Download Ruling Out Chance

Navigating Statistics and
Data Analysis in the Secondary
Mathematics Curriculum
Roxy Peck
[email protected]
Quito, Ecuador
2012
A Bit of History…
 In many countries, statistics has been
a recommended part of the secondary
school mathematics curriculum for a
LONG time.
 For example, in the United States
 Curriculum and Evaluation Standards
(NCTM, 1989), Standard 10
 Principles & Standards for School
Mathematics (NCTM, 2000), Data Analysis
and Probability Standard
Quito, Ecuador
2012
A Bit of History…continued
BUT…
 Impact? Not so much.
 Implementation has been elusive.
 Where statistics has been
incorporated, it has often been done
in an ad hoc way.
 All things considered, disappointing
and little impact.
Quito, Ecuador
2012
A Bit of History…continued
So what has changed?
In the United States
Common Core State Standards in
Mathematics.
Statistics and probability standards give
statistics and probability a more prominent role
and place much more emphasis on conceptual
understanding.
Expected to have major impact on K-12 curriculum
beginning in 2013 -2014.
Will also impact AP Statistics course and university
level introductory statistics course.
Quito, Ecuador
2012
Secondary Statistics Standards Include:
(Subset of the Common Core Standards)
Summarize, represent, and interpret data on
a single count or measurement variable
S-ID.1. Represent data with plots on the real
number line (dot plots, histograms, and box
plots).
S-ID.2. Use statistics appropriate to the shape of
the data distribution to compare center
(median, mean) and spread (interquartile
range, standard deviation) of two or more
different data sets.
S-ID.3. Interpret differences in shape, center, and
spread in the context of the data sets,
accounting for possible effects of extreme data
points (outliers).
Quito, Ecuador
2012
The Standards – There’s More!
Summarize, represent, and interpret data on two
categorical and quantitative variables
S-ID.5. Summarize categorical data for two categories in twoway frequency tables. Interpret relative frequencies in the
context of the data (including joint, marginal, and
conditional relative frequencies). Recognize possible
associations and trends in the data.
S-ID.6. Represent data on two quantitative variables on a
scatter plot, and describe how the variables are related.



a. Fit a function to the data; use functions fitted to data to
solve problems in the context of the data. Use given functions
or choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and
analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear
association.
Quito, Ecuador
2012
The Standards – But Wait!
There’s Even More!
Make inferences and justify conclusions from sample
surveys, experiments, and observational studies
S-IC.3. Recognize the purposes of and differences among
sample surveys, experiments, and observational studies;
explain how randomization relates to each.
S-IC.4. Use data from a sample survey to estimate a population
mean or proportion; develop a margin of error through the
use of simulation models for random sampling.
S-IC.5. Use data from a randomized experiment to compare
two treatments; use simulations to decide if differences
between parameters are significant.
S-IC.6. Evaluate reports based on data.
Quito, Ecuador
2012
And not just in the U.S.
 Similar content is showing up in the
secondary curriculum of many other
countries, including




U.K.
Australia
New Zealand
And others!
Quito, Ecuador
2012
The Challenge…
 Integration of this content presents a
challenge for many secondary
mathematics teachers.
 How is statistical reasoning different
from mathematical reasoning?
 How can statistical reasoning be
developed?
Quito, Ecuador
2012
Consider the map of counties shown below. The number
in each county is last month’s incidence rate for a disease
in cases per 100,000 population.
(Dick Schaeffer, 2005)
Quito, Ecuador
2012
Statistical Thinking Versus
Mathematical Thinking
 Mathematical Thinking
 Explain patterns
 Often a deterministic way of thinking
 Statistical Thinking
 Search for patterns in the presence of
variability
 Acknowledge role of chance variation
Often involves “Ruling Out Chance” as an
explanation
Quito, Ecuador
2012
The BIG Idea…
 Ruling Out Chance!
 The plan for the rest of today’s
session:
Three classroom activities that
 illustrate the connections between
probability and statistical inference
 help students develop statistical thinking
Quito, Ecuador
2012
Ruling Out Chance…
Activity 1: The Cookie Game
(My favorite classroom activity!)
Quito, Ecuador
2012
Discussion Points
 Cookie Game illustrates the thought
process that underlies almost all of
statistical inference
 Could this have happened by chance when…
 Competing claims about a population, one of
which is initially assumed to be true (the null
hypothesis)
 Sample from the population
 A decision based on whether the observed
outcome would have been likely or unlikely to
occur BY CHANCE when the null hypothesis is
true
Quito, Ecuador
2012
Discussion Points
 Convincing evidence vs. proof
 Relationship between probability
assessment and choice of significance
level
Quito, Ecuador
2012
Ruling Out Chance…
 Activity 2: Inappropriate Dress
 A CareerBuilders (www.careerbuilders.com)
press release dated June 17, 2008 claims
more than one third of employers have sent
an employee home for inappropriate attire.
Suppose that in a random sample of 40
employers, 15 report that they have sent
an employee home to change clothes. Do
the data provide convincing evidence that
the CareerBuilders claim is correct?
Quito, Ecuador
2012
15
Sample proportion = = .38
40
Is chance variation from sample to sample
(sampling variability) a plausible
explanation for why the sample proportion
is greater than 1/3?
Quito, Ecuador
2012
 To be convinced, we must see a sample
proportion not just greater than 1/3, but
one that is enough greater than 1/3 that it
is not likely to have occurred just by chance
due to sampling variability.
 Ruling Out Chance: What kind of sample
proportions would not be convincing? What
kind of sample proportions would we expect
to see just due to chance when the
population proportion is 1/3 ?
Quito, Ecuador
2012
The dotplot
0.12
0.18
0.24
0.30
0.36
0.42
Simulated Prportions
Quito, Ecuador
2012
0.48
0.54
Discussion Points
 Can’t rule out chance…
 What does this conclusion mean?
Have we shown that the proportion is
NOT greater than 1/3?
Quito, Ecuador
2012
Ruling Out Chance…

Activity 3: Duct Tape to Remove Warts

Some people seem to believe that you can fix anything with
duct tape. Even so, many were skeptical when researchers
announced that duct tape may be a more effective and less
painful alternative to liquid nitrogen, which doctors
routinely use to freeze warts. The article “What a Fix-It:
Duct Tape Can Remove Warts” (San Luis Obispo Tribune,
October 15, 2002) described a study conducted at Madigan
Army Medical Center. Patients with warts were randomly
assigned to either the duct tape treatment or the more
traditional freezing treatment. Those in the duct tape group
wore duct tape over the wart for 6 days, then removed the
tape, soaked the area in water, and used an emery board to
scrape the area. This process was repeated for a maximum
of 2 months or until the wart was gone.
Data on handout

Quito, Ecuador
2012
 Duct tape was more successful (84% successes) than
liquid nitrogen (60% successes), but is this convincing
evidence that the duct tape treatment is superior?
Could this have happened by chance just due to the
random assignment?
 Ruling Out Chance: Suppose that there is no
difference between the treatments and that these 36
people would have had successful removal no matter
which treatment was applied. If this is the case, the
difference between the 15 successes for the liquid
nitrogen group and the 21 successes for the duct tape
group is just due to the “luck of the draw” when the
random assignment to groups was done.
Quito, Ecuador
2012
The Frequency Distribution
Based on 100 trials
Number
Of Successes
15
16
17
18
19
20
21
22
Count
3
8
26
24
26
10
2
1
Percent
3.00
8.00
26.00
24.00
26.00
10.00
2.00
1.00
Quito, Ecuador
2012
Discussion Points
 Observed result “unlikely” to have
occurred by chance just due to the
random assignment of subjects to
experimental groups
(reference to Cookie Game activity
for what constitutes “unlikely”)
 “Ruled out” chance. What does this
conclusion mean? Can we be certain?
Quito, Ecuador
2012
Later, in AP Statistics or
University Level Statistics…
 Can move from activities like these to the
more formal approaches, linking to the
activities
 Sampling distributions become our way of
deciding whether we can rule out chance in
situations where our intuition isn’t adequate (as
it was in the cookie game) or when we get tired
of simulating (in the case of proportions).
 Can then move from situations involving
proportions where simulation is more straight
forward to situations involving means that don’t
lend themselves to simulation. Theoretical
results rather than simulation provide the
information needed to decide if we can “rule out
chance”.
Quito, Ecuador
2012
Linking to Traditional Methods
 Hypothesis testing logic and types of conclusions that
can be drawn—students remember the cookie game
activity and the logic involved
 Convincing evidence and significance levels (the
cookie game activity makes the customary choices for
significance levels seem intuitively reasonable—What
does it take to rule out chance?)
 P-Values as a basis for drawing a conclusion (the
activities motivate why it is reasonable to base
conclusions about a population based on the P-Value
as a measure of whether it is likely or unlikely that we
would observe a sample result as extreme as what
was observed just by chance if the null hypothesis is
true)
Quito, Ecuador
2012
Thanks for coming!
Feel free to contact me with any questions.
[email protected]
Quito, Ecuador
2012