Download Statistics - Algebra 2 Teaching Suggestions Unit 9 9.1 To identify

Statistics - Algebra 2 Teaching Suggestions
Unit 9
9.1
To identify and perform an appropriate method of gathering data (experiment,
simulations, observational studies including sample surveys).
9.1A. SUGGESTION: List several examples of topics of study and have students identify
whether census, sample survey, simulation, experiment or observational survey
would be the best method for collecting the data.
9.1B. SUGGESTION: Have students bring in examples of experiments and observational
studies to analyze. These can be found in online newspapers, magazines, research
papers, scientific journals, etc. Determine if bias exists. If so, identify and suggest
possible corrections of the bias. Make sure to discuss sampling errors (random
sampling error, undercoverage, bad sampling – VRS, convenience) and non-sampling
errors (processing errors, response bias, question wording and nonresponse bias)
and make sure students know the difference between the two.
9.1C. SUGGESTION: Describe surveys and have students identify the sampling flaws.
The following website has examples of survey templates that can be used in the
classroom and analyzed.
https://www.surveymonkey.com/mp/survey-templates/
9.1D. SUGGESTION: Have students design an experiment that minimizes sampling error.
Students should be able to set up a simple experiment and blocking, blindness and
matched pairs should be investigated.
9.1E. SUGGESTION: Have students create their own sample survey. They will develop
their own questions (discussing question wording bias) and carry out the survey.
Students will then report the results along with biases they encountered. Students
can pick a topic that interests them and can pick their population (students at their
school, high school teachers, adults in their neighborhood, etc.)
9.1F.
9.2
SUGGESTION: Give students a list of questions and ask them what would be the
best method for gathering the data choosing from: census, experiment, simulation
observational study or sample survey. Possible questions: How many hours of
television does a high school student watch per day? How much pressure can be
exerted on a chicken egg before it breaks? What percentage of American League
baseball players had a batting average about 0.300 this season? Is there a
relationship between the amount of physical activity a person gets and his or her
perceived level of stress? Do the seniors at your school do less homework than the
sophomores? Are oral medications better than lotions in killing lice? Does smoking
2 packs of cigarettes a day vs 1 pack of cigarettes a day increase your chances of
getting lung cancer? What is the likelihood of having no girls in 6 births?
To understand the importance of randomization and the difference between random
sampling and random assignment.
9.2A. SUGGESTION: Have students distinguish between random sampling and random
assignment. Random sampling occurs when a study is set up so that the sample
that is selected will represent the population in all its diversity. Random assignment
refers to when a scientific study (experiment) with a goal of inferring a cause and
effect relationship is set up. Random assignment tends to produce treatment groups
with the same mix of values for variables extraneous to the study.
9.2B. SUGGESTION: Have students use a random number generator (computer or
graphing calculator) or random number table to draw a simple random sample
(SRS).
9.2C. SUGGESTION: Have students do the Gettysburg Address activity. Provide students
a copy of the Gettysburg Address and have them scan it for 5 seconds. Students
should then try to estimate the average length of the words used in the speech.
After collecting all of the students guesses, now have students randomly select 10
words from the passage (use a random generator on their calculator) and find the
average length of those ten words. Compare these random averages to the guessed
average. Our eyes are “biased” and will see longer words and not identify how many
small words there are in the passage. The guessed average will always be higher
than the actual, which is 4.3, found from finding the average of all 268 words of the
Gettysburg Address. This illustrates how important randomization is. One formal
write-up of this activity can be found at the website below. The pdf also includes
other activities to help illustrate the importance of randomization and how it reduces
bias.
http://express.lander.edu/stats/images/materials/rossmanmyrtlebeachhandout.pdf
9.2D. SUGGESTION: Have students discuss different ways of obtaining random samples
(SRS, stratified, cluster, multistage, systematic) and the merits of each of those
methods. Also have students look at samples that do not employ randomness and
are biased like convenience and voluntary response samples. These types of
samples systematically favor some parts of the population over others in choosing
the sample.
9.2E. SUGGESTION: Have students set up a random assignment for an experiment. Give
them an example of an experiment and ask how they would randomly assign the
treatments. Students might suggest putting all names in a hat, shaking the hat and
then picking out the treatment groups or they might number all participants and use
a random number table or random number generator to pick the groups. Discuss
how these methods will give us random, equal (in number) groups for our
experiment. Explain how this is different from using a coin or dice to randomize our
treatments (this would not result in equal number of participants in the groups).
Students should then discuss whether to block, do a matched pairs or a completely
randomized design. Have them diagram the experiment showing each stage of the
experiment, starting with random assignment and ending with the response variable
measured at the end.
9.2F.
9.3
SUGGESTION: Have students do the “Rolling Down the River” activity which has
students explore the importance of random sampling by creating convenience,
stratified and simple random samples. Find the activity at the website below.
http://courses.ncssm.edu/math/Stat_inst01/PDFS/river.pdf
To organize data.
9.3A. SUGGESTION: Students should be familiar with different graphical representations
of data including histograms, bar graphs, circle graphs, boxplots, dotplots, and stem
and leaf plots.
9.3B. SUGGESTION: Give students a list of variables, like the ones below. Have the
students decide whether each variable is categorical (qualitative) or quantitative
(numerical) and decide which graph type would be best to display the data.
Exs: height, eye color, shoe size, GPAs, SAT scores, a team’s jersey numbers,
amount of sleep (in hours), pulse rate, number of letters in first name.
9.3C. SUGGESTION: Have students work with spreadsheets and graphing calculators to
have experience with scaling graphs and recognizing distortions or bias in graphs.
9.3D. SUGGESTION: Have students find measures of center and variability for data sets
using the appropriate measures with respect to its shape.
9.3E. SUGGESTION: Have students describe the data distribution by noting its center,
shape and spread and identifying any unusual features like outliers, gaps and
clusters.
9.3F.
9.4
SUGGESTION: Have students organize data in frequency tables and describe the
distribution by finding mean, standard deviation and shape (uniform, skewed left,
skewed right, symmetric)
To know and apply the empirical rule.
9.4A. SUGGESTION: Have students define standard deviation in the context of different
situations. Students must have an in depth understanding of standard deviation as a
measurement tool in order to apply the empirical rule.
9.4B. SUGGESTION: Have students complete the standard deviation activity below. This
activity helps students develop a better intuitive understanding of what is meant by
variability in statistics.
https://www.causeweb.org/repository/StarLibrary/activities/delmas2001/
9.4C. SUGGESTION: Have students investigate data in the real world that take on
approximately normal shapes like heights of adults, size of things produced by
machines, errors in measurements, blood pressure, IQ scores, marks on a test, and
SAT scores.
9.4D. SUGGESTION: Have students use the empirical rule to assess if the data is normally
distributed. Find data that is approximately normally distributed and calculate the
mean and standard deviation of the data set. Determine if approximately 68% of
the data falls one standard deviation away from the mean, if 95% of the data falls
two standard deviations away from the mean and if almost all of the data falls within
three standard deviations.
9.4E. SUGGESTION: Have students place a variable trait into a graphing calculator. This
variable could be their height, weight, shoe size, last test score, how many hours of
sleep they got last night, pulse rate, etc. The data may or may not be approximately
normal, but by applying the empirical rule, students will be able to evaluate whether
or not the data is symmetric, unimodal and bell-shaped. If heights are used and are
approximately normal, this could be illustrated visually by having students make a
human histogram by lining up in the classroom.
9.5
To use and apply normal distributions when appropriate (z-score, percentile).
9.5A. SUGGESTION: Have students define normal distributions as unimodal, symmetric
and bell-shaped. Look at normal distributions with the same mean and different
standard deviations, noting their different shapes and placement of their inflection
points.
9.5B. SUGGESTION: Remind students that not all symmetric or “bell-shaped” curves are
 x  μ 
normal, only those with the following equation: f  x  
2σ 2
2
e
σ 2π
9.5C. SUGGESTION: Have students use the mean and standard deviation of a data set to
fit it to a normal distribution and to estimate population percentages with z-scores.
Recognize that there are data sets for which such a procedure is not appropriate.
Use calculators, spreadsheets and tables to estimate areas under the normal curve
and to work backward to find an observation for a given percentile.
9.5D. SUGGESTION: Use technology with the normalcdf and invNorm commands on the
graphing calculator to find the area under the normal curve or to find the observation
that matches a particular percentile.
9.5E. SUGGESTION: Have students graph approximately normal data as a histogram with
technology and compare that graph with the normal probability plot of that
distribution. If the data is approximately normal, the normal probability plot (NPP)
should be very close to linear. Also plot skewed data that is not normal and notice
the differences in the normal probability plots. If the data as skewed right, the NPP
will “drop off to the right” from the linear pattern and if the data is skewed left, it will
“drop off to the left”.
9.5F.
9.6
SUGGESTION: Have students work on the “Number of Hours of Sleep” activity.
Students work in groups and answer how many hours they slept in the past 24
hours. Students answer questions based upon their amount of sleep answers and a
previously conducted survey regarding the number of hours of sleep of high school
students. The activity and worksheet can be found at the following website:
http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalResource.htm
To know the difference between a parameter and a statistic.
9.6A. SUGGESTION: Have students distinguish between parameters and statistics. A
population is a big word for a “group of things we are studying”. A population can be
big or small, but it is the group we want information about. Parameters are the
“things we want to measure about our group – the population”. Note parameter and
population both start with a “p”. Since it is usually impossible to get information
about the population (if it is possible, then we can do a census), it is important that
we do proper random sampling of our population. The data we get from this sample
is called a statistic, and will accurately represent the population parameter if we have
a random, unbiased sample. Sample and statistic both begin with an “s”, helping to
keep the terms identified correctly. Our sample is a subset of the population chosen
to represent that group.
9.6B. SUGGESTION: Have students identify the population, the parameter of interest, the
sample and the sample statistic for the following reports.
Exs: 1) A magazine asked all subscribers whether they had used alternative
medical treatments and, if so, whether they had benefited from them. For almost all
of the treatments, approximately 20% of those responding reported cures or
substantial improvement in their condition. 2) A company packaging snack foods
maintains quality control by randomly selecting 10 cases from each day’s production
and weighing the bags, and then inspecting the contents. The weight of a case
should be 2 lbs. One day they found that the weight of the 10 cases was 2.08 lbs.
3) State police set up a roadblock to estimate the percentage of cars with up-todate registration and insurance. They found problems with 10% of the cars they
stopped. 4) The Environmental Protection Agency took soil samples at 20 locations
near a former industrial waste dump and checked each for evidence of toxic
chemicals. The found no elevated levels of any harmful substances. 5) The
telephone company says that 62% of all residential phone numbers in Los Angeles
are unlisted. A telephone survey contacts a random sample of 1000 Los Angeles
telephone numbers, of which 58% are unlisted.
9.7
To understand that inference is drawing a conclusion about a population parameter
based on a random sample from that population.
9.7A. SUGGESTION: Have students perform a hands-on activity to investigate a claim
about a population proportion. Students are divided into small groups and presented
with M & M’s to check the claim that the company produces 16% green M & M’s in its
packages. Students will be exploring the relationships between a population,
population parameters, random samples and statistics. By the end of the
investigation, students will be able to informally relate a sample statistic to a known
population parameter. The activity can be found at the following website:
http://www.amstat.org/education/STEW/pdfs/PopulationParameterswithMMs.pdf
For data on M&Ms and in depth study of colors of 48 packages, visit:
http://joshmadison.com/2007/12/02/mms-color-distribution-analysis/
9.7B. SUGGESTION: Have students investigate the Reese’s Pieces applet which can be set
with a known parameter and then draws random samples from that population to
illustrate the conclusions that can be made about parameters based on unbiased
statistics. This is also an informal introduction to a sampling distribution.
http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html
9.7C. SUGGESTION: Have students find sample means and sample proportions from data
collected for a study or experiment. Calculated the estimated population proportion
or mean by calculating the sample proportion or mean and discuss how this estimate
will closely resemble the parameter if the sample is unbiased.
9.8
To determine whether empirical results are consistent with the theoretical model.
9.8A. SUGGESTION: Have the students roll two dice 50 times and record their results in a
table. Ask: What is the empirical (experimental) probability of rolling a 7? What is
the theoretical probability of rolling a 7? How do the empirical and theoretical
probabilities compare? Are the data consistent or cause us to call into question the
model that was proposed? Have students place values in a probability distribution
table and graph the data with a bar graph. Students should be able to determine the
shape of the distribution from the table or the graph.
9.8B. SUGGESTION: Have each student flip a coin 10 times and record the percentage of
heads. Combine students to make 50 trials and record the percentage. Continue this
activity for 100 trials, 150 trials, etc. until the entire class is totaled. In each case,
compare the result to the theoretical probability of 50 percent. Make a graph of
relative frequency versus number of trials. Discuss if the results are close to the
theoretical probability or if the results make us question the model.
9.8C. SUGGESTION: Have students use manipulatives like coins, dice, cards and spinners
to find simple probabilities through experimentation and then find the theoretical
probabilities of those events occurring.
9.8D. SUGGESTION: Have students come up with their own probability game. Determine
probabilities needed to win their game. Then, students take turns playing other
student’s games so that empirical results can be compared to the theoretical
probabilities stated for that game.
9.8E. SUGGESTION: Have students work on the random M & M activity which reinforces
randomness and empirical/theoretical models.
http://apstatsmonkey.com/StatsMonkey/m&m_Activities_files/mnmRandom.pdf
9.9
To develop the concept of margin of error through the use of simulation models for
random sampling.
9.9A. SUGGESTION: Have students investigate the margin of error which is an amount
that is allowed and accounted for due to sampling variability. Students need to
calculate 95% margins of error for a sample proportion using the quick formula:
E
1
, which illustrates the larger the sample, the smaller the margin of error.
n
Intuitively, students should notice that in order to cut our margin of error in half, we
must use a sample four times as large.
9.9B. SUGGESTION: Have students perform the experiments in Suggestion 9.9A or 9.9B
and find the margin of error for those two simulations. Discuss whether the intervals
captured the true parameters.
9.9C. SUGGESTION: Have students informally assess margins of error in a real-world
setting. Students should be able to interpret a margin of error statement.
Ex: We would like to find out if a majority of adults in the U.S. voted in the last
election. We take a poll of 1700 people selected at random and ask them if they
voted in the last election. 795 said yes. So for our sample, the statistic is 795/1700
or roughly 47%. We could estimate that 47% of the adult population voted in the
last election. Now, we would like to say something about how confident we are that
our estimate is correct. By calculating the margin of error for 95% confidence as
1/sqrt(1700), we get approximately 2.4%. We are 95% confident that between
44.6% and 49.4% of the adults voted in the last election. A majority(50%) is not
included in our confidence interval, so no, a majority of U.S. adults did not vote in
the last election.
9.9D. SUGGESTION: Have students define a margin of error plus or minus 3 percentage
points as, “if we were to take many samples using the same method we used to get
this one sample, 95% of the samples would give a result within plus or minus 3
percentage points of the truth about the population.”
9.9E. SUGGESTION: Have students informally investigate sampling distributions and
notice how the margin of error is reduced as the sample size increases. This can be
done with simulation models online.
http://demonstrations.wolfram.com/TheCentralLimitTheorem/
http://demonstrations.wolfram.com/ConfidenceIntervalsForAMean/
http://demonstrations.wolfram.com/RandomlyFillingAnArray/
9.10
To use data from a randomized experiment to compare two treatments.
9.10A. SUGGESTION: Have students work on an activity which interprets data, evaluates
statistical summaries and then critique someone else’s interpretations of the data.
This activity will help introduce students on how to decide if there is a significant
difference between two statistics.
http://map.mathshell.org/materials/lessons.php?taskid=217#task217
9.10B. SUGGESTION: Have students compare two experimental treatments by comparing
box plots and comparing their means and variability to assess if there are differences
between the two treatments.
https://learnzillion.com/lessons/3413-compare-experimental-treatments-bycomparing-box-plots
9.10C. SUGGESTION: Have the students complete Activity 2: Dolphin Therapy at the
website below. Students look at an experiment of 30 subjects that is trying to
assess if swimming with dolphins can be therapeutic for patients suffering from
clinical depression. Students look at a proportional difference between the two
groups and decide if there is evidence to say the therapy is more effective.
http://www.math.ucla.edu/~radko/circles/lib/data/Handout-366-439.pdf
9.11
To use simulations to decide if the difference between two statistics is significant.
9.11A. SUGGESTION: Have students informally investigate hypotheses testing. The applet
below can assist in this at the link below.
http://demonstrations.wolfram.com/HypothesisTestsAboutAPopulationMean/
9.11B. SUGGESTION: Have students learn how to change a research question into a
statistical hypothesis by stating the null and alternative hypotheses. Then, have
students decide how they would interpret their decision if the decision is to reject the
null hypothesis or if the decision is to fail to reject the null hypothesis.
Ex: A school publicizes that the proportion of its students who are involved in at
least one extracurricular activity is 61%. We believe that the proportion is lower.
Ex: A manufacturer claims that there are, on average, 50 matches in a box. We
think the claim is incorrect.
Ex: The USDA limit for salmonella contamination for chicken is 20%. A meat
inspector reports that the chicken produced by a company exceeds the USDA limit.
You perform a hypothesis test to determine whether the meat inspector’s claim is
true.
Ex: A car dealership announces that the mean time for an oil change is less than 15
minutes. You perform a hypothesis test to determine whether the mean inspector’s
claim is true.
https://learnzillion.com/lessons/3415-state-the-null-and-alternative-hypotheses
9.11C. SUGGESTION: Have students work on this activity where they will have the
opportunity to explore real data collected on the completion of a maze. Students will
test to see if the mean time to complete the maze is significantly different for males
compared to females. Students can read the evidence (t-test) and draw a
conclusion.
http://www.amstat.org/education/STEW/pdfs/AnAmazingComparison.pdf
9.11D. SUGGESTION: Have students watch these lessons on hypothesis tests and
comparing statistics by identifying the basic components of a hypothesis test and a
comparative experiment.
https://learnzillion.com/lessonsets/402-use-data-from-a-randomized-experiment-tocompare-treatments-evaluate-reports-based-on-data
9.11E. SUGGESTION: Have students complete the activity below to illustrate how
randomization is used in making inferences from data.
file:///C:/Users/SNRDPD/Downloads/Halvorsen_Changing_Face_of_Statistics.pdf
9.12
To evaluate reports based on data.
9.12A. SUGGESTION: Provide statistical abstracts/reports for the students to read and
critique. Here are some good questions to ask the students and facilitate
understanding.




What is the purpose of the study?
How did the authors go about collecting data?
Were the collection methods appropriate for this particular study?
Were they randomized?








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Were the significance levels appropriate?
What was the null hypothesis?
What was the alternative hypothesis?
Were the correct statistical tests used for the data?
Was the data fully presented?
Who were the study participants?
Who funded the study? Can bias arise from that?
What were the results?
Why are the results significant even if there were no significant findings?
9.12B. SUGGESTION: Aside from asking questions about already completed studies, have
students come up with hypothetical studies to help develop a deeper understanding
of the core concepts. Students can choose topics that interest them and think about
how they would go about testing aspects of these topics to ensure statistical validity.
9.12C. SUGGESTION: Have students evaluate published reports in the media by learning to
ask questions about the source of the study, the design of the study, possible biases,
and the way the data was analyzed and displayed. Students should use technology
to recognize and analyze distortions in data displays.
9.12D. SUGGESTION: Have students evaluate experiments with the following experiment
analysis template:
http://apstatsmonkey.com/StatsMonkey/TPS3e_files/AnalyzeExperimentsTemplate.p
df.