# Teaching and learning plan on inferential statistics

```Teaching & Learning Plan
Inferential Statistics
for Proportions
Leaving Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have
and also to knowledge which may be necessary in order to support them in
accessing this new topic.
Learning Outcomes outline what a student will be able to do, know and
understand having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/
or Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching
and learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing
the topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on possible lines
of inquiry and gives details of the key student tasks and teacher questions which
move the lesson forward.
ii.
Student Activities – Possible Responses: Gives details of possible student
reactions and responses and possible misconceptions students may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
2
Teaching & Learning Plans:
Inferential Statistics for
Proportions
Aims1
The aim of this series of lessons is:
• To understand why sampling is important.
• To identify that there is a link between statistics and probability.
• To understand the phrase “inferential statistics”.
• To understand the link between the 95% confidence and the empirical rule.
• To recognise how sampling variability influences the use of sample
information to make statements about the population.
• To understand what factors must be kept in mind when sample information
is used to make statements about the population.
• To apply the idea of a confidence interval.
• To understand that a sample proportion may not be the same as the
population proportion.
• To evaluate margin of error for a population proportion.
• To analyse that increasing the sample size decreases the size or radius of
the margin of error.
• To observe that doubling the sample size does not halve the size or radius
of the margin of error.
• To analyse the idea of hypothesis testing.
• To understand how to conduct a hypothesis test on a population
proportion using the margin of error.
formalises the intuitive notion about the size of a
• To understand that
95% confidence interval for a population proportion.
• To apply knowledge and skills relating to statistics to solve problems.
• To use mathematical language, both written and spoken, to communicate
understanding effectively.
1
This Teaching & Learning Plan illustrates a number of strategies to support the implementation of
Literacy and Numeracy for Learning and Life: the National Strategy to Improve Literacy and Numeracy
among Children and Young People 2011-2020 (Department of Education & Skills 2011). Attention to the
recommended strategies will be noted at intervals within the Lesson Interaction Section of this Teaching and
Learning Plan.
© Project Maths Development Team 2014
www.projectmaths.ie
3
Teaching & Learning Plan: Inferential Statistics for Proportions
Prior Knowledge
Students have prior knowledge of:
• Quantifying probabilities from Teaching and Learning Plan 1:
Introduction to Probability
• Task on Household Sizes from page 2 of the Workshop 10 booklet on
www.projectmaths.ie
• The Empirical Rule
• Sampling Variability
• The difference between a population and a sample.
• Simple random sampling
• Describing the shape, centre and spread of distributions
• The Data Handling Cycle.
Learning Outcomes
As a result of studying this topic, students will be able to:
• Calculate the margin of error for a 95% confidence interval for a
population proportion using .
• Make a statement about the population proportion using a 95%
confidence interval.
• Conduct a hypothesis test on a population proportion using the margin
of error.
• Understand how inferential statistics might be applied in every-day
situations.
Catering for Learner Diversity
In class, the needs of all students, whatever their level of ability level,
are equally important. In daily classroom teaching, teachers can cater
for different abilities by providing students with different activities and
assignments graded according to levels of difficulty so that students can
work on exercises that match their progress in learning. Less able students,
may engage with the activities in a relatively straightforward way while the
more able students should engage in more open-ended and challenging
activities.
In interacting with the whole class, teachers can make adjustments to meet
the needs of all of the students.
Apart from whole-class teaching, teachers can utilise pair and group work to
encourage peer interaction and to facilitate discussion. The use of different
grouping arrangements in these lessons should help ensure that the needs
of all students are met and that students are encouraged to articulate their
mathematics openly and to share their learning.
© Project Maths Development Team 2014
www.projectmaths.ie
4
Teaching & Learning Plan: Inferential Statistics for Proportions
Relationship to Leaving Certificate Syllabus
Sub-Topic
Learning Outcomes
Students
Students working at OL
should be able to
1.4 Statistical
reasoning
with an aim
to becoming
a statistically
aware
consumer
• discuss populations and
samples
Students working at HL
should be able to
• decide to what extent
conclusions can be
generalised
• construct 95%
1.7 Analysing, • recognise how sampling
confidence intervals for
variability influences the use
interpreting
the population mean
of sample information to
and drawing
from a large sample
inferences
and for the population
population
from data
proportion, in both cases
• use appropriate tools to
using z tables
describe variability drawing
population from the sample
• interpret the analysis and
relate the interpretation to
the original question
• make decisions based on the
empirical rule
• recognise the concept of a
hypothesis test
• calculate the margin of
error ( ) for a population
proportion*
• conduct a hypothesis test
on a population proportion
using the margin of error
* The margin of error referred to here is the maximum value of the radius of
the 95% confidence interval.
Resources Required
Formulae and Tables, whiteboards, rulers, GeoGebra and calculators.
60 yellow unifix cubes, blocks or pieces of card.
140 non-yellow unifix cubes, blocks or pieces of card.
© Project Maths Development Team 2014
www.projectmaths.ie
5
Teaching & Learning Plan: Inferential Statistics for Proportions
Lesson Interaction
Input
Student Activities: Possible Teacher’s Supports and
and Expected Responses
Actions
Checking Understanding
Teacher Reflections
Section A – Sampling variability and confidence intervals
»» In today’s lesson we are going to
carry out a statistical investigation.
From the investigation we would
question: “What proportion of Irish
post-primary students keep their
mobile phone under their pillow at
night?”
»» On one half of the
board write the
question “What
proportion of Irish
post-primary students
keep their mobile
phone under their
pillow at night?”
»» When we say “Irish post‑primary
• Many.
students” how many Irish
post‑primary students do we mean? • 50,000 students.
»» Do students understand that
when we say “Irish postprimary students” we mean
all of them?
• 300,000 students.
• All of the post-primary
students.
• All post-primary students
in Ireland.
»» In statistics when we refer to “all”
or “everybody”, what name do we
give to this group?
»» So we would like to know what
proportion of the population of
Irish post-primary students keep
their mobile phone under their
pillow at night? We are interested
population.
© Project Maths Development Team 2014
• The population.
www.projectmaths.ie
»» Write the word
“population” on the
board and encourage
students to write
an explanation of
the term in their
copybooks.
»» Do students understand that,
in statistics, the complete set
of people/items is known as
"the population"?
»» Do students understand
that when we say “Irish
post-primary students” we
mean the population of Irish
post‑primary students?
KEY: » next step
6
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning
Student Activities: Possible
and Expected Responses
Teacher’s Supports and Actions
Assessing the Learning
»» Can anybody
suggest how we
question?
• We need to survey some
people.
»» Write the second stage of the datahandling cycle “Collect Data” on the
board. Link it to the first stage by means
of an arrow.
»» Can students identify
the second stage of
the data-handling
cycle?
»» Add to the diagram of the datahandling cycle to highlight the two
general approaches to gathering dataconducting a census and sampling.
»» Do students
understand that,
in general, when
gathering data you
can survey the entire
population or a subset
of the population?
• We need some data.
Teacher Reflections
here in the room.
»» If we were to
gather the data
ourselves, how
many students
• We could ask them all.
students.
• We could take a sample of
students.
• 100.
• 1,000.
• All the students in our
school.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
7
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning Tasks: Student Activities: Possible and
Teacher Input
Expected Responses
Teacher’s Supports Assessing the Learning
and Actions
»» Can you explain why
you might choose one
approach over the
other?
and takes a long time.
the important
of sampling vs.
conducting a
census to the
flow chart.
• It wouldn’t be possible to ask
every post‑primary student.
»» Do students recognise that there are
both approaches to collecting data?
Teacher Reflections
»» Can students identify the advantages
to collecting data?
• Sampling is faster and cheaper.
• If you sample you mightn’t get
• When you sample you have to
be careful to make sure the
sample is representative.
»» Do students recognise that sampling
is used in the majority of statistical
investigations?
»» For many reasons you
have just discussed,
question in statistics,
we usually use data
of from the entire
population.
»» Do students understand why
sampling is used in the majority of
statistical investigations?
»» Do students understand that the
use of sampling raises the question
of how accurate the results of a
statistical investigation are?
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
8
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher’s Supports and Actions
Student Activities:
Possible and
Expected Responses
Assessing the Learning
»» You also pointed out a major disadvantage
to sampling – that of accuracy.
»» Do students
understand that
I have created a
population so that I
can investigate how
reliable a sample
is in describing a
population?
»» Because of this, we are going to carry out
a small investigation to see if it’s possible
to use the result from a sample to answer
»» For the investigation, I have created
(simulated) my own population of
students using coloured counters.
»» Show students the container of counters.
»» Distribute a key to each group of students explaining what
each colour counter represents.
»» There are 300 students (or 300 counters)
in my population. Each colour counter
represents a different location in which
students keep their mobile phone at night.
»» I have purposely set up my population
to have a specific proportion of students
who keep their mobile phone under their
pillow at night.
»» I’ve written this proportion on a piece of
paper in this envelope.
»» We are now going to see if, by choosing a
sample from my population, I can find out
what this proportion is.
»» Show students the envelope with the population proportion
sealed in it. Pin it to the board.
»» On one side of the board, write the heading “Population”.
Underneath it write “No. of students in population = 300” and
“Proportion of students in population who keep their mobile
phone under their pillow = _____”.
»» Encourage each group of students to replicate what’s written
on the board on their own miniature whiteboard.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
»» Do students
understand that
each unit of my
population is
represented by a
coloured counter?
»» Do students
understand that
different colours
represent different
locations in which
students keep their
mobile phone?
»» Do students
understand that we
are going to use the
simulated population
to see if a sample can
be used to determine
the population
proportion?
9
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and Actions
Assessing the Learning
»» In turns, I would like each
group to choose a simple
random sample of 25
students (counters) from
the container and calculate
the proportion of the
sample who keep their
mobile phone under their
pillow at night.
• Students draw 25 counters
from container and record
results.
»» Across from the heading “Population” write a
second heading “Sample”. Underneath it write
“Number of students in my sample = 25” and
“Proportion of students in my sample who
keep their mobile phone under their pillow at
night = _____ “
»» Do students understand that they
are interested in the proportion of
counters which are yellow?
• Students calculate the
proportion of their sample
which is yellow.
»» Do students understand how to
choose a simple random sample?
»» Do students understand how to
calculate a proportion?
»» This is stage three of the
data-handling cycle –
analyse the data.
»» Encourage each group of students to replicate
what’s written on the board on their own
miniature whiteboard.
»» Add in the third stage of the data-handling cycle
to the flow chart on the board.
»» Circulate to make sure students are completing
»» Encourage students to write their proportion in
the appropriate space on their whiteboard.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
10
Teaching & Learning Plan: Inferential Statistics for Proportions
Input
»» Group 1, could you tell me the
proportion of students in your
sample who keep their mobile
phone under their pillow at
night?
»» Is there any other way in which
this result could be written?
Student Activities: Possible and Teacher’s Supports and Actions
Expected Responses
•
Note: This is only one of the
possible proportions calculated
from the sample.
•
or 40% or 0.4.
• As a fraction or as a decimal or
as a percentage.
»» I am now going to use the
result from Group 1’s sample
to make a statement about the
population of 300 students.
This is the final stage in the
data-handling cycle – interpret
the results.
»» The proportion of students
in the population who keep
their mobile phone under their
pillow at night is 0.4.
• Yes.
• Yes, I got the same result.
»» Write Group 1’s result in the appropriate
space on the board in the form in which they
reported it.
»» Do students recognise that a
proportion may be written in
different ways?
»» Encourage students to convert Group 1’s
proportion to different representations.
»» Do students understand that
fractions, decimals and percentages
are equally valid ways of
representing a proportion?
»» Add the different ways in which this
proportion could be written to the board in
the appropriate location.
»» On the flow-chart showing the data-handling
cycle, add in the final step of “Interpret the
results”.
• Well I got a different answer.
• We all got different answers.
• Why are we using Group 1’s
www.projectmaths.ie
»» Can students easily change between
the different ways of representing a
proportion?
»» Do students recognise that each
group got a different sample
proportion?
»» Do students recognise that this
makes it difficult to make any firm
based on the results of a single
sample?
• No, our group got a different
proportion.
»» Are you happy with this
statement?
© Project Maths Development Team 2014
Assessing the Learning
»» Add Group 1’s result to the appropriate
»» Write each group’s sample proportion in the
correct location under the heading “Sample”.
KEY: » next step
11
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» The fact that we all get
different proportions when
we sample is known as
"sampling variability".
• Our samples were
randomly chosen.
»» On the side of the board
write the key term
“sampling variability”.
»» Can students explain what
sampling variability is?
»» Can you explain why we all
get different proportions
i.e. can you explain why
sampling variability occurs?
• We all chose different
samples from the container. »» Encourage students to
discuss with each other
• We chose our samples
what sampling variability
randomly so you wouldn’t
means and to write the
term and its description
the same.
into their journals.
Teacher Reflections
»» Do students understand
why sampling variability
occurs?
»» Can students explain why
sampling variability occurs?
»» The aim of this activity was
to see if we can use a single
sample to determine the
proportion of students in a
population who keep their
mobile phone under their
pillow at night.
»» Because I simulated the
population, I know what
the population proportion
is: remember it’s sealed in
the envelope on the board.
»» Given what we’ve just
discovered, how confident
would you be that Group
1’s proportion is the
same as the population
proportion?
© Project Maths Development Team 2014
• Not very confident.
• I’d say it’s around the right
• Reasonable confident.
• I don’t think it’s likely to be
the same.
www.projectmaths.ie
KEY: » next step
12
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» Can you explain to me why
you’re not very confident
with Group 1’s result?
• Well, it’s just one of the
possible results we could
get.
»» Encourage students to
discuss their ideas with
each other.
• Because of sampling
variability.
»» Encourage each group to
share their thinking with
the other groups in the
classroom.
»» Do students understand
that Group 1’s result is
only one of the possible
we sample a population?
• Different groups got
different values to Group 1.
Teacher Reflections
• There’s nothing special
• Maybe our result is the
correct one.
»» Would you have more
confidence in the result
• Not really.
• All the results are as good
as each other.
• One of the results is
probably correct.
»» Encourage students to
discuss their ideas with
each other.
»» Encourage each group to
share their thinking with
the other groups in the
classroom.
»» Do students understand
that while some results are
better than others we have
no way of knowing which
are better?
• Some of the results are
probably closer to the real
value than others.
• There’s no way to know
which result is best.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
13
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» So we agree that we are
• The proportion of students
not very confident using a
in the population who
proportion from a single
keep their mobile phone
sample to make conclusions
under their pillow at night
is around 0.4.
»» With this in mind
and based on all the
information we have up on
the board, could you come
the population proportion
in which you’d have
greater confidence?
• The proportion of students
in the population who
keep their mobile phone
under their pillow at night
is around 0.32.
• The proportion of students
in the population who
keep their mobile phone
under their pillow at night
is somewhere between 0.24
and 0.52.
• The proportion of students
in the population who
keep their mobile phone
under their pillow at night
is the average of all our
results.
© Project Maths Development Team 2014
www.projectmaths.ie
Teacher’s Supports and
Actions
Assessing the Learning
»» Encourage students to
come up with a statement
discussing it in groups.
»» Do students understand
that they cannot assume
that their sample
proportion is the same
as the population
proportion?
»» Encourage students to
justify their statement.
Teacher Reflections
»» Do students understand
that the chance of their
sample proportion being
equal to the population
proportion is low?
»» Do students understand
that they can make a more
the population proportion
(i.e. a statement in
which they have more
confidence) using a range
or interval of values?
KEY: » next step
14
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» Can you explain why you
have more confidence in a
statement which is based
on a range of values?
• It says that the population
proportion is around 0.4
not that it’s exactly equal
to 0.4.
“Sampling variability", this
idea should be discussed with
the class.
• It says that the population
proportion could be lots of
values, not just one.
Teacher’s Supports and
Actions
Assessing the Learning
»» Can students explain why
they have more confidence
in the last statement
compared to previous
statements?
Teacher Reflections
• It takes into account the
fact that different samples
• Although different
groups got different
to each other and the last
statement takes this into
account.
• It recognises the existence
of sampling variability.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
15
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
»» Our class has just come
up with one of the most
important ideas in statistics
and that is: the emergence
of a range of values when
a population based on a
single sample.
»» On the board cross off the statement
about the population and replace it with
the following “The proportion of students
in the population who keep their mobile
phone under their pillow at night is
between 0.24 and 0.52”.
»» By using an interval we can
have more confidence that
the population is true.
»» Write the term “confidence interval” on
the board.
»» Using an interval takes into
account the existence of
sampling variability.
»» If I were to widen my
interval to make the
following statement “The
proportion of students
in my population who
keep their mobile phone
under their pillow at
night is between 0.1 and
0.7 – would you be more
or less confident in this
statement?
»» Encourage students to come up with an
explanation of what a confidence interval
is and to write this into their journal.
• Less confident (wrong).
• More confident.
»» Draw this confidence interval on the
board and compare it to our original
interval.
• More confident because
we’re including more
possible values.
Assessing the
Learning
»» Do students
understand that by
using an interval
I can be more
confident that the
statement I am
population is true?
Teacher Reflections
»» Do students
understand
what the term
“confidence
interval” means?
»» Can students
verbalise what
a "confidence
interval" is?
»» Do students
understand
that a narrower
interval affects
how confident
we are in our
the population?
• More confident because
a wider interval means
it’s more likely to be true.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
16
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
»» If I were to make my
• Less confident.
interval narrower
with the following
• Less confident – we’re
statement “The
only looking at a small
proportion of students
in my population who
we could get.
keep their mobile
phone under their
• Less confident – there’s
pillow at night is
a smaller chance that we
between 0.35 and 0.45
are capturing the real
– would you be more
population proportion.
or less confident in this
statement?
»» Add the narrower confidence interval to the
diagrams on the board.
Assessing the
Learning
»» Do students
understand
that a narrower
interval affects
how confident
we are in our
statement
population?
Teacher Reflections
»» Explain your
reasoning.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
17
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Activities:
Teacher’s Supports and Actions
Possible and
Expected Responses
»» So when we make a statement about
a population based on a single sample,
how wide should we make our interval?
And how confident should we be that
the interval captures the population
proportion?
»» Do students
understand
that while we
understand
the need for
a confidence
interval
when making
statements
population we
have yet to
discuss how to
construct this
interval using a
single sample?
»» Luckily for us, statisticians have already
decided this. Using the empirical rule,
they developed a simple method for
creating this interval based on the results
of a single sample.
»» If we take the proportion calculated from • Students complete
the calculation of
our single sample and subtract
the confidence
from it we get the lower end of the
interval, for the
interval.
example on the
board.
»» If we take the proportion calculated from
to
it we get the upper end of the interval.
»» When we do this we create an interval
for which we can be 95% confident that
what we are saying about the population
is true.
Assessing the
Learning
»» Write the expression
on the board.
Teacher Reflections
»» Go through a sample calcultion on the
board using any sample proportion
»» Are students
able to
(preferably one which wasn’t calculated by
complete the
any of the groups in the classroom).
confidenceinterval
»» Encourage students to complete the
calculation?
calculation using their calculators.
»» This is known as the 95% confidence
interval.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
18
Teaching & Learning Plan: Inferential Statistics for Proportions
Input
Student Activities: Possible Teacher’s Supports and
and Expected Responses
Actions
Assessing the Learning
»» Now, using your own sample
proportion, I would like you to
create a 95% confidence interval
and use this to make a statement
• Students complete their
»» Circulate around the
own confidence – interval
room to ensure students
calculation.
are completing it correctly.
»» Do students understand
that they are using their
own sample proportion
to construct their own
confidence interval?
»» We introduced the idea of a
confidence interval because we
realised that, due to sampling
variability, we cannot simply use the
result from a single sample to make
»» Encourage students to
the population using their
confidence interval.
»» Do students know how
to construct a 95%
confidence interval?
»» Get each group to write
their confidence interval
on the board beside their
sample proportion.
»» Can students use their
95% interval to make
population?
Teacher Reflections
»» Open the envelope to
reveal the proportion of
students in the population
who keep their mobile
phone under their pillow
at night.
»» Let’s see if our confidence-interval
approach has worked i.e. does it
allow us to make correct statements
»» Because I simulated the population
we investigated I know what the
proportion of students in the
population who keep their mobile
phone under their pillow at night is:
it’s 0.3.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
19
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» If you had used your
sample proportion only to
the population, would
correct?
• No.
»» Highlight each group's
result on the board and
the fact that none of these
equal the population
proportion.
»» Do students see that using
just the sample proportion
to make a statement
an incorrect statement
and that if it doesn’t it is
merely due to chance?
»» Go through each group’s
confidence interval on the
board and use it to make
population.
»» Do students understand
that by creating an
interval around our sample
proportion we are now
able to make a statement
which is very likely (95%)
to be true?
• It would have been close.
• It wouldn’t have been too
Note: Because the sample
size is 25, it is not possible
for a student to get a sample
proportion of 0.3 since all the
sample proportions must be a
multiple of 0.04.
»» If you use your confidence
interval to make a
population, is your
statement correct?
• Yes.
• Yes the answer lies within
my interval.
• Not for mine.
»» Encourage students to
correct or incorrect.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
Teacher Reflections
20
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» Would you expect every
the population to be
true / would you expect
every group’s confidence
interval to contain the true
population proportion?
• Yes (incorrect).
• No.
• Only 95% of the time.
• There’s a small chance some
won’t.
Teacher’s Supports and
Actions
Assessing the Learning
»» Do students understand
that by creating an interval
in this way there is still a
chance that the interval
will not contain the true
population proportion/the
statement that they make
not be true?
Teacher Reflections
»» Do students understand
that you would expect
population to be true only
95% of the time?
»» Do students understand
that there is only a 95%
chance of their statement/
confidence interval being
correct?
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
21
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning Tasks: Student Activities: Possible Teacher’s Supports and Actions
Teacher Input
and Expected Responses
»» Open the GeoGebra file “Confidence Interval. »» Do students
ggb”.
understand
that the line
»» Click the button “Show population
represents the
proportion”.
population
proportion of
»» Enter one group’s sample proportion in
0.3?
the input box labelled “Single Sample
Proportion” and click “Return” to plot it.
»» Do students
understand
»» Repeat the previous step to input and plot
that the graph
every group’s sample proportion.
reinforces
the notion
of sampling
variability?
»» Let’s look at this idea
using a graphical
representation.
»» The dashed line
represents the
proportion of
students in our
population who keep
their mobile phone
under their pillow
at night i.e. the line
represents what we
were trying to find
out using our single
samples.
»» Do any of the sample
proportions match
the population
proportion?
• No.
• Some are close to it.
• Most of them are close
to it but not a perfect
match.
• Most are close to it but a
few are far away.
population using our
sample proportion
would it have been
correct?
• No.
• Some of us would have
been close.
• Some groups’ statements
would have been way off.
© Project Maths Development Team 2014
Assessing the
Learning
www.projectmaths.ie
Teacher Reflections
»» Do students
understand
that none of
their sample
proportions is
equal to the
population
proportion?
KEY: » next step
22
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
»» Does every group’s 95%
confidence interval
cover the same range of
values?
• No.
»» Can you explain why this
is?
• Some do but most do not.
• Because of sampling
variability.
»» On the GeoGebra file, click the
button “Show 95% confidence
interval”.
»» Encourage students to explain
why each group got a different
confidence interval.
• Because they were all
created using a different
sample.
Assessing the
Learning
»» Do students
understand that each
group’s confidence
interval is different?
Teacher Reflections
»» Can students explain
why each group’s
confidence interval is
different?
• Because each group’s
sample proportion was
different.
»» Does each group’s
confidence interval
contain the population
proportion we were
looking for?
• Yes.
»» Can you explain how this
can be, given that the
confidence intervals are
all different?
• The intervals overlap.
© Project Maths Development Team 2014
• Most of them do.
• 95% of them do.
»» Encourage students to point out any
intervals which do not capture the
population proportion.
»» Do students
understand that
although each
group’s confidence
interval is different
»» Encourage students to discuss how
they all have a great
different intervals can all capture the
deal of overlap?
population proportion.
• Although the intervals are
different they capture a
lot of the same values.
www.projectmaths.ie
KEY: » next step
23
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» Would you expect each
group’s confidence interval
to contain the population
proportion?
• Yes.
»» Ask students to explain to
each other what it means
when we say the interval
we construct is a 95%
confidence interval.
»» Do students understand
that because we have
chosen to construct a
95% confidence interval,
sometimes the interval will
not include the population
proportion?
»» Encourage students to
point (on the board) those
population that are correct
and those that are not.
»» Do students understand
that when the 95%
confidence interval
captures the population
proportion the resulting
population will be true?
• No.
• I’d expect most of them to.
• It’s a 95% confidence
interval so I’d expect 95%
of the intervals to contain
the population proportion.
»» If each group makes a
population using their
confidence interval, will
they all be correct?
• Yes.
• No.
• Most of them will be.
• I’d expect 95% of them
to be correct in their
statement.
• There’s a 95% chance that
each statement will be
correct.
© Project Maths Development Team 2014
www.projectmaths.ie
Teacher Reflections
»» Do students understand
that using this confidenceinterval approach to
a population based on
a single sample will be
correct 95% of the time?
KEY: » next step
24
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities:
Teacher’s Supports and Actions
Possible and
Expected Responses
Assessing the
Learning
»» We have only looked at
a small number of the
different samples that could
have been chosen from the
population of 300 students.
• Very few.
»» Can students
predict how
many confidence
intervals will
capture the
population
proportion?
• None.
• There’s a tiny
chance that one
of the sample
»» Using ICT I can very quickly
proportions
take many more samples and
will be equal to
see if what we’ve learned so
the population
far still holds true.
proportion.
»» If I took 100 different
• Most of them.
samples how many of these
would you expect to be
• Nearly all of them.
the same as the population
proportion?
• 95% of them.
»» If I took 100 different
samples and used the results • Hardly any.
to construct 100 different
• Less than 5% of
95% confidence intervals,
them.
how many of these intervals
would you expect to capture
the population proportion?
»» On the GeoGebra file click the “Reset” button.
»» Click on the button “Show population proportion”
»» Drag the slider “No. of samples” to create additional samples of size 25.
»» Encourage students to identify any sample proportions which are identical to
the population proportion.
»» Click on the button “Show 95% confidence interval”.
»» Encourage students to identify any confidence interval which does not
capture the population proportion.
»» How many confidence
intervals would you
expect not to capture the
population proportion?
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
»» Do students
understand that
when we use a
95% confidence
interval we
would expect
our interval
to capture the
population
proportion 95%
of the time?
»» Do students
understand
that by using
this approach,
we expect the
statement we
population to be
correct 95% of
the time?
25
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible and Expected Responses
»» Let’s take a minute and
review what we’ve just
discovered.
Teacher’s Supports and
Actions
Assessing the Learning
»» Encourage students to
discuss these questions
with each other and to
explain their thinking
to each other.
»» Do students understand that due to
sampling variability it is highly unlikely
that a result from a sample will match the
result for the population?
»» If I take a sample from a
population will the results
from the sample match
the results for the entire
population?
• No.
• It’s highly unlikely.
»» Can you explain why this
is so?
• Because of sampling variability.
• Because you’re only looking at a piece of the population.
»» Given this, is it possible to
make a correct statement
based on the results from
a single sample?
• Yes.
• Yes, but not just by using the result from our sample.
• We can’t say for definite what the result for the population
»» Where needed,
is but we can say that it’s likely to be in a certain range.
use appropriate
• Yes we can, by using a confidence interval.
questioning to help
• Yes, by using a 95% confidence interval.
students formulate
• It’s very likely that the interval we create captures the
question.
population proportion.
• There’s a 95% chance that our interval holds the correct
»» Write a summary
value for the population.
• There’s a 95% chance that the statement we make about
question on the board
the population is correct.
– based on the class’s
work and encourage
• Yes.
students to take
• Yes, but it’s very unlikely.
note of the questions
• Yes, but there’s only a 5% chance of that happening.
copybooks.
»» When we create a 95%
confidence interval, what
does this mean?
»» Is it possible that when we
use a confidence interval,
the statement we make
not be true?
© Project Maths Development Team 2014
»» Encourage each
group to write an
each question on their
whiteboards.
www.projectmaths.ie
»» Circulate the room
to see if students
understand the
questions.
KEY: » next step
»» Do students understand that because of
this we use a confidence interval when
making a statement about a population
based on a sample?
»» Do students understand that the interval
we choose to create is a 95% confidence
interval?
»» Do students understand that a 95%
confidence interval means that there is a
95% chance that the interval captures the
»» Do students understand that by using a
95% confidence interval, there is a 95%
chance that the statement we make
»» Do students understand that there is a
small (5%) chance that the confidence
interval does not capture the correct
»» Do students understand that sometimes
(5% of the time) the statement we make
about the population based on the results
from a sample will not be true?
26
Teaching & Learning Plan: Inferential Statistics for Proportions
Input
Student Activities: Possible
and Expected Responses
•
N and n.
•
•
P and p (incorrect).
p and p^.
»» Could we write the idea of
a confidence interval using
mathematical notation?
•
•
•
•
Yes, using an inequality.
0.28 ≤ p ≤0.28 +
0.28 - 0.2≤ p ≤0.28 + 0.2
0.08 ≤ p ≤ 0.48
»» We’ve just written down the
confidence interval for one
specific sample proportion from
a sample of size 25. Could we
write down the general form
of a confidence interval for any
sample proportion from any sized
sample?
• Yes.
• No.
• That sounds hard.
»» What symbols might we use
to represent the population
proportion and the sample
proportion?
© Project Maths Development Team 2014
Assessing the Learning
»» Do students
»» On the diagram on the
understand that it
whiteboard showing the
makes sense to use
population and sample
short-hand notation
information, re-write each
to represent the
statement using the correct
various quantities?
notation.
»» The statements we made on the
board take up a lot of space and
take a long time to write down.
In maths we like to represent
variables using symbols or letters,
for this very reason. We will do
the same here.
»» What notation might we use the
represent the number of units
in our population and in our
sample?
Teacher’s Supports and
Actions
• p^ -
www.projectmaths.ie
≤ p ≤ p^ +
»» Use suitable questioning
to help students write
down the 95% confidence
interval as an inequality.
Teacher Reflections
»» Can students write
down the 95%
confidence interval
as an inequality?
»» Use suitable questioning to »» Can students write
down the general
help students write down
form of a 95%
the general form of a 95%
confidence interval
confidence interval for a
for a proportion?
proportion.
»» Encourage students to
record this in their journals.
KEY: » next step
27
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» To construct a 95%
confidence interval we
quantity from our sample
proportion. What is this
quantity?
• 0.2.
•
•
•
»» This quantity
is known
as the margin of error.
Could you explain why it is
so called?
Teacher’s Supports and
Actions
Assessing the Learning
on the board and
»» Circle
label it as the margin of
error.
»» Encourage students to
write this term into their
journals.
»» Do students understand
that the margin of error
for a proportion is
?
Teacher Reflections
»» Do students understand
why this is known as the
margin of error?
• No.
• It takes into account
the fact that our sample
proportion may not be
exactly right.
• It allows for some error
in relating our sample
proportion to the
population proportion.
• It allows for the fact that
the population proportion
is probably different to the
sample proportion.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
28
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Supports and Assessing the Learning
Actions
»» Can students apply their
understanding of sampling
and confidence intervals to
complete Section A: Student
Activity 1?
»» We will now apply our
newfound knowledge
of sampling to another
statistical investigation.
»» Distribute copies of
Section A: Student
Activity 1 to all
students.
»» In groups, I would like you
to complete Section A:
Student Activity 1.
»» Circulate around the
room to ensure that
students understand
what they are
supposed to do and
»» Is the proportion of the
regularly speeding 0.42?
• Yes.
»» How did you confirm this?
• Yes,
»» Did you think the
was fair? Explain your
reasoning.
• Yes, the population proportion
mightn’t be exactly 42% but it’s
likely to be close.
• No, the proportion of our
sample is 42%. We don’t know
what the proportion is for all
lorry drivers.
• No, if a different sample had
been taken it would have given
a different result.
• No, this is the result for a
single sample, not the entire
population.
• There is a 95% chance that
the population proportion lies
between 0.42 and 0.42 +
.
© Project Maths Development Team 2014
=0.42.
www.projectmaths.ie
»» Encourage students
to discuss the
questions with each
other and to explain
the reasoning behind
»» Encourage students
to use the relevant
notation when
writing down their
Teacher Reflections
»» Can students calculate the
sample proportion from the
information given?
»» Do students recognise that
we cannot say the population
proportion is equal to the
sample proportion because of
sampling variability?
»» Do students understand that
to make a fair statement
should use a confidence
interval?
»» Can students calculate the
95% confidence interval?
»» Where needed, use
suitable questioning
to help students
complete the activity.
»» Can students calculate the
margin of error?
»» Are students comfortable
using mathematical notation
to represent various
quantities?
KEY: » next step
29
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible and
Expected Responses
»» What is the 95%
confidence interval for the
population proportion?
≤ p ≤ 0.42 +
• 0.42• 0.3753 ≤ p ≤ 0.4647
»» What is the margin of error
for our sample population
proportion?
•
• 0.4472
Teacher’s Supports and Assessing the Learning
Actions
Teacher Reflections
»» What statement can you
• I‘m 95% confident that the
proportion of all truck drivers who
regularly speed is between 0.3753
confidence interval.
and 0.4647.
• It is most likely that the proportion
of all truck drivers who regularly
speed is somewhere between
37.53% and 46.47%
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
30
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Supports and
Actions
»» Can students
complete Section A:
Student Activity 2?
»» When we calculate the
margin of error or the 95%
confidence interval we use
sample size to do so.
• Students complete Section A:
Student Activity 2.
Note: This activity could also be
given as a homework exercise
which would then be reviewed and
discussed at the start of the next
lesson.
»» Can you describe what
happens to the margin
of error as sample size
increases?
• The margin of error gets smaller.
»» Can you describe what
happens to the 95%
confidence interval as the
sample size increases?
• It gets narrower.
© Project Maths Development Team 2014
Teacher Reflections
»» Can students
calculate the margin
of error for a given
sample size?
»» It is important to
understand how sample
size affects the margin of
error or how sample size
affects the 95% confidence
interval.
»» To this end, in groups, I
would like you to complete
Section A: Student Activity
2.
Assessing the Learning
• It decreases.
»» Distribute Section A:
Student Activity 2 to all
students.
»» Circulate around the
room to ensure students
correctly.
»» Where needed, help
students complete the
activity by using suitable
questioning.
»» Can students
calculate the 95%
confidence interval
for a given sample
size?
»» Can students plot a
graph to show the
relationship between
the margin of error
and sample size?
»» Can students describe
the relationship
between the margin
of error and sample
size?
• The 95% confidence interval
covers a smaller range of values.
www.projectmaths.ie
KEY: » next step
31
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning Tasks: Student Activities: Possible
Teacher Input
and Expected Responses
Teacher’s Supports and Actions
Assessing the Learning
»» Can you explain why
this relationship exists
between margin of
error (or the 95%
confidence interval)
and sample size?
»» Select different groups of students to fill in each row of the
table on the board.
»» Can students describe the
relationship between the
95% confidence interval and
sample size?
»» Can students explain why
these relationships exist?
»» Do students understand
how margin of error (or the
width of the 95% confidence
interval) affects our ability
to make useful statements
»» Do students understand how
sample size affects our ability
to make useful statements
»» Do students understand the
sample sizes?
© Project Maths Development Team 2014
• If you used a small
number of people in your
sample they might not be
reflective of the general
population.
• Because of randomness,
a small sample could
give a result which is
very different from the
population.
• A larger sample is likely to
produce a more reliable
result.
• A large sample is more
likely to give a result
which is the same as the
population.
• A small sample could,
by chance, give you an
population proportion so
to capture the population
proportion you would
need a wide interval.
www.projectmaths.ie
»» Ask one group of students to sketch the graph showing the
relationship between margin of error and sample size.
KEY: » next step
32
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
»» If I were to decrease the
sample size, what effect
would that have on the
margin of error (95%
confidence interval)?
• The margin of error will
increase.
»» On the board highlight
the population based on
a sample size of 10 to
demonstrate how limited
the statement is.
• The width of the 95%
confidence interval will
increase.
Assessing the Learning
Teacher Reflections
• The 95% confidence
interval will get wider.
»» Can you identify an
larger sample size when
population?
definite.
useful.
»» Can you identify any
larger sample size?
• No.
• Not really.
• It costs more money.
• It takes more time.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
33
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning Tasks: Student Activities: Possible
Teacher Input
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» Distribute Section A:
Student Activity 3 to all
students.
»» Move around the room
to ensure all students
»» If students are having
difficulties completing
questioning to guide
them on their way.
»» Encourage students to
write a description of
how to use a sample to
a population into their
copybooks.
»» Can all students complete Section A: Student Activity 3?
»» We have learned a
how to use the results
from a sample to
make statements
»» To review the
learning, in groups,
I would like you to
complete Section A:
Student Activity 3.
• Students complete Section
A: Student Activity 3.
»» What information
would you need to
• The size of the sample.
• The number of students
who said they intended to
continue into third-level
education.
• Use it to calculate a sample
»» What would you do
with this information?
proportion.
• Use it to calculate a sample
proportion and a 95%
confidence interval.
• Calculate a 95% confidence
interval to take sampling
variability into account.
• Construct a 95% confidence
interval and use it to make
population.
© Project Maths Development Team 2014
www.projectmaths.ie
»» Can students describe what information they would need to
»» Can students describe the process of using the results from a single
sample to make a statement about a population?
»» Can students describe the process of constructing a 95% confidence
interval and how to use it to make a suitable statement about a
population?
»» Can students explain why it is important to use a confidence interval
when making a statement about a population based on a single
sample?
»» Can students explain what they would need to do to make a more
definite statement about the population using a single sample?
»» Do students understand that a larger sample size enable a more
KEY: » next step
34
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» What type of statement
population?
• The proportion of Junior
Certificate students who
intend going to third level
is between x and y.
Teacher’s Supports and
Actions
Assessing the Learning
Teacher Reflections
• I am 95% confident that
the proportion of Junior
Certificate students who
intend entering third-level
education is between a and
b.
• I am 95% confident that:
p^ - ≤ p ≤ p^ + .
»» What could you do to
improve the quality of the
the population?
• Use a larger sample.
• Increase my sample size.
• Reduce the margin of error.
• Make the 95% confidence
interval narrower.
• We could increase n.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
35
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Activities: Possible and
Expected Responses
Teacher’s Supports Assessing the
and Actions
Learning
Section B – Hypothesis Testing
»» Write some
examples of
claims on the
board.
»» We are now going to look at another
area of statistics which is important in
everyday life and that is determining if
or by anybody are accurate.
»» Could you give me an example of a claim
you’ve seen or heard in the media?
»» Is it important to know if a claim is
accurate or not? Explain.
Teacher Reflections
»» Do students
understand
what a claim
is?
»» Can students
recall
examples of
claims that
they have
seen or heard
in everyday
life?
• Taking a certain supplement will help
you lose weight.
• Different creams can get rid of
wrinkles.
• Some yoghurts help boost your
immune system.
• Smoking causes cancer.
• Eating fatty foods causes heart
disease.
• Support for a political party is at a
certain level.
»» Can students
explain the
importance
of checking a
claim?
• Not really.
• Yes, so that we don’t waste our
money.
• Yes, especially if it’s to do with your
health.
»» We would like to decide a fair way to
determine if a claim is true or not.
»» We’re going to look at a particular
type of claim that is a claim about a
proportion.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
36
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Supports and
Actions
»» There is an example of
such a claim in Section B:
Student Activity 1.
»» In groups, I would like you
to complete this activity.
• Students complete Section B:
Student Activity 1.
»» Distribute Section B:
Student Activity 1 to all
students.
»» To test a claim what is the
first thing we need?
• Gather some evidence.
• Survey some customers.
• Get some data.
»» In Question 2 of Section B:
Student Activity 1 you are
of all of the airline’s
customers. Is there another
word used to describe “all
of the airline’s customers”?
• The population.
• The population of customers.
• No.
»» If I made the following
statement: “The proportion • No, 0.64 is the sample
proportion, not the population
of the population that is
proportion.
satisfied with the service
• No, this is the proportion from
provided by the airline
a single sample. A different
is 64%” would you be
sample could give a different
happy with it? Explain your
result.
reasoning.
• No. Because of sampling
variability we cannot say that.
• No, the chance of that being
true is tiny.
© Project Maths Development Team 2014
www.projectmaths.ie
»» Move around the room
to make sure all students
understand what they
are supposed to do.
»» Use suitable questioning
strategies to help
students who are having
difficulty completing the
»» Encourage students to
discuss each question
and to come up with an
»» Encourage students to
explain their reasoning
to each other.
Assessing the Learning
Teacher Reflections
»» Do students understand
that to test a claim we
need to gather data?
»» Do students understand
that when we refer
to all customers we
the population of
customers?
»» Do students understand
that we cannot say
that the proportion
of customers in our
sample that is satisfied
is unlikely to equal the
proportion of customers
in the population that is
satisfied?
KEY: » next step
37
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning Tasks: Student Activities: Possible and Expected Teacher’s Supports and Actions
Teacher Input
Responses
»» What statement
did you make
levels amongst the
population of the
airline’s customers?
• It is very likely that between 0.6084 and
0.6716 of the population are satisfied
with the service provided by the airline.
• There’s a 95% chance that between
0.6084 and 0.6716 of the population are
satisfied with the service provided by the
airline.
• I am 95% confident that between
0.6084 and 0.6716 of the population are
satisfied with the service provided by the
airline.
• 0.6084 ≤ p ≤ 0.6716.
»» Based on our
evidence do you think
the airline is correct
to claim that 70%
of their customers
are satisfied with
the service they
provide? Explain your
reasoning.
• No.
• Yes. Their claim is close to the result we
got.
• No. There is a more than a 95% chance
that they are wrong.
• No. There is a less than 5% chance that
their claim is right.
• They could be right but it’s very unlikely.
• No. We know there is a 95% chance
that the true population proportion lies
between 0.6084 and 0.6716 so their claim
of 0.7 is unlikely to be true.
© Project Maths Development Team 2014
www.projectmaths.ie
Assessing the Learning
»» Sketch a proportion line on the »» Can students make a
board and mark in the sample
proportion.
the proportion of
the population which
»» On the same diagram, mark in
is satisfied with the
the airline’s claim.
airline’s service?
»» Do students recognise
»» On the same diagram shade in
the need for a
the 95% confidence interval.
confidence interval
when making a
population using a
single sample?
»» Can students construct
a 95% confidence
interval correctly?
Teacher Reflections
»» Do students recognise
that the chance of
the true population
proportion lying
outside the 95%
confidence interval is
very low?
»» Do students understand
that, based on the 95%
confidence interval, it
is extremely unlikely
that the airline’s claim
is correct?
KEY: » next step
38
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» Is it possible that the
true proportion of the
population that is satisfied
is not between 0.6084 and
0.6716?
• Yes.
• Yes, but it’s very unlikely.
• Yes, but the chance of that
is less than 5%.
Teacher’s Supports and
Actions
Assessing the Learning
Teacher Reflections
»» Do students understand
that if a claim lies outside
the 95% confidence
interval we reject the claim
(because it only has a 5%
chance or less of being
true)?
»» When we test a claim we
use a 95% confidence
interval to determine if the
claim is fair or not.
»» If a claim lies outside the
95% confidence interval
constructed using our data,
we reject the claim.
»» Does this mean we are
rejecting Go Fast Airline’s
customer-satisfaction claim
of 70%? Explain.
© Project Maths Development Team 2014
»» On the board write “I
• Yes.
reject the airline’s claim”.
• Yes, because their claim
lies outside our 95%
confidence interval.
• Yes, because 0.7 lies outside
the 95% confidence
interval we constructed.
www.projectmaths.ie
»» Do students understand
that because the airline’s
claim of 70% lies outside
our 95% confidence
interval we reject their
claim?
KEY: » next step
39
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» What do you think our
conclusion would have
claim lay within our 95%
confidence interval.
• We would have accepted
the company’s claim.
Teacher’s Supports and
Actions
Assessing the Learning
»» Do students understand
that if the claim lies within
the 95% confidence
interval our conclusion will
be different?
• We’d conclude that the
company’s claim is true.
Teacher Reflections
• We’d conclude that it is
reasonable to say that the
proportion of all customers
that is satisfied is 70%.
»» Let’s look at Section B:
Student Activity 2 to see if
sense.
• Students complete Section
B: Student Activity 2.
»» Distribute Section B:
Student Activity 2 to all
students.
»» Move around the room to
make sure that all students
are on task and know what
to do.
»» I would like you to
complete the activity,
working in groups.
»» Help students who are
having difficulties with
questioning.
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
40
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning
Student Activities: Possible and
Expected Responses
Teacher’s Supports and Actions
»» To test a claim, what • Gather some evidence.
is the first thing we • Survey some customers.
• Get some data.
need to do?
»» What statement
did you make
levels amongst the
population of the
airline’s customers?
• There’s a 95% chance that
between 0.5584 and 0.7216 of
the population are satisfied
with the service provided by the
airline.
• I am 95% confident that the
proportion of the population
satisfied with the airline’s service
is between 0.5584 and 0.7216.
would you reject
the airline’s claim of
a 70% satisfaction
rating? Explain.
• No.
• No, because the airline’s claim
lies within my 95% confidence
interval.
• No, because given my 95%
confidence interval runs from
0.5584 to 0.7216 it is possible that
the true proportion of customers
satisfied with the service is 70%.
• No, I’d accept the claim.
would you accept
the airline’s claim of
a 70% satisfaction
rating? Explain.
• Yes.
• Yes, because the claim lies within
my confidence interval.
© Project Maths Development Team 2014
www.projectmaths.ie
Assessing the Learning
»» Can students identify that
the first thing needed to
test a claim is some data?
»» On the board, below the
proportion line used in Section B:
Student Activity 1, draw another
proportion line.
»» On the new proportion line, mark
in the airline’s claim.
»» On the new proportion line mark
in the sample proportion.
»» On the new proportion line shade
in the 95% confidence interval.
»» Can students construct a
95% confidence interval
using the data and use
it to make a statement
»» Write the conclusion “I accept the
airline’s claim” on the board.
»» Do students naturally
use the word “accept”
when making a conclusion
lies within the 95%
confidence interval?
Teacher Reflections
»» Can students make
a suitable conclusion
regarding the airline’s
claim?
KEY: » next step
41
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and Actions
Assessing the Learning
»» We need to be a little
careful here with how we
describe our conclusion
regarding the company’s
claim. Let’s understand
why.
Teacher Reflections
»» Section B: Student
Activity 1 and Section B:
Student Activity 2 are
similar to each other.
Can you identify the
similarities between the
two activities?
same airline.
»» The claim is the same in
both cases.
»» The sample proportion we
calculate is the same (0.64)
for both.
»» Even though the sample
proportion is the same
for both activities our
conclusions are very
different for each. How
can this be?
»» Because the confidence
intervals are different sizes.
»» The margin of error is not
the same for each.
»» Because the confidence
interval is wider in Section
B: Student Activity 2, the
airline’s claim lies within in.
»» Because the confidence
interval is narrower in
Section B: Student Activity
1, it fails to capture the
airline’s claim and so we
reject the claim.
© Project Maths Development Team 2014
www.projectmaths.ie
»» Point out that the confidence intervals
from Section B: Student Activity 1
and Section B: Student Activity 2 have
different widths.
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
»» Do students recognise
that the confidence
intervals in Section B:
Student Activity 1 and
in Section B: Student
Activity 2 are different
widths?
0.85
KEY: » next step
42
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Activities: Possible and Expected
Responses
»» Why is the confidence interval wider in
Section B: Student Activity 2 compared
to Section B: Student Activity 1?
• Because the sample size is smaller.
• Because we didn’t sample as many customers.
• Because the margin of error gets bigger as the
number in your sample gets smaller.
• Because the smaller n is, the wider your 95%
confidence interval will be.
Teacher’s Supports Assessing the Learning
and Actions
»» Do students understand what causes the
difference in the confidence-interval widths?
»» In Section B: Student Activity 2, the
• Probably not.
consumer agency had originally planned • No, because the 95% confidence interval
to sample 1000 customers but only
would have been much narrower.
managed to sample 150. If they had
• No, because we would have gotten a similar
sampled 1000 customers do you think
result to Section B: Student Activity 2.
the 95% confidence interval would still
have captured the airline’s claim of 0.7?
Explain.
»» With this in mind, for Section B: Student
Activity 2, can you explain why the
airline’s claim falls within the 95%
confidence interval? Is it because the
claim is now true?
•
•
•
•
Yes.
Not necessarily.
Maybe.
It’s because the confidence interval is now
wider.
• It’s because we had a much smaller sample.
»» Do students understand that the reason we
get two different conclusions for Section B:
Student Activity 1 and Section B: Student
Activity 2 is because of the different
confidence-interval widths?
»» When we test a claim and find that it
lies within the 95% confidence interval,
what could the reason be?
• Because the claim is true.
• Because a small sample was used making the
95% confidence interval very wide.
• It could be for two reasons: The claim could be
true or we mightn’t have used a big enough
sample to find out that it’s false.
»» Do students understand that a claim may
lie inside our 95% confidence interval, not
because it is true but because the confidence
interval is really wide?
»» Do students understand that a claim may lie
inside our 95% confidence interval because
we used a small sample to test it?
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
43
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
»» For this reason when a
claim lies within the 95%
confidence interval we
don’t say we accept the
claim, rather we say we fail
to reject the claim.
»» While each of these
statements may seem to
say the same thing, they do
not.
»» If we say “we accept the
claim” what does this
mean?
• It means the claim is true.
• It means we accept the
claim to be true.
»» If we say “we fail to reject
the claim” what does this
mean?
• It means the claim could be
true.
• It means the claim could be
true or we haven’t used a
big enough sample to find
out.
»» When we are evaluating
a claim using a sample
what are the two possible
conclusions we can make?
• We can reject the claim or
we can fail to reject the
claim.
© Project Maths Development Team 2014
www.projectmaths.ie
Teacher’s Supports and
Actions
Assessing the Learning
»» Draw a line through the
statement “I accept the
company’s claim” and
replace it with “I fail
to reject the company’s
claim”.
»» Underline the two possible
conclusions when testing a
claim “reject” and “fail to
reject”.
»» Do students understand
that “accept a claim” and
“fail to reject a claim” do
not mean the same thing?
Teacher Reflections
»» Do students understand
why we use the term
“accept” when making a
»» Do students understand
that when making a
the two options are “reject
the claim” or “fail to reject
the claim”?
KEY: » next step
44
Teaching & Learning Plan: Inferential Statistics for Proportions
Student Learning Tasks: Student Activities: Possible and Expected Responses
Teacher Input
Teacher’s Supports and Actions
Assessing the Learning
»» In groups, I would
like you to complete
Question 1 of Section
B: Student Activity 3.
• Students complete Question 1 of Section B: Student
Activity 3.
»» Distribute Section B: Student Activity 3
to all students.
»» Can students test the
claim contained in the
Activity Sheet?
»» What was your
Department of the
Environment’s claim?
• I rejected the claim (wrong).
• I accepted the claim (incorrect language).
• I failed to reject the claim.
»» Move around the room, using
suitable questioning to test students’
understanding of the activity.
»» Do students use the
correct language when
making their conclusion
on the claim?
»» Can students justify
their work?
»» Draw a proportion line on the board
and fill in the claim, the sample
proportion and the 95% confidence
interval.
»» Why did you fail to
reject the claim?
• Because the claim lies within our 95% confidence
interval.
• Because the claim of 0.85 lies within our 95% confidence
interval of 0.7984 and 0.8616.
»» Can students justify the
use of “fail to reject”
when making their
conclusion?
»» Why is it incorrect
to say we accept the
claim?
• Because the fact that the claim lies within my 95%
confidence interval does not necessarily mean it’s true.
• The existence of the claim within my 95% confidence
interval might be due to a small sample and not because
the claim is true.
• If I took a larger sample, my margin of error would
be smaller and the claim might lie outside my 95%
confidence interval.
»» Can students explain
why using the word
“accept” is not correct?
© Project Maths Development Team 2014
www.projectmaths.ie
KEY: » next step
45
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» As with all areas of
mathematics, there is formal
language to describe the
process of testing a claim.
Teacher Reflections
»» I am going to describe the
formal language used and
2 of Section B: Student
Activity 3.
as I do so I would like you to
B: Student Activity 3.
»» On one side of the board
write the term “Hypothesis
Test”.
»» The process of testing
a claim is known as a
hypothesis test.
is known as the null
hypothesis. The shorthand
notation for the null
hypothesis is H0.
»» What is the null hypothesis
• The proportion of Irish
in Section B: Student Activity
households who pay the
3?
local property tax is 0.85.
• 85% of Irish households
pay the local property tax.
•
© Project Maths Development Team 2014
p = 0.85.
www.projectmaths.ie
»» Do students understand that
the null hypothesis means
the claim being tested?
»» Can students identify the
null hypothesis?
»» Do students understand that
the alternative hypothesis is
the counter claim?
KEY: » next step
46
Teaching & Learning Plan: Inferential Statistics for Proportions
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» For every claim that’s made
we can make a counter
claim. The counter claim is
known as the alternative
hypothesis. The shorthand
notation for the alternative
hypothesis is HA or H1.
Teacher Reflections
»» What is the alternative
hypothesis in Section B:
Student Activity 3?
• 85% of Irish households do
not pay the local property
tax (wrong).
»» When testing a claim (or
carrying out a hypothesis
test) it is usual to start
off by stating the null
hypothesis and the
alternative hypothesis.
• The proportion of Irish
households that pays the
local property tax is not
0.85.
© Project Maths Development Team 2014
•
»» Can students identify the
alternative hypothesis?
p ≠ 0.85.
www.projectmaths.ie
KEY: » next step
47
Teaching & Learning Plan: Inferential Statistics for Proportions
Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Assessing the Learning
»» We have learned a huge
claim or about carrying out a
hypothesis test.
Teacher Reflections
»» Let’s take some time to review
what we’ve learned.
»» Working in pairs, I would like
you to complete Question 1 of
Section B: Student Activity 4.
• Students complete
Question 1 of Section B:
Student Activity 4.
»» Distribute Section B:
Student Activity 4 to all
students.
»» When you have completed
Question 1 of Section B:
Student Activity 4, I want you
the group next to you.
»» Move around the room to
ensure all students are fully
engaged and that they
understand what they are
meant to do.
other group’s worksheet, I
want you to answer Question 2
of Section B: Student Activity 4
using their instructions on how
to carry out a hypothesis test.
»» Emphasise that, having
swapped worksheets with
another group, students
must use the other group’s
Question 2 of Section B:
Student Activity 4.
Question 2, I want you to
correct the other group’s
Question 1 and give them
feedback on their description
of a hypothesis test.
»» Encourage each group to
give constructive feedback
to each other in relation
to their description of a
hypothesis test.
© Project Maths Development Team 2014
www.projectmaths.ie
»» Can students accurately
describe the steps
involved in carrying out
a formal hypothesis
test?
»» Can students complete
a formal hypothesis
test, using appropriate
language?
»» Can students critically
evaluate other
students’ description of
a hypothesis test?
KEY: » next step
48
Teaching & Learning Plan: Inferential Statistics for Proportions
Section A: Student Activity 1
The Road Safety Authority of Ireland (Údaras Um Shábháilteacht
Ar Bhóithre) is interested in how many lorry drivers speed on
a regular basis. To answer this question they choose a simple
random sample of lorry drivers from the members list of the Irish
the speed limit. Of the 500 drivers who replied, 210 stated that
they regularly speed.
1. Using suitable calculations, confirm that the proportion of
the sample that admit to regularly speeding is 0.42.
2. A national newspaper includes the following headline on its
front page: “42% of all Irish lorry drivers admit to regularly
speeding”. Is this a fair statement? Explain your reasoning.
3. Construct a 95% confidence interval for the proportion of
Irish lorry drivers who admit to speeding.
4. Write down the margin of error for this sample.
all lorry drivers which you consider fair.
© Project Maths Development Team 2014
www.projectmaths.ie
49
Teaching & Learning Plan: Inferential Statistics for Proportions
Section A: Student Activity 2
The government is interested in finding out what proportion of
the Irish population is in favour of introducing water metering.
To find this out, they commission RED C Research and Marketing
Ltd. to carry out a survey on a sample of the population. RED C
find that 0.37 of their sample are in favour of introducing water
metering.
1. The table below shows some of sample sizes RED C may have
used to calculate the proportion of 0.37. For each sample size
presented:
a Calculate the margin of error (correct to three decimal places).
b Construct a 95% confidence interval for the population
proportion.
c Shade in the 95% confidence interval on the proportion line.
d Make a statement about the proportion of the Irish
population which is in favour of introducing water metering.
Sample
size
Margin 95%
The Proportion Line
of error confid­
ence
interval
10
I can be 95% confident that between __ and __
of the population are in favour of water metering.
50
100
200
500
700
1000
© Project Maths Development Team 2014
www.projectmaths.ie
50
Teaching & Learning Plan: Inferential Statistics for Proportions
Section A: Student Activity 2 (continued)
2. Draw a graph of margin of error versus sample size.
3. By referring to your table and graph complete the following
statements:
a As the size of a sample increases, the margin of error _________.
b As the size of a sample increases, the 95% confidence interval
________________.
4. By referring to the fourth column of the table, can you see
any advantage to using a larger sample size? Explain your
reasoning.
© Project Maths Development Team 2014
www.projectmaths.ie
51
Teaching & Learning Plan: Inferential Statistics for Proportions
Section A: Student Activity 3
For future planning, The Central Application Office (CAO) are
interested in what proportion of current Junior Certificate
students think it likely that they will continue to third-level
education after leaving second-level education. Due to cost and
time concerns they decide to use a sample of students to help
Describe how the CAO should use a single sample of students
to make a fair statement about all students. As part of your
description you should consider the following:
a What information would the CAO need?
b What the CAO would do with this information and why would
they do this? (Include a description of any calculations needed.)
c The type of statement would the CAO make about the
population.
d What the CAO could do to improve the quality of the
statement they make about the population?
© Project Maths Development Team 2014
www.projectmaths.ie
52
Teaching & Learning Plan: Inferential Statistics for Proportions
Section B: Student Activity 1
Go Fast Airlines advertise that 70% of their customers are satisfied
with the service they provide. A consumer agency wants to
determine if the company’s claim is true or not.
1. Describe how the consumer agency could go about
determining if the claim is true or not?
2. The consumer agency survey 1,000 customers and find that 640
of these are satisfied with the service provided by the airline.
Using this information make a fair statement about satisfaction
levels amongst all of the airline’s customers.
3. Based on your answer to Question 2, do you think the airline’s
claim of a 70% satisfaction rating is correct? Explain your
reasoning.
© Project Maths Development Team 2014
www.projectmaths.ie
53
Teaching & Learning Plan: Inferential Statistics for Proportions
Section B: Student Activity 2
Go Fast Airlines advertise that 70% of their customers are satisfied
with the service they provide. A consumer agency wants to
determine if the company’s claim is true or not.
1. Describe how the consumer agency could go about
determining if the claim is true or not?
2. The consumer agency plan to survey 1000 customers to
help them test this claim, however due to time and money
constraints they only manage to survey 150 customers. Of
these, 96 say that they are satisfied with the service provided
by the airline. Use this information to make a fair statement
about satisfaction levels amongst the population of the airline’s
customers.
3. Based on your answer to Question 2 what would you conclude
© Project Maths Development Team 2014
www.projectmaths.ie
54
Teaching & Learning Plan: Inferential Statistics for Proportions
Section B: Student Activity 3
The Department of the Environment, Community and Local
Government (Roinne Comhshaoil, Pobail agus Rialtais Áitiúil)
claims that 85% of households now pay the local property tax
(LPT). An opposition party commissions research to test the
validity of this claim. This research finds that, out of a sample of
1,000 households surveyed, 830 confirm that they pay the LPT.
1. Based on the evidence presented, what would you conclude
about The Department of the Environment’s claim?
2. Fill in each of the following terms for the above claim.
Null Hypothesis
Alternative Hypothesis
© Project Maths Development Team 2014
www.projectmaths.ie
55
Teaching & Learning Plan: Inferential Statistics for Proportions
Section B: Student Activity 4
1. In the space below, write a brief, step-by-step guide of how to
carry out a hypothesis test. As part of this you should include
explanations of why you carry out each step.
Step 1
Step 2
Step 3
© Project Maths Development Team 2014
www.projectmaths.ie
56
Teaching & Learning Plan: Inferential Statistics for Proportions
Section B: Student Activity 4 (continued)
2. Using the above instructions, answer the following question:
Manchester United claims that 90% of fans were in favour of
replacing David Moyes as manager. The Manchester United
fanzine, www.rednews.com carries out a survey of 2,000 of
its members, 1,700 of whom state that they were in favour of
replacing David Moyes.
Conduct a hypothesis test of Manchester United’s claim using
the evidence presented.
Step 1
Step 2
Step 3
© Project Maths Development Team 2014
www.projectmaths.ie
57
```

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project