Michelle Dalrymple
Use statistical methods to make an inference.
TEACHING ACTIVITIES TOWARDS
ACHIEVEMENT STANDARD 91264
(2.9) INTERNAL
4 CREDITS
Use statistical methods to make
an inference.
Population
Population
parameter
What we’re trying to
estimate
Sample
Sample
statistics
Historical development
 This standard replaces the old sampling
standard with making an inference about a
single population
 Extends development of the curriculum
material developed by Chris Wild and his
team at Auckland University
 Follows on from 91035 (1.10) Multivariate
Data
What is new/changed?
 Use of exploratory data analysis.
 Statistical inference comparing two
populations (or two groups within one
population).
 Informal confidence intervals for population
medians.
 Sampling variability.
 Using relevant (given) contextual knowledge.
Approaches
The approach you take will depend on
 Type of course offered
 Time allowed for the topic
 Incorporating Stat Lit (reports) material, or
material from other Statistics standards
 Background of students
Access to ICT
Key ideas…
 Statistical literacy
 Correct vocabulary
 Sampling variability
 Impact of sample size
 Impact of spread of population
 Informal confidence intervals
 Level 7 guide
 Making a call based on these intervals
Sequence of learning experiences:
Based on work by Lindsay Smith and Pip Arnold
1. Introduction to making an inference
2. Sampling methods
3. Using a sample to make a point estimate &
sampling variability
4. Sampling variability: effect of sample size
Sequence of learning experiences:
Based on work by Lindsay Smith and Pip Arnold
5. Sampling variability: effect of spread of
population
6. Developing the formula for informal confidence
interval for the population median
7. PPDAC for summary & checking how well our
intervals capture the population median
8. PPDAC for comparison (clear difference)
9. PPDAC for comparison (not a clear difference)
Handout
Original resources available on…
Lindsay Smith (University of Auckland)
http://www.censusatschool.org.nz/2011/statisticsteachers-day-years-12-and-13/
Pip Arnold (Cognition)
http://seniorsecondary.tki.org.nz/Mathematics-andstatistics/Achievement-objectives/Achievementobjectives-by-level/AO-S7-1
http://seniorsecondary.tki.org.nz/Mathematics-andstatistics/Achievement-objectives/Achievementobjectives-by-level/AO-S7-2
http://nzstatsedn.wikispaces.com/Gisborne+2012
Reminder – PPDAC cycle…
Lesson 3 & 4
PPDAC REVIEW
SAMPLING VARIABILITY
POINT ESTIMATES OF
POPULATION PARAMETERS
Population
Population
parameter
What we’re trying to
estimate
Sample
Sample
statistics
Problem
 What sort of question
is this?
 How would we have
 I wonder what the
worded this question
median weight of
last year? (Level 1)
Stage 1 Statistics
 What other sort of
students at Auckland
investigative questions
University is?
are there?
 What makes a good
question?
Reminders…
Question types
Good questions
 SUMMARY
 Can be answered with the
 Description of one variable
 COMPARISON
 Comparing two (or more)
subsets of data across a
common numeric variable
 RELATIONSHIP
 Looking at the
interrelationship between
two paired numeric variables
data
 Population of interest is clear
 Variable(s) of interest is clear
 Intent (summary,
comparison, relationship) is
clear
 Someone is interested in the
answer
Comparative question progression –
ASIDE
from Pip Arnold
Level 6
Level 7
I wonder if
heights of NZ
Yr 11 boys
tend to be
greater than
heights of NZ
year 11 girls
 looking for a
tendency, do
the boxes
overlap or not,
if they do is it
too much
 I wonder if

the median
height of NZ
year 11 boys
tends to be
greater than
the median
height of NZ
year 11 girls
 seeing if the
informal
confidence
interval overlap
or not
Level 8 – under
development still…
 I wonder what
the difference
in heights is
between NZ
year 11 boys
and NZ year 11
girls.
 finding an interval
for the difference
– if zero in the
interval then
probably not
making the call
I wonder what the median weight
of Stage 1 Statistics students at
Auckland University is?
 What do you think the typical weight will be?
 Why?
 Sketch the shape of the distribution of
weights of Stage 1 Statistics students from
Auckland University.
 Population information
provide a point
estimate of the
population
parameter
 Use sample median to
Conclusion
 From my sample data I estimate
that the median weight for all
Stage 1 statistics students at
Auckland University is….
Conclusion
 But they’re all different!
 Who is right?
 From my sample data
I estimate that the
median weight for all
Stage 1 statistics
students at Auckland
University is….
Everyone’s plots
 How can we use our sample to predict what is
going on back in the population?
 The sample median is our best idea of the
population median
Sampling error
 The process of taking a sample and using the
median of the sample to predict the
population median will never produce the
exact value of the population median.
 This is called sampling error
 The difference between the sample median and
the true value back in the population
Lesson 5 & 6
SAMPLING VARIABILITY –
THE EFFECT OF SAMPLE SIZE
Using technology…
 Sampling kiwis
 Collecting the medians
from repeated sampling
Remember we’re in
TEACHING WORLD in the ‘real world’ we
wouldn’t be able to
take lots and lots of
samples to see what
happens!
Showing this with technology
 One sample
 Collecting medians
Your collection of medians...
Analysis
 For each sample size:
A. I notice that the sample median
weights of kiwis for samples of
size ___ vary from ___ to ___
B. I notice that the bulk of the
sample median weights of kiwis
for samples of size ___ ranged
from ___ to ___
C. I notice that the median for the
sample median weight of kiwis
for samples of size ___ is ___
and that the median for the
sample IQR is ___
Analysis
Sample
size
I notice that the
sample median
weights of kiwis
for samples of size
___ vary from ___
to ___
I notice that the
bulk of the sample
median weights
of kiwis for
samples of size
___ ranged from
___ to ___
I notice that the
median for the sample
median weight of kiwis
for samples of size ___
is ___ and that the IQR
for the sample
medians is ___
n=15
from ___ to ___
from ___ to ___
Median-median = ___
IQR-median = ___
n=30
from ___ to ___
from ___ to ___
Median-median = ___
IQR-median = ___
n=50
from ___ to ___
from ___ to ___
Median-median = ___
IQR-median = ___
n=100
from ___ to ___
from ___ to ___
Median-median = ___
IQR-median = ___
Analysis
 I notice that the variation of the median
weights of kiwis ________ as the sample size
_________.
 For samples of size 15 the median weight ranged
from ____ to ____, a difference of _____,
 Whereas for samples of size 100 the median
weight ranged from ____ to ____, a difference of
____.
Conclusion
As the sample
size increases,
• the variation of
the medians
__________
 What is a sensible and
reliable sample size to use to
make inferences about the
population?
Conclusion
Remember
 Our best point estimate of the population
parameter – the population median is our
sample median
 The estimates vary, even with n = 100
 It is better to provide a range of possible values for
the parameter, based on our estimate, rather than
stating one value
Developing a reflex…
 Chris Wild movie - n = 30
We want to plant a reflex…
Movies – one sample - summary
Box plot with memory…
Lesson 7
SAMPLING VARIABILITY –
THE EFFECT OF SPREAD OF
POPULATION
The scenario
Intermediate School
Year 7 & 8
Middle School
Year 7 – 10
 An intermediate school
 A middle school wants to
wants to purchase new
furniture for their students,
based on the median
height of students in years
7 and 8.
 A teacher takes a sample
of 30 intermediate
students from C@S to
make an estimate of the
population median
purchase new furniture for
their students, based on
the median height of
middle school students.
 A teacher takes a sample of
30 middle school students
from C@S to make an
estimate of the population
median
Which teacher is likely to get a better
estimate of the students heights?
WHY?
Incorporating
sample size
Lesson 8
DEVELOPING THE FORMULA FOR
INFORMAL CONFIDENCE
INTERVALS FOR THE
POPULATION MEDIAN
So far…
Population
Population
parameter
What we’re trying to
estimate
Sample
Sample
statistics
Median weight of kiwis is somewhere between ___ and ___
Samples of size ___ were reliable enough
Distribution of sample medians…
The median weight of kiwis was somewhere
between ___ and ___
(90% ish of our sample medians)
Measures from Sample of Kiw ipop
2.2
2.4
2.6
2.8
m edian
Dot Plot
3.0
3.2
However in real life …
We don’t get to take multiple samples so this process
WON’T work
We need to find an informal confidence interval for the
population median based ON A SINGLE SAMPLE
Our informal interval needs…
To take into account both
•Sample size and
•spread
More kiwis…
Your turn…
Handout
Now…
Add your SAMPLE MEDIANS TO THE SHEET
Student worksheet
Add your IQR (box) TO THE SHEET
Student worksheet
Complete Q3 – Q5 on the worksheet
Q3: I notice that the width of the IQR for
sample medians when the sample size is 30
is approximately 1/5 of the width of the
population IQR
WIDTH
0.138 kg
WIDTH
0.6805 kg
Q4: I notice that the width of the IQR for
sample medians when the sample size is 400
is approximately __________
of the width of
1/20
the population IQR
WIDTH
0.0349 kg
WIDTH
0.6805 kg
Q5: Relationship between the width of the IQR
for sample medians of sample size n and the
population IR and the sample size…
 IQR for sample medians (sample size = n) is
approximately of 1
the population IQR
n
 When n = 400 the IQR of the sample medians
is approximately ________________ of
population IQR
 When n = 30 the IQR of the sample medians is
approximately ________________ of
population IQR
How wide should
our interval be?
Lesson 8
DEVELOPING THE FORMULA FOR
INFORMAL CONFIDENCE
INTERVALS FOR THE
POPULATION MEDIAN
Kiwi kapers 3
Developing an informal
confidence interval for the
population median…
 For our informal confidence interval for the
population median we want to use
 Sample median
 Sample IQR/n
 We need to see how big to make this interval
so we’re pretty sure the interval includes the
population median
 We want it to work about 90% of the time
 Remember we’re in TEACHING WORLD
 We’re going to explore how wide our intervals
should be when we can work backwards from
a given population.
 Informal confidence intervals…
sample median  k x sample IQR/n
Dot Plot
Kiw ipop
3 different samples n = 30
3 different medians
3 different IQRs
1.5
2.0
2.5
3.0
3.5
w eight
Movable line is at 2.53
 What would be the ideal number (k) of
sample IQR/ n to use all the time to be
pretty sure the interval includes the
population median?
4.0
That is…
 We know what the population median
actually is
 We can look and see how far away from the
population median this is:
IQR
n
Worksheet 2
Deciding how many sample IQR/n we need
for the informal confidence interval
(finding k)
For each example…
1. Mark the sample median on the big graph and draw
a line to the population median
2. Find the distance the sample median is from the
population median (2.529kg)
3. Divide by sample IQR/n
 This gives the number of sample IQR /n that the
sample median is away from the population median
 THIS IS THE NUMBER WE ARE INTERESTED IN
Handout
1.
2.
3.
Mark the sample
median on the big
graph and draw a
line to the
population
median
Find the distance
the sample
median is from
the population
median (2.529kg)
Divide by sample
IQR/n
0.113
3. Divide by sample IQR/n
0.113/0.12689
= 0.89
This gives the number of
sample IQR/n that the
sample median is away from
the population median
0.159
0.159/0.1075
= 1.479
0.212
0.212/0.1479
= 1.433
EG 4) 0.1222
EG 5) 1.0399
EG 6) 1.0005
EG 7) 1.3007
EG 8) 2.2880
EG 9) 1.3370
EG 10) 1.4119
0.113
3. Divide by sample IQR/n
0.113/0.12689
= 0.89
This gives the number of
sample IQR/n that the
sample median is away from
the population median
0.159
0.159/0.1075
= 1.479
0.212
0.212/0.1479
= 1.433
From our 10
samples it
would appear
±1.5 x
IQR/sqrt(n)
would be
most
effective.
That is… it
should
capture the
population
median most
of the time
Final formula for informal
Confidence interval
The final formula for the informal confidence interval is :
I’d lost them…
Prezi recap [if time]
Lesson 9
PPDAC FOR SUMMARY &
CHECKING HOW WELL OUR
INTERVALS CAPTURE THE
POPULATION MEDIAN
Problem
 What is the median
weight of New
Zealand kiwis?
FORMULA FOR INFORMAL CONFIDENCE INTERVAL
FOR THE POPULATION MEDIAN
Plan & Data
 Simple random samples of 30 kiwis
 I sampled for you
Handout
Analysis
 Box plot
 Summary statistics
 I did this for you as well
YOU NEED TO…
 Use the formula to construct an informal
confidence interval for the population median
for each sample of 30 kiwis
Analysis
Conclusion
Use your interval from SAMPLE A to complete
the conclusion
 From my sample, I am pretty sure
that the median weight of New
Zealand kiwis is
between ____ and ____
Teaching and learning world
 How many of our informal confidence
intervals captured the population median?
Population median = 2.529 kg
Box Plot
Sample of Kiw ipop
1.6
1.8
2.0
2.2
2.4
2.6
2.8
w eight
3.0
3.2
3.4
3.6
3.8
How many “lots” of IQR/sqrt(n) our samples are
away from the population median
5
4
3
2
1
0
1
-1
-2
-3
-4
4
7
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100
Lesson 11 & 12
PPDAC FOR COMPARISON
(CLEAR COMPARISON)
Handout
Investigative question
 I wonder if the median
height of NZ kiwi females
tends to be greater than
the median height of NZ
kiwi males
Population parameter
Variable of interest
Groups/sub-populations
Population
Data
 I sampled for you
 Have a look how…
 http://www.censusatschool.org.nz
Analysis
Select and use appropriate displays and measures.
Construct the informal confidence intervals
Analysis
Discuss sample distributions by comparing features of them.
Compare
- shape
- overlap
- shift
- spread
- middle 50%
- unusual or
interesting
Conclusion - inference
 From my informal confidence intervals, I am
pretty sure that the population median height of
NZ kiwi females is between ____ and ____.
 Similarly, I’m pretty sure that the population
median height of NZ kiwi males is between ____
and ____.
Teaching and learning world NOTE – Matt Regan
In the conclusion…
We used "sure" rather than "confident" as we should reserve the use
of the term 'confident' to ideas about the confidence we have in our
interval estimate (i.e., our confidence interval) which is different from
the confidence we have about the 'pattern repeatability' and we don't
want students to get muddled.
Conclusion
 Based on these samples I would make
the call that the population median
height of NZ kiwi females is greater
then the population median height of
NZ kiwi males. That is, I would make the
call that NZ kiwi females tend to be
taller than NZ kiwi males back in the
two populations.
Conclusion
- justification
 The informal confidence interval for the
population median height of NZ kiwi females
is (much) further up the scale than the
informal confidence interval for the
population median height of NZ kiwi males
and these informal confidence intervals do
not overlap.
 I am quite sure that if I were to take another
sample of NZ kiwi females and another
sample of NZ kiwi males girls this nonoverlapping pattern in confidence intervals
for the population medians would persist,
thus giving the same conclusion.
Conclusion
Further thoughts…
 What would happen if you took another
sample and completed this process again?
 What would happen if to the informal
confidence intervals if you increased the
sample size?
Use statistical methods to make
an inference - ASSESSMENT