Teaching the High School Common Core Statistics Standards Po

```The New Illinois Learning Standards for High School
Statistics and Probability
Dana Cartier
Illinois Center for School Improvement
Julia Brenson
Lyons Township High School
Tina Dunn
Lyons Township High School
The New Illinois Learning Standards
Agenda
 Resources Available Through ISBE
 Algebra II/Math III
– Normal Distribution, Random Sampling, Experimental
Design, and Comparing Two Treatments

Algebra I/Math I (and above)
– Assessing the Fit of a Function to Data

Algebra II/Math II
– Conditional Probability
The New Illinois Learning Standards
ILStats
http://ilstats.weebly.com/
 All materials from this session are available at this
website.
 This website is currently under construction, but
The New Illinois Learning Standards
Algebra II and Math III
Statistics Standards for Algebra II/Math III
Normal Distribution
Algebra II & Math III
Standard
PBA
EOY
S.ID.4
X
X
Statistics Standards for Algebra II/Math III

S-ID.4 Use the mean and standard deviation
of a data set to fit it to a normal distribution
and to estimate population percentages.
Recognize that there are data sets for which
such a procedure is not appropriate. Use
estimate areas under the normal curve.
Statistics Standards for Algebra II/Math III
Normal Distribution
The Normal Distribution is:
 “Bell-shaped” and symmetric
 mean = median = mode
 Larger standard deviations produce a distribution with greater spread.
μ = 10, σ = 1
μ = 10, σ = 2
Statistics Standards for Algebra II/Math III
Normal Distribution
The Empirical Rule
68%
95%
97.5%
Statistics Standards for Algebra II/Math III
Normal Distribution
Example: The ACT is normally distributed with a mean of 21 and
a standard deviation of 5.
.0235 + .0015 = .0250
6
11
16
21
26
31
36
1) Using the Empirical Rule, estimate the probability that a randomly
selected student who has taken the ACT has a score greater than 31.
2) What percent of students score less than or equal to 31.
3) What does this tell you about an ACT score greater than 31?
Statistics Standards for Algebra II/Math III
Normal Distribution
Animal Cracker Lab
The label on a 2.125 oz. Barnum’s
Animal Cracker box says that there
are 2 servings per box. A serving size
is 8 crackers.
How many crackers do we typically
expect to find in a box?
How do you think Nabisco determined
this number?
Will every box have exactly this many
animal crackers?
Statistics Standards for Algebra II/Math III
Normal Distribution
Animal Cracker Lab
The graph at right shows the
n = 28 boxes
distribution of the number of
crackers in a sample of 28 Barnum’s
Collection 2
Animal Crackers boxes.
The label on the box indicated that
we should expect 16 cookies in a
box. Based on the graph and
statistics at right, how likely is it that
a box contains less than 16 cookies?
12
mean
Why does Nabisco tell the consumer
there are 16 cookies in a box?
stdDev
count
14
16
= 20.0357
= 0.912146
= 28
Dot Plot
18
20
22
Anim al_Crackers
24
26
28
Statistics Standards for Algebra II/Math III
Normal Distribution
Activities:
Animal Cracker Lab
 See Illustrative Mathematics Activities:
 SAT Scores
 Should We Send Out a Certificate?
 Do You Fit In This Car?

Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Algebra II & Math III
Standard
PBA
EOY
S.IC.1
X
X
Statistics Standards for Algebra II/Math III

S-IC.1 Understand statistics as a process for
parameters based on a random sample from
that population.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
A statistic is a numerical summary computed from a
sample. A parameter is a numerical summary computed
from a population. A statistic will vary depending on
the sample from which it was calculated, but a
population parameter is a constant value that does not
change.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Suppose we wish to know something about a
population. For example we might want to know
the average height of a 17 year old male, the
proportion of Americans over 70 who send text
messages, or the typical number of kittens in a litter.
It is often not possible or practical to collect data
from the entire population, so instead, we collect
data from a sample of the population. If our
sample is representative of the population, we can
make inferences, or in other words, draw
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Activity: Random Rectangles
What is the size (area) of a typical rectangle in
our population of 100 rectangles?
Random Rectangles is used with permission from Richard L. Scheaffer
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Judgment Sample
First ask students to take a quick look at the population of rectangles
and then select 5 rectangles that they think together best represent
the rectangle population. This is a judgment sample. Students
record the rectangle number and the area of the rectangle for each
of the five rectangles in the table provided and calculate the mean
of the sample. Each student records their mean on the class dot plot
on the chalkboard. Repeat this process 4 more times.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
How do we ensure that we select a sample that is
representative of the population? We choose a method
that eliminates the possibility that our own preferences,
favoritism or biases impact who (or what) is selected. We
want to give all individuals an equal chance to be chosen.
We do not want the method of picking the sample to
exclude certain individuals or favors others. One method
that helps us to avoid biases is to select a simple random
sample. If we want a sample to have n individuals, we use
a method that will ensure that every possible sample from
the population of size n has an equal chance of being
selected.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Suppose we wanted a simple random sample of size 4
from a class of 20 students. The class has 10 juniors and
10 seniors. Which of the following sampling methods
would result in a simple random sample?
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
A) Write the names of each of the 20 students
on a separate slip of paper, place the slips
in a hat, mix the slips, and without looking
selects four slips of paper.
B) Beginning with the first row, use a calculator
to pick a random number from 1 to 5. Count
back to the student sitting in the seat
designated by the random number and select
this student for the sample. Repeat for each
row.
C) First puts the names of the 10 juniors in one
box and the names of the 10 seniors in
another box. Randomly selects 2 juniors from
the first box and 2 seniors from the second.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Back to Random Rectangles
Use a calculator or a random digits table to select a
simple random sample of size 5 from the rectangles.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Random Digits Table
There are 100 rectangles. First select a row to use in the
table. Select two digits at a time, letting 01 represent 1, 02
represents 2, and so on with 00 representing 100. Skip
repeats.
Our Sample: 36, 79, 22, 62, 33
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Calculator
Reseed: Enter a four digit number
STO
MATH  PRB
1: rand
ENTER.
Generate five random numbers
from 1 to 100 inclusive.
MATHPRB
5: randInt(1, 100, 5)
ENTER.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Sampling Distribution
100 Samples of Size 5
mean = 7.762
Dot Plot
Measures from Sample of Rectangles
2
4
mean
= 7.762
count
= 100
stdDev
= 2.45438
6
Sample Distribution
500 Samples of Size 5
mean = 7.356
8
10
Sam pleMean
12
14
Dot Plot
Measures from Sample of Rectangles
2
16
4
mean
= 7.3564
count
= 500
stdDev
6
8
10
Sam pleMean
12
= 2.39474
As the number of samples increases, the mean of the sampling
distribution gets closer and closer to the mean of the population.
14
16
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Sampling Distribution
100 Samples of Size 10
mean = 7.67 square units
4
6
8
10
Sam pleMean
12
14
16
2
4
mean
= 7.67
mean
= 7.5814
count
= 100
count
= 500
stdDev
= 1.84826
Dot Plot
Measures from Sample of Rectangles
Dot Plot
Measures from Sample of Rectangles
2
Sample Distribution
500 Samples of Size 10
mean = 7.581 square units
stdDev
6
8
10
Sam pleMean
12
14
= 1.71477
As the sample size increases, the spread of the sampling distribution
decreases. (The standard deviation gets smaller.)
16
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Big Ideas:




When multiple samples are taken from the population,
the values of the sample statistics vary from sample to
sample. This is known as sampling variability.
If the population distribution is not too unreasonably
skewed, as more and more samples are taken from the
population, the mean (center) of the sampling distribution
approaches the population parameter.
As the sample size increases, the spread of the sampling
distribution decreases.
The shape of the sampling distribution is approximately
normal.
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Activities:
Random Rectangles
 Reese’s Pieces

What proportion of Reese’s Pieces are Orange?
(http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html
Permission to share with Illinois math teachers has been given by Beth Chance
and Allan Rossman .)
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Algebra II & Math III
Standard
PBA
EOY
S.IC.3
X
X
S.IC.6
X
X
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Three types of statistical studies are surveys, observational
studies, and experiments.
 In a survey the researcher gathers information by asking
the subjects questions.
 In an observational study, the researcher observes and
 In an experiment, the research randomly assigns subjects
to treatment groups and notes their response.
For each of these three types of studies, if we want to make
inferences (draw conclusions) that we can generalize from the
sample to the population, the subjects must be selected
randomly. If the sample of subjects is not randomly selected,
we can only make conclusions about the sample
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
More on Observational Studies
There are times when it is unethical or impractical to assign subjects to
a treatment group. For example, if we wanted to measure the longterm effects of smoking, it would not be good to ask subjects to take up
smoking. If we want to decide which math text book is best at
improving student performance, it might be impractical to ask a group
of teachers to teach one group of students using textbook A and
another group of students using textbook B. In situations like these,
rather than randomly assigning subjects to treatments (smoking,
textbook A), we instead make observations of groups that subjects are
already a part of. For example, we randomly select a group of
smokers and randomly select a group of non-smokers and record our
observations for both groups. Since the subjects are not assigned
randomly to a treatment group, we may not conclude a cause-andeffect relationship from an observational study.*
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
More on Experiments
An experiment allows us to study the effect of a treatment, such as a
drug or some type of experience, on the subjects. For example to
investigate if a new cholesterol medicine is more effective than a current
brand, subjects could be randomly assigned to the treatment new
medicine or old. In an experiment other factors that might also have an
effect on the response are identified. Starting cholesterol level,
exercise, diet, and weight might all have an effect on the subject’s final
cholesterol level. The researcher may try to control some of these
factors so that they are the same for both treatment groups. For
example, all participants may be given the same diet. Randomization
(randomly assigning subjects to treatments) helps to ensure that factors,
such as being overweight or not exercising, are likely to be present in
both treatment groups. A randomized, controlled experiment allows us
to conclude that the treatment caused an effect (response). To be able
to make inferences from the sample to the population, an adequate
number of observations must be collected. This is called replication.
What conclusions may we draw from statistical studies?
How were subjects selected?
Random Sampling
Random
How were Assignment
subjects
assigned to
treatment
groups?
No Random
Assignment




No Random Sampling
 May infer cause and effect,
but
May infer cause and effect
 Cannot generalize findings
AND
from sample to the
May generalize findings
population. (We can
from sample to the
conclude that the treatment
population
caused a response for this
sample only.)
May generalize findings  Cannot generalize findings
from sample to population,
from sample to population
but
AND
Cannot infer cause and
 Cannot infer cause and
effect.
effect
(adapted from Ramsey and Schafer’s The Statistical Sleuth)
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity :Chocolate Taste Test
(http://www.today.com/video/today/54076112#54076112)
Guided Classroom Discussion
 What was the population of interest?
 How were subjects selected?
 Is this a survey, an observational study, or an
experiment?
 If this is an experiment, what are the treatment
groups? How were the subjects assigned to the
treatment groups?
 What conclusions did the investigator make as a
result of this study? Were these conclusions
appropriate? Explain.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity :Baseball and Break-Away Bases
Read: Study finds break-away bases effective in professional baseball.
http://www.noinjury.com/articles/bases.htm.
Guided Classroom Discussion
 What was the population of interest?
 How were subjects selected?
 What are the treatments?
 Is this a survey, an observational study, or an experiment?
 How were the subjects assigned to the treatment groups?
 What conclusions did the investigator make as a result of
this study? Were these conclusions appropriate? Explain.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activities:
 Chocolate Taste Test (http://www.today.com/video/today/54076112#54076112)
 Did You Wash Your Hands?
http://www.amstat.org/education/msss/pdfs/MSSS_SampleInvestigation.pdf Activity used with permission from
Roxy Peck.)

Duct Tape Therapy
(The Efficacy of Duct Tape vs Cryotherapy in the Treatment of Verruca Vulgaris (the Common War) available as a
http://archpedi.jamanetwork.com/article.aspx?articleid=203979&resultClick=1)

Break-Away Bases
http://www.noinjury.com/articles/bases.htm)

High blood pressure
(www.illustrativemathematics.org)
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Algebra II & Math III
Standard
PBA
EOY
S.IC.5
X
X
Statistics Standards for Algebra II/Math III

S-IC.5 Use data from a randomized
experiment to compare two treatments; use
simulations to decide if differences between
parameters are significant.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Sleep Deprivation
Does the effect of sleep deprivation linger or can we
“make up” for lost sleep? To test this, 21 volunteer
subjects ages 18 to 25 were randomly assigned to one
of two treatment groups. Both groups first received
training on a visual discrimination task. One group
was deprived of sleep for the first night following this
training, but were allowed unlimited sleep on the next
two nights. The second group was allowed unlimited
sleep on all three nights. Both groups were retested
on the third day.
Used with permission from Beth Chance and Allan Rossman.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Based on the graph and statistics below, do you think there is
evidence that sleep deprivation on the first night might have
had an effect on a subject’s improvement on the visual
What is the difference between the means for the two groups? 15.92
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Primary Question of Inference:
If the treatment had no effect, is it possible
that we would see this great a difference
simply by chance (random assignment)?
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Let’s investigate.

Here is the data for the 21 subjects:
(A negative value indicates a decrease in performance.)
Sleep Deprivation Group
Unrestricted Sleep Group
-10.7
9.6
25.2
45.6
4.5
2.4
14.5
11.6
2.2
21.8
-7.0
18.6
21.3
7.2
12.6
12.1
-14.7
10.0
34.5
30.5
-10.7
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Simulation

Write each of the improvement scores on a separate
card. Shuffle the cards and deal them into two
groups. The first group will be sleep deprivation. The
second is unrestricted sleep. Find the mean of each
group, then find the difference in the two means. Plot
this value on a class dot plot and record the value on
the class table. Repeat the process 4 more times.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Simulation Continued
Now let’s let technology take over and create the
sampling distribution as more and more samples are
selected.
A Sleep Deprivation simulation, can be found in the
Rossman/Chance Applet Collection. Look under
Statistical Inference


•
Randomization Test for quantitative response (two groups)
http://www.rossmanchance.com/applets/randomization20/Randomization.html
Used with permission from Beth Chance and Allan Rossman.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Computer Simulation of 1000 Trials
How likely is it to get a
difference in the mean
improvement score that is
15.92 or higher by chance
(random assignment)?
10 out of 1000
samples (1.0%) were
15.92 or higher.
Not very likely!
Used with permission from Beth Chance and Allan Rossman.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Conclusion:
Can we conclude cause and effect?

Can we generalize our findings for the sample to a
population? (Who is the population of interest in this
study?)
From the simulation we can see that the difference between
the means of the two treatment groups (sleep deprived
and unrestricted sleep) is very unlikely to happen simply by
chance (random assignment). We conclude that sleep
deprivation, even when followed by two nights of
unrestricted sleep, did have an effect on subjects’
improvement on the visual discrimination task.

Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Distracted Driver
Are drivers more distracted when using a cell phone
than when talking to a passenger in the car?
In a study involving 48 people, 24 people were randomly
assigned to drive in a driving simulator while using a cell
phone. The remaining 24 were assigned to drive in the driving
simulator while talking to a passenger in the simulator. Part of
the driving simulation for both groups involved asking drivers
to exit the freeway at a particular exit. In the study, 7 of the
24 cell phone users missed the exit, while 2 of the 24 talking to
a passenger missed the exit. (from the 2007 AP* Statistics exam, question 5)
Activity created by Peck, R., Starnes, D., & Rowland, C. (2007 NCSSM Summer Statistics Writing Program
Permission given by Roxy Peck to share with Illinois Math Teachers
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Distracted Driver
Activity created by Peck, R., Starnes, D., & Rowland, C. (2007 NCSSM Summer Statistics Writing Program
Permission given by Roxy Peck to share with Illinois Math Teachers
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Primary Question of Inference:
If the treatment (cell phone vs. passenger) had
no effect, is it possible that we would see this
great a difference simply by chance (random
assignment)?
Activity created by Peck, R., Starnes, D., & Rowland, C. (2007 NCSSM Summer Statistics Writing Program
Permission given by Roxy Peck to share with Illinois Math Teachers
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
The simulation



How might we use a deck of cards to represent the response: 9
people missing the exit (distracted) and 39 people who were not
distracted?
One possibility
Distracted: A-9 clubs
Not Distracted:
Shuffle the cards and deal them into two piles. The first pile will be
the cell phone treatment group. The second pile is the passenger
treatment group. Count the number of clubs in each group. This
represents the number of people who missed the exit (distracted
drivers) that occur by chance. Each group repeats 9 more times.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Computer Simulation of 1000 Trials from Teacher Notes
In the original experiment 7
members of the cell phone group
were distracted and missed the
exit. In the simulation, how often
“just by chance” did the cell
phone group have 7 or more
distracted drivers?
With random reassignment, 6.8%
of the cell phone group missed
the exit 7 or more times.
Activity created by Peck, R., Starnes, D., & Rowland, C. (2007 NCSSM Summer Statistics Writing Program
Permission given by Roxy Peck to share with Illinois Math Teachers
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activities:

Sleep Deprivation
(Permission given by Beth Chance and Allan Rossman)
 First read the abstract from Stickgold, R., James, L., & Hobson, J. A. (2000). Visual
discrimination learning requires sleep after training. Nature Neuroscience, 3(12), 12371238.
•
After physical (hands-on) simulation, use applet titled Randomization Test for quantitative
response (two groups) available from
http://www.rossmanchance.com/applets/randomization20/Randomization.html

Distracted Driving
(Permission given by Roxy Peck)
Peck, R., Starnes, D., & Rowland, C. (2007 NCSSM Summer Statistics Writing Program

Student
(http://courses.ncssm.edu/math/Stat_Inst/Stats2007/Distracted%20Driver/Distracted%20driving%20
final.pdf)

Teacher Notes
(http://courses.ncssm.edu/math/Stat_Inst/Stats2007/Distracted%20Driver/Distracted%20d
riving%20Teacher%20version%20final.pdf)
The New Illinois Learning Standards
Algebra I & Math I
Standard
S.ID.6a
S.ID.6b
PBA EOY
PARCC Model Content Frameworks and Blue Prints
i) Tasks have a real-world context.
Algebra I ii) Exponential functions are limited to those with domains in
X
the integers.
i) Tasks have a real-world context.
Algebra II ii) Tasks are limited to exponential functions with domains not X
X
in the integers and trigonometric functions.
Math I
X
ii) Tasks are limited to linear functions and exponential
functions with domains in the integers.
X
Functions (See EOY Evidence Table S-ID.Int.2 and S-ID.6a-1)
i) Tasks have a real-world context.
Math III ii) Tasks are limited to exponential functions with domains not X
X
in the integers and trigonometric functions.
Possibly quadratic functions (See EOY Evidence Table – SAlgebra I
X
ID.Int.2)
Math II and exponential functions with domains in the integers.
X
Course
Math III
i) Tasks have a real-world context.
ii) Tasks are limited to exponential functions with domains not
in the integers and trigonometric functions.
X
The New Illinois Learning Standards

S-ID.6 Summarize, represent, and interpret data on two
categorical and quantitative variables. Represent data
on two quantitative variables on a scatter plot, and
describe how the variables are related.*



S-ID.6a 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
S-ID.6b Informally assess the fit of a function by plotting and
analyzing residuals.*
S-ID.6c Fit a linear function for a scatter plot that suggests a
linear association.*
Statistics Standards for Algebra I/Math I
Interpret Linear Models
Algebra I & Math I
Standard
PBA EOY
S.ID.6
X
S.ID.7
X
S.ID.8
X
S.ID.9
X
The New Illinois Learning Standards

S-ID.7 Interpret linear models. Interpret the
slope (rate of change) and the intercept
(constant term) of a linear model in the context
of the data.*
 S-ID.8 Interpret linear models. Compute (using
technology) and interpret the correlation
coefficient of a linear fit.*
 S-ID.9 Interpret linear models. Distinguish
between correlation and causation.*
Statistics Standards for Algebra I/Math I
Interpreting Scatterplots
Below is a graph of the heights and
weights of currently rostered
Chicago Bears (April 2014).
Which is the best interpretation
of the scatterplot?
A. As heights go up, weight
increases.
B. As heights go up, weight
tends to increase.
C. If you get taller, you will get
heavier.
D. Taller football players tend
to be heavier football
players.
http://chicagosports.sportsdirectinc.com/football/nfl-teams.aspx?page=/data/nfl/teams/rosters/roster16.html
Statistics Standards for Algebra I/Math I
Assessing the Fit of a Linear Function
To assess the fit of a linear function to a
scatterplot consider all of the following:
1. Look at the scatterplot.
Does the data appear linear?
2. Look at r-value.
What is the strength of the linear relationship
between x and y?
3. Look at the residuals.
Are they scattered with no discernable pattern?
Statistics Standards for Algebra I/Math I
Correlation
When interpreting the correlation coefficient there are
FOUR things that should be discussed:
1. The strength of the linear relationship: strong,
moderate, weak, or no linear relationship.
Statistics Standards for Algebra I/Math I
Correlation
2. Whether the relationship between x and y is
positive or negative. If the slope of the best fit line is
positive, then the r value is also positive. If the slope of
the best fit line is negative, then the r value is negative.
Statistics Standards for Algebra I/Math I
Correlation
3. The relationship we are evaluating is a linear
relationship between x and y.
For example, the two graphs below show a relationship
between x and y that is something other than linear.
Statistics Standards for Algebra I/Math I
Correlation
4. CONTEXT!
Be sure to interpret the correlation coefficient in the context
of the problem. The linear association is between what two
variables?
Interpretation: r = 0.7
There is a moderate,
positive linear relationship
between the height (in
inches) and the weight (in
pounds) for the currently
rostered Chicago Bears.
Statistics Standards for Algebra I/Math I
Correlation vs. Causation
Example:
Do fresh lemons cause a lower highway fatality rate?
www.grossmont.edu/johnoakes/s110online/Causation%20versus%20Correlation.pdf
Statistics Standards for Algebra I/Math I
Residuals

The residual is the vertical distance from the data
point to the line of best fit.
(5, 28)
actual
(5, 21)
predicted
Residual = Actual – Predicted
Residual = 28 – 21
Residual = 7
Statistics Standards for Algebra I/Math I
Residuals

To create a residual plot, graph each x-coordinate
with its corresponding residual. (x-value, residual)
(x-value, residual)
(5, 7)
Residual = 7
This horizontal axis
corresponds to the
regression line.
Statistics Standards for Algebra I/Math I
Residuals


Patterns of a residual plot are used to assess
the fit of a function to the data.
Patterns should appear scattered with no
discernible pattern.
Outlier
Curved pattern
Larger residuals for
larger values of x
Statistics Standards for Algebra I/Math I
Residuals
Why do we need to look at residual plots?
Here is an example from Engage NY:
The temperature (in degrees Fahrenheit) was measured
at various altitudes (in thousands of feet) above Los
Angeles. The scatter plot (next slide) seems to show a
linear (straight line) relationship between these two
quantities.
http://www.engageny.org/sites/default/files/resource/attachments/algebra_i-m2-teacher-materials.pdf
Statistics Standards for Algebra I/Math I
Assessing the Fit of a Function to Data
Activity: Altitude vs. Temperature
The outside air temperature (in degrees Fahrenheit) for various
altitudes (in thousands of feet) was measured for a plane flying
above Los Angeles. Is there a linear relationship between altitude
and temperature?
http://www.engageny.org/sites/default/files/resource/attachments/algebra_i-m2-teacher-materials.pdf
Statistics Standards for Algebra I/Math I
Assessing the Fit of a Function to Data
Altitude vs. Temperature
Part I: Scatterplot.
Does the data appear linear?
Part II: Correlation Coefficient
What is the strength of the linear relationship
between x and y?
Statistics Standards for Algebra I/Math I
Assessing the Fit of a Function to Data
Altitude vs. Temperature
Part III: Residuals
Are they scattered with no discernable pattern?
Statistics Standards for Algebra I/Math I
Residuals
Example – Snakes! (www.insidemathematics.org)
Snake 1
Snake 1
Rita catches 5 more snakes.
She wants to know whether they belong to species A or to species B.
The measurements of these snakes are shown in the table below.
Statistics Standards for Algebra I/Math I
Fit a Function to Data and Interpret Linear Models

Altitude vs. Temperature
Adapted from an activity featured at www.engageny.org.

Snakes
– A residual activity from Inside Mathematics
The New Illinois Learning Standards
Algebra II and Math II
Statistics Standards for Algebra II/Math II
Making Inferences & Justifying Conclusions
Algebra II & Math II
Standard
PBA
EOY
S.CP.1
X
S.CP.2
X
S.CP.3
X
S.CP.4
X
S.CP.5
X
S.CP.6
X
S.CP.7
X
Statistics Standards for Algebra II/Math II
Conditional Probability and the Rules of Probability
Big Ideas




Provide opportunities for students to independently
determine if a problem is asking for an AND, OR, or
conditional probability.
Provide data from real life problems. Ask students to
calculate probabilities and interpret probabilities in the
context of the problem.
Students should be able to create and interpret two-way
frequency tables, diagrams, probability trees, and other
graphical displays.
Determine if two events are independent or dependent,
and interpret what this means in the context of the
problem.
Independence
Two events E and F are said to be independent if:
= ()
If E and F are not independent, they are said to be dependent events.
If    = (), it is also true that    = (), and vice versa.
If two events E and F are independent then the Multiplication Rule becomes:
∩  = () ∙ ()
Examples of events which are likely to be dependent:
 The event that a student spends more time on homework and studying and the event that the
student’s test average goes up.
 The event that the amount of snow is above average in a given winter and the event that the
number of show shovel sales is also above average for that winter.
 The event that a randomly selected passenger on the Titanic was a woman and the event that
the passenger survived.
Example of events which are independent
 The event that a 2 is rolled on the first roll of a dice and the event that a 2 is rolled on a second
roll of a dice.
 The event that a couple’s first child is a boy and the event that a couple’s second child is a girl.
Statistics Standards for Algebra II/Math II
Conditional Probability and the Rules of Probability
Activity: Conditional Probability Four Square
Acknowledgements and Resources
Chance, B. & Rossman, A. (Preliminary Edition). Investigating Statistical Concepts, Application
and Methods. Duxbury Press.
http://www.rossmanchance.com/.
http://chicagosports.sportsdirectinc.com/football/nflteams.aspx?page=/data/nfl/teams/rosters/roster16.html
Daily Mail. (2012, December 2). What are the odds? New study shows how guessing heads
Duggan, M. & Brenner, J. (2013, February 14). The Demographics of Social Media Users –
Focht, D., Spicer, C, and Fairchok, M. (2002). The Efficacy of Duct Tape vs Cryotherapy in the
Treatment of Verruca Vulgaris (the Common Wart). 156 (10) pp. 971-974. Retrieved
from http://archpedi.jamanetwork.com/article.aspx?articleid=203979&resultClick=1.
Acknowledgements and Resources
Franklin, C., Kader, G., Mewborn, J. M., Peck, R., Perry, M. & Schaeffer, R. (2007) Guidelines
for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K-12
Curriculum Framework. Alexandria, VA: American Statistical Association.
McCallum, B., et al. (2011, December 26). Progressions for the Common Core State
http://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bi
s.pdf.
McCallum, B., et al. (2012, April 21). Progressions for the Common Core State Standards in
Moore, D. & McCabe, P. (1989). Introduction to the Practice of Statistics. New York, NY: W.
H. Freeman.
Oakes, J. “Causation verses Correlation” Grossmont. Retrieved July 7, 2013, from
www.grossmont.edu/johnoakes/s110online/Causation%20versus%20Correlation.pdf
Peck, R., Gould, R., & Miller, S. (2013). Developing Essential Understand of Statistics for
Teaching Mathematics in Grades 9-12. Reston, VA: The National Council of Teachers of
Mathematics, Inc.
Acknowledgements and Resources
Peck, R., Olsen C. & Devore J. (2005). Introduction to Statistics and Data Analysis. Belmont,
CA: Brooks/Cole.
Peck, R. & Starnes, D. (2009). Making Sense of Statistical Studies. Alexandria, VA: American
Statistical Association.
Ramsey, F. & Schafer, D. (2002). The Statistical Sleuth: A Course in Methods of Data
Analysis. Boston, MA: Brooks/Cole, Cengage Learning.
Rossen, J. (2014, January 15). Taste Test Pits Fine Chocolate Against Cheaper Brands.
http://www.today.com/video/today/54076112#54515111.
Rossman, A. (2012). Interview With Roxy Peck. Journal of Statistics Education, 20(2). pp. 1
Rossman, A., Chance, B., & Von Oehsen, J. (2002). Workshop Statistics Discovery With Data
and the Graphing Calculator. New York: Key College Publishing.
Acknowledgements and Resources
Scheaffer, R., Gnanadesikan, M., Watkins, A., & Witmer, J. (1996). Activity-Based
Statistics. New York: Springer-Verlag.
Stickgold, R., James, L. & Hobson, J. (2000). Visual discrimination learning requires
http://www.nature.com/neuro/journal/v3/n12/pdf/nn1200_1237.pdf.
Strayer, D. and Johnston, W. (2001, November 6) 12(6). Driven to Distraction: DualTask Studies of Simulated Driving and Conversing on a Cellular Telephone. Pp. 462466Retrieved from http://www.psych.utah.edu/AppliedCognitionLab/PSReprint.pdf.
The Institute for Preventative Sports Medicine. (2001) Study finds break-away bases
http://www.noinjury.com/articles/bases.htm.
Online Resources
Census at School. http://www.amstat.org/censusatschool/
http://causeweb.org/
Engage NY. http://www.engageny.org/mathematics
Illustrative Mathematics. http://www.illustrativemathematics.org/
Inside Mathematics. http://www.insidemathematics.org
Mathematics Assessment Project. http://map.mathshell.org/
Math Vision Project. http://www.mathematicsvisionproject.org/
NCSSM Statistics Institutes.
NCTM Core Math Tools – Data Sets
http://www.nctm.org/resources/content.aspx?id=32705
Online Resources
PARCC Model Content Frameworks.
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovemb
er2012V3_FINAL.pdf
PARCC Mathematics Evidence Tables. https://www.parcconline.org/assessmentblueprints-test-specs
Smarter Balanced Assessment Consortium. http://www.smarterbalanced.org/
Statistics Education Web (STEW). http://www.amstat.org/education/STEW/
The Data and Story Library (DASL). http://lib.stat.cmu.edu/DASL/
The High School Flip Book Common Core State Standards for Mathematics.
http://www.azed.gov/azcommoncore/files/2012/11/high-school-ccss-flipbook-usd-259-2012.pdf
The New Illinois Learning Standards for Algebra I / Math I
Statistics and Probability
Thank you for joining us!
Dana Cartier
Julia Brenson
Tina Dunn
[email protected]
[email protected]
[email protected]
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