of the equations.
The graphs of the two equations are drawn and the solution to the equations is the
coordinates of the point of intersection of the graphs, since these values of and y will
satisfy both equations.
~ 13 ~
Example:
The world’s human overall population is growing at about 1.13% per year. That is each year
the population will be 1.13% more than it was at the start of the year.
The population on 6 November 2015 was about 7.3 billion.
Based on this:
By 2016 it would be 7.3 billion × (1.0113) ≈ 7.4 billion
By 2025 it would be 7.3 billion × (1.0113)10 ≈ 8.2 billion
By 2065 it would be 7.3 billion × (1.0113)50 ≈ 12.8 billion
By 2075 it would be 7.3 billion × (1.0113)60 ≈ 14.3 billion
As you can imagine these figures have huge implications for how countries manage their
resources, food supply, water, energy etc.
Remember: Exponential equations look like
xy b c
Example:
Graph the exponential equation y = 2x .
x
y = 2x
All exponential graphs are similar in shape.
2
y
x
3 2 1 1 3 4
4
8
6
2
2
y
x
3 2 1 1 3 4
4
8
6
2
2
Almost doubled in 60 years
~ 14 ~
Example: Sketch y = 2x + 5.
Example: Sketch y = 2x – 3
Example: Sketch y = (
1
2
)
. We could consider this as y = 2− .
Your sketch must have:
labelled the y-intercept with its exact value
labelled the x-intercept, if it exists, with its exact value
indicated the minimum or maximum value that the graph
is approaching
the correct shape.
~ 15 ~
Growth and Decay Models.
Growth model – exponential increases and exponent is positive
Decay model – exponential decreases (or negative growth) and exponent is negative.
Applications
Example: Population Growth
In 2009 India had a population of around 1.15 billion, and
it is estimated that the population will double in 44 years.
If the population growth continues at the same rate using
the growth model P = P0 2
what will be the population in:
(a) 10 years time (b) 35 years time (c) 2050?
Example: Radioactive Decay
Radioactive gold-198 (198Au) is used in imaging the structure of
the liver, has a half-life of 2.67 days and its decay model is A =
A0 (2)
−
ℎ
If a dose of 50 milligrams of the isotope is given to a patient, how
many milligrams will be left after (a)
1
2
day (b) 1 week?
Give your answers to 2 decimal places.
~ 16 ~
The exponential equation with base ℮
On the calculator ℮ ≈ 2.718 281 828 235 …………..
So, y = ℮ defines the exponential equation with base ℮.
℮ is a natural base that nature often follows for growth and decay.
The graph of y = ℮ : it looks similar to any exponential graph.
Example:
The current flowing in an electrical circuit after it is switched on is given by:
I(t) = 75℮0.015 amps, where I is the current and t is time measured in seconds.
How much current is flowing in the circuit after 1 minute?
●
x
y
1
y = ℮
~ 17 ~
Drawing graphs of Exponential Growth or Decay.
Example: Marketing: Nutrition
A company wants to market a new energy drink.
Their market research suggests that the sale of this new product will follow a limited growth
model.
The research finds:
S = 200(1 − ℮−0.05 ), where
S represents the sales in thousands of dollars and t represents the time in weeks.
The marketing team now has to report back their findings, so that the company can decide
whether to produce the new drink or not.
(a) Calculate the number of drinks expected to be sold after the first week.
(b) Calculate the number of drinks expected to be sold after 5, 20, 35, 45, 50 and 52
weeks.
(c) Sketch a graph of the sales expected over the first year.
(d) Comment on the general trend in sales and what the company can expect to happen.
(a) S = 200(1 − ℮−0.05(1) )
S = 9.754…
The company can expect to sell about $ 9, 754 worth of drinks.
(b) Using the formula and your calculator to calculate the start point, end point and at
least three other points on your graph.
t 5 20 35 45 50 52
S
(×1000)
44 126 165 179 184 185
~ 18 ~
(c)
(d) General trend is an increase in sales.
Initially the company can expect this increase to rise quite quickly until about week
35. Then the sales continue to rise but at a much slower rate.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40 50 60
Expected Sales of Energy Drinks
S
(×1000)
t
●
●
●
● ●
●
S = 200(1 − ℮−0.05 )
~ 19 ~
Pre Tutorial Exercise: complete before the tutorials on this topic.
1.
Plot the above points on the grid below.
2.
Check your answers with your class mates and the teacher.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
y
A (3,5) B (2,4) C (–3,2) D (7,0) E (–4,–2) F (0,–3) G (5, 3) H (0,0)
Find the coordinates of the points labeled in the grid below:
C
A
B
E
D
G
H
F
~ 20 ~
Week 8 Lecture: Probability
Probability
Probability is the science of certainty and uncertainty.
It is used in psychology, weather forecasting, finance, physics, economics and many other
areas including law.
In Mathematics, probability is a measure of how likely it is that something will happen. It is a
measure of the chance of something happening.
Some language:
Experiment: An experiment is an activity where observations are made.
Outcome: An outcome is the result of an experiment.
Sample Space: The Sample Space, S , is the set of all the possible outcomes of an
experiment.
Event: An event is a subset of the sample space. It is usually something that we are interested
in measuring
The measure of the probability of event E happening is
This formula assumes that all events in the sample space have the same chance of happening
(are equally likely).
Example: There are two counters. One side of each is red (R), the other blue (B). One is
round and the other is an ellipse. The two counters are tossed and the colours seen are
recorded.
(i) List the sample space.
(ii) List the event E: both counters are the same colours.
(iii) What is the probability of E happening, P(E)?
P (E) =
=
( )
( )
~ 21 ~
A cube with a number of dots, from 1 to 6, on each face is called a die.
Example: A fair die is thrown and the number on the uppermost face is recorded.
(a) List the sample space.
(b) E is the event that the number recorded is greater than 4.
(i) List E.
(ii) What is P (E)?
Probability values are always between 0 and 1.
0 1
0 ≤ P(E) ≤ 1
Example: Bag A has 3 cards numbered 1, 2 and 3. Bag B has 3 cards labelled a, b, and c. The
experiment is to choose one card from bag A and one from bag B.
(i) List the sample space.
(ii) E is the event that 2 and c are chosen. What is P(E)?
The probability is zero. The
event cannot happen. It is
impossible.
The probability is one. The event
will definitely happen. It is a
certainty.
The bigger the probability, the more likely the event is to happen.
~ 22 ~
(iii) D is the event that an odd number and a vowel are chosen. What is P(D)?
English Alphabet
5 vowels 21 consonants
a, e, i, o, u
Two events A and B: Union of Events
Let two events be A and B, then P (A ∪ ) represents the probability of A union B.
It measures the probability that at least one of the two events will happen. That is union
measures the probability that event A happens, event B happens or both happen.
Example: A survey was carried out of all the students in a Foundation Year class. The survey
results were that 22 students like Tin-Foil Dance music (TDM), 18 like Heroine Pop music
(HP) and 7 like both.
a) Draw a Venn diagram to represent this information.
b) What is n(S)?
c) What is the probability that student surveyed liked TDM music only?
( ) ( ) ( ) ( )P A B P A P B P A B
A
B
46
( ∩ )
~ 23 ~
Example: A survey of Health Centres in a city shows that 750 centres offer patients access to
a physiotherapy programme, 640 offer access to a diabetes management programme and 280
offer access to both.
a) Draw a Venn diagram to represent this information.
b) What is the probability that a Health Centre offers its clients access to only one of
these programmes?
Example: A small town has two radio stations, an AM station and an FM station. As part of a
critical incident response survey residents of the town were asked which radio station they
listened to regularly. 100 residents were surveyed and produced the following results.
66 residents listened to the AM station regularly, 47 listened to the FM station and 31 listened
to both.
a) Draw a Venn diagram to represent this information.
Use your Venn diagram to calculate the probability that a resident in the survey
b) listens to the AM station, but not the FM station regularly
c) did not listen to either station regularly.
~ 24 ~
Complement of A
Complement of A is written A′. P(A′) is a measure of the probability that event A will not
happen.
Example: S = {5, 6, 7, 8, 9, 10, 11, 12} A = {5, 6, 10, 11, 12}
Find P(A′)
Application: Odds of an Event: Often in chance situations it is customary to speak of odds
for (or against) an event E happening rather than the probability of it happening.
If ( )P E is the probability of event E , then we define:
Odds for E as ‘( ) : ( )P E P E
Odds against E as ‘( ) : ( )P E P E
Express as a ratio in its lowest terms.
Example: If you roll a fair die once, the probability of rolling a 3 is
1
6
, whereas the odds in
favour of rolling a 3 is:
‘( ) : ( )
1 5
:
6 6
1 : 5
P E P E
The odds against the event happening is:
‘( ) : ( )
5 1
:
6 6
5 : 1
P E P E
P(A) + P(A′) = 1
P (A’) = 1 – P (A)
A
A’
~ 25 ~
Example: From a survey involving 1000 people in a hot country, it was found that 500
people had tried diet cola, 600 had tried zero cola and 200 had tried both.
a) Draw a Venn diagram for this data.
b) A person is chosen at random, what is the probability that they have tried the diet or
the zero cola?
c) What are the odds for the event in b)?
d) A person is chosen at random, what is the probability that they have tried one of the
colas but not both?
e) What are the odds against the event in d)?
Successive Events: Events that happen one after the other are called successive events.
The probability of both events happening is found by multiplying the probabilities of the
events together. This is known as the multiplication law.
Example: A fair counter, with one red face and one blue face, is tossed and then a fair die is
thrown. Find the probability that the red face is up on the counter and there is a six on the die.
Example: A …