Transcript Statistics Standards for Algebra II/Math III Normal Distribution

The New Illinois Learning Standards for Algebra II / Math III
Statistics and Probability
Julia Brenson
The Four Components of a
Statistical Investigation*
1) Formulate a question
2) Design and implement a plan to collect data
3) Analyze the data by measures and graphs
4) Interpret the results in the context of the
original question
*Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report
American Statistical Association
http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf
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
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
Mean = 20.04 cookies
Standard Deviation = 0.91 cookies
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
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
conclusions about the population.
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
More about simple random samples.
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
of your choice into your TI-84 then
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
100 Rectangles
7.7
9.6
5.6
5.6 6.4
5
5.5 6 6.5
6.4
7.7
7 7.5 8 8.5 9
9.6
9.5 10
Simulation Process Model shared with permission from Sharon Lane-Getaz
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
Understand and Evaluate Random Processes
Algebra II & Math III
Standard
PBA
EOY
S.IC.2
X
X
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Big Idea

Use simulation to determine probabilities or verify a probability
model.
Example: Spinning Pennies

Stanford professor Persi Diaconis’ research indicates a spinning
penny lands tails up approximately 80% of the time. Can this
be true?
See “What are the odds? New study shows how guessing heads or tails isn’t really a 50-50 game,”
(Daily Mail http://www.dailymail.co.uk/news/article-2241854/What-odds-New-study-showsguessing-heads-tails-isnt-really-50-50-game.html )

Questions:
 What is the probability of a spinning penny landing tails up?
 What is the probability that a spinning penny lands tails up
at least 80% of the time?
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Number of Tails in 40 Spins
?
Spins
Statistics Standards for Algebra II/Math III
Understand and Evaluate Random Processes
Activities:
 Penny Spinning
 Sarah the Chimpanzee
(www.illustrativemathematics.org)

Block Scheduling
(www.illustrativemathematics.org)
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
records characteristics about the subjects.
 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.
The Institute for Preventative Sports Medicine. (2001) Retrieved from
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?
(from Making Sense of Statistical Studies and posted for free download at
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
free download from JAMA Pediatrics)
http://archpedi.jamanetwork.com/article.aspx?articleid=203979&resultClick=1)

Break-Away Bases
(Study finds break-away bases effective in professional baseball. Retrieved from
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.4
X
X
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
In our work with S.IC.2 (Random Rectangles), we found that the center
of the sampling distribution provides a reasonable estimate of the
population parameter. In S.IC.4, we use a sample statistic +/- a
margin of error to make inferences about the population parameter.
Here are the key ideas:
 We take a random sample from the population. The sample statistic
will be our estimate of the population parameter.
 We know that our sample proportion is likely to differ from the
population proportion. How much do we expect it to differ? The
margin of error is the anticipated difference between the sample
proportion and the true population proportion.
 S.IC.4 states that students should be able to find a margin of error
using simulation.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Margin of Error
Part I Do you Tweet?
The Pew Internet Research Project, in an internet
article titled The Demographics of Social Media
Users – 2012, reported the results from a
landline and cellphone survey of internet users.
The table below is taken from the online article.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Part I Do You Tweet?
Questions:
1. What was the population
of interest?
2. How many people were in
the sample?
3. What percent of all
internet users use Twitter?
4. What margin of error is
reported?
5. What do you think this
margin of error means?
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Margin of Error
Part II Paper Bag Population - Exploring the Margin of
Error
I have an entire population of colored
beads in my paper bag.
What questions might we ask about my population?
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Margin of Error (Paper Bag Population)
Let’s conduct a statistical investigation.
Formulate a question
What proportion of the beads in the bag is blue?
Design and implement a plan to collect data
Take random samples of size 25 from the paper bag
population. The sample proportion of blue beads will
be our estimate of the proportion of blue beads in the
paper bag population. We use the symbol 𝑝 (p-hat)
to represent our estimate of the population proportion.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Activity: Margin of Error (Paper Bag Population)
Analyze the data by measures and graphs
A possible sampling distribution:
Mean =0.61
Std Dev = 0.10
n = 28
Paper Bag Population
0.2
0.3
0.4
0.5
0.6
Proportion_of_Blue
0.7
Dot Plot
0.8
0.9
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Margin of Error
One method of constructing an interval is to use a
sample statistic ± a margin of error. The margin of error
is 2 times the standard deviation of the sampling
distribution.
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ± 𝑚𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟
𝑥 ± 2 ∙ 𝑠𝑡𝑑 𝐷𝑒𝑣
𝑜𝑟
𝑝 ± 2 ∙ 𝑠𝑡𝑑 𝐷𝑒𝑣
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
We use the reasoning that if the sample statistic is likely to
be within 2 standard deviations of the center of the
sampling distribution, then the center of the distribution is
also likely to be within 2 standard deviations of the sample
statistic. Remember that the population parameter is
approximately equal to the mean of the sampling
distribution, so the population parameter is likely to be
within two standard deviations of our sample statistic. In
other words, we conclude that the population parameter is
likely to be in this interval. (Or at least we expect 95% of
the intervals created by this method to include the
population parameter.)
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Let’s look at this mathematically.
We anticipate that 95% of our sample proportion, 𝑝 will fall
within 2 standard deviation of the population parameter, 𝜋 .
𝜋 − 2𝜎𝑝 ≤
𝑝 ≤ 𝜋 + 2𝜎𝑝
−𝜋
−𝜋
−𝜋
−2𝜎𝑝 ≤ 𝑝 − 𝜋 ≤ 2𝜎𝑝
−𝑝
−𝑝
−𝑝
−𝑝 − 2𝜎𝑝 ≤ −𝜋 ≤ −𝑝 + 2𝜎𝑝
𝑝 + 2𝜎𝑝 ≥ 𝜋 ≥ 𝑝 − 2𝜎𝑝
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Back to our Paper Bag Population!
Interpret the results in the context of the original question

Find the margin of error
Margin of error = 2 ∙ 𝑠𝑡𝑑 𝐷𝑒𝑣
= 2 ∙ 0.10
= 0.20

Report a margin of error
Suppose my sample proportion was 0.52, then
𝑝 ± 𝑚𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟
0.52 ± 0.20

Interpret the margin of error
The true population proportion of blue beads is estimated to
be 0.52 ± 0.20. I anticipate that the proportion of blue
beads in the paper bag population is between 0.32 and 0.72.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
Back to Twitter!

Interpret the margin of error
Based on the Pew Research survey, the true percentage of
internet users who use Twitter is estimated to be 16% ±
2.6%. We anticipate that the percentage of internet users
that use Twitter is between 13.4% and 18.6%.
Statistics Standards for Algebra II/Math III
Making Inferences & Justifying Conclusions
How might S.IC.4 be tested?
Speculation…
 Interpret a given margin of error
 Estimate a margin of error from a given sampling
distribution.
 Describe how a sampling distribution might be
created for a given situation.
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
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
discrimination task? Explain.
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)
Acknowledgements and Resources
Chance, B. & Rossman, A. (Preliminary Edition). Investigating Statistical Concepts, Application
and Methods. Duxbury Press.
Chance, B., et al. Rossman/Chance Applet Collection. Retrieved from
http://www.rossmanchance.com/.
Chicago Tribune. (2014, April). Chicago Bears. Retrieved from
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
or trails isn’t really a 50-50 game. Retrieved from
http://www.dailymail.co.uk/news/article-2241854/What-odds-New-study-showsguessing-heads-tails-isnt-really-50-50-game.html.
Duggan, M. & Brenner, J. (2013, February 14). The Demographics of Social Media Users –
2012. Retrieved from http://www.pewinternet.org/2013/02/14/the-demographicsof-social-media-users-2012/.
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
Standards in Mathematics (draft) 6-8 Statistics and Probability. Retrieved from
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
Mathematics (draft) High School Statistics and Probability. Retrieved from
http://commoncoretools.me/wpcontent/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf.
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.
Retrieved from http://www.today.com/video/today/54076112#54301611.
Rossen, J. (2014, February 26). Underage Alcohol Buys. Retrieved from
http://www.today.com/video/today/54076112#54515111.
Rossman, A. (2012). Interview With Roxy Peck. Journal of Statistics Education, 20(2). pp. 1
– 14. Retrieved from http://www.amstat.org/publications/jse/v20n2/rossmanint.pdf.
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
sleep after training. 3(12) pp. 1237-1238. Retrieved from
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
effective in professional baseball. Retrieved from
http://www.noinjury.com/articles/bases.htm.
Online Resources
Census at School. http://www.amstat.org/censusatschool/
Consortium for the Advancement of Undergraduate Statistics Education.
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.
http://courses.ncssm.edu/math/Stat_Inst/links_to_all_stats_institutes.htm
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 High School
Statistics and Probability
Julia Brenson
Lyons Township High School
[email protected]