# Dana Cartier Illinois Center for School Improvement Julia Brenson

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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 This PowerPoint is intended to be used as a walk-through of the New Illinois Learning Standards for Statistics and Probability allocated to Algebra I / Math I. In addition to providing example activities, guidance of key points to emphasize is provided for many of the standards.

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 All materials from this session are available at this website. This website is currently under construction, but please keep checking back for more information about the Statistics Standards.

The New Illinois Learning Standards
Algebra II and Math III

Statistics Standards for Algebra II/Math III Normal Distribution
PBA EOY S.ID.4 X The PARCC Evidence Tables indicate that all statistics standards for Algebra II and Math III will be assessed on both the Performance Based Assessment and the End-Of-Year Assessment. (The exception is the conditional probability standards which will be assessed on the End-Of-Year Assessment in Algebra II and Math II.) Work on S.ID.4 and the normal distribution lays the foundation for making inferences in the S.IC standards. 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 calculators, spreadsheets, and tables to estimate areas under the normal curve.

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 calculators, spreadsheets, and tables to 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 In statistics we use the Greek alphabet to represent population parameters and the Roman alphabet (our alphabet) to represent sample statistics. The standards do not indicate that students are expected to use the correct notation when writing about population parameters and sample statistics. On the next slide, you can see that the words mean and standard deviation are used rather than symbols. μ (lower case mu) is the population mean σ (lower case sigma) is the population standard deviation μ = 10, σ = 2

Statistics Standards for Algebra II/Math III Normal Distribution
The Empirical Rule Empirical Rule If a distribution can be well approximated by the normal curve, then we expect: 68% of observations to be within 1 standard deviation of the mean. 95% of observations to be within 2 standard deviations of the mean. 97.5% of observations to be within 3 standard deviations of the mean. Using the empirical formula we can estimate probabilities. Probabilities must have a value 0≤ prob. ≤1. The area under the normal curve is 1.00, so the percentages above provide us with the approximate probabilities. For example: 1) The probability that an observation is within one standard deviation of the mean is .68. 2) The probability that an observation is more than one standard deviation from the mean is .32 ( ). 3) The probability that an observation is less than the mean is .50. The probability that an observation is more than one standard deviation above the mean is .16. The probability that an observation is less than 2 standard deviations below the mean is (Note that this is a very low probability. We call observations that are more than 2 standard deviations from the mean, outliers.) 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. 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? = .0250 21 Answers: .0250 = .975 of 97.5% Only 2.5% of ACT test takers have a score greater than 31. A score greater than 31 is unusual and is an outlier! (Sometimes it is great to be an outlier!) The probability of a score greater than 31 is equal to a probability of a score greater than or equal to 31. p(x>31) = p(x≥31) = .025 The normal distribution is a continuous distribution. We can use it to find areas under the curve (probabilities) for intervals, but not for isolated (discrete) values. We cannot use the normal distribution to determine the probability of a student scoring a 31. A score of 31 would not have an area (imagine a rectangle with a height, but no width) and therefore no probability can be determined.

Statistics Standards for Algebra II/Math III Normal Distribution
Animal Cracker Lab The label on a 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 = cookies Standard Deviation = 0.91 cookies n = 28 boxes The graph at right shows the distribution of the number of crackers in a sample of 28 Barnum’s 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? Why does Nabisco tell the consumer there are 16 cookies in a box? Additional Questions: Approximately how many standard deviations above or below the mean were the number of crackers in your box? What is the shape, center and spread of the distribution of animal crackers per box?

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
PBA EOY S.IC.1 X The PARCC Evidence Tables indicate that all statistics standards for Algebra II and Math III will be assessed on both the Performance Based Assessment and the End-Of-Year Assessment. (The exception is the conditional probability standards which will be assessed on the End-Of-Year Assessment in Algebra II and Math II.) S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Statistics Standards for Algebra II/Math III
S-IC.1 Understand statistics as a process for making inferences about population 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 conclusions about the population.

Activity: Random Rectangles
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? The Random Rectangles activity is adapted from Scheaffer et al. (1996, Activity-Based Statistics). A similar version of this task is featured on the Illustrative Mathematics website. Dick Sheaffer has given permission for this activity to be shared with Illinois math teachers. 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. Generally, students tend to select a sample of rectangles with a mean larger than the actual population mean. Their own biases cause them to overestimate the area of the rectangles.

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. We do not want the method of picking the sample to produce samples with characteristics that are different than the population characteristics.

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
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. 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. 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. Students will at first define a simple random sample as a sample where every element of the population has an equal chance of being selected. This is not the precise definition. In the example above, methods A, B and C all result in each student having a 1/5 chance of being selected. However, a simple random sample means that every possible sample of size 4 has a chance of being selected. In B, it is impossible to have a sample that includes everyone in the first row. In C, it is impossible to have a sample that consists of all juniors. Only method A offers a sampling method where every sample of size 4 has an equal chance of being selected. Example A is the method that will produce a simple random sample. This is not to say that method B and C are not good. Sampling methods B and C are actually examples of stratified sampling. Strata are subgroups that are usually selected because they are homogeneous for a characteristic that we think might influence the response. Simple random samples are then taken from each strata. This method can be highly desirable and sometimes easier to implement than taking a simple random sample. Students do not need to know about stratified sampling, but they should know that when selecting a simple random sample of size n, every sample of size n from the population has an equal chance of being selected.

Back to Random Rectangles
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. Be sure to show students how to select random samples using a random digits table and a calculator. The Random Rectangles activity that has been provided uses a random digits table for select samples of rectangles. Generating random numbers by calculator appears in the Margin of Error – A Guided Activty.

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 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)

Sampling Distribution 100 Samples of Size 5 mean = 7.762
Statistics Standards for Algebra II/Math III Understand and Evaluate Random Processes Sampling Distribution 100 Samples of Size 5 mean = 7.762 Sample Distribution 500 Samples of Size 5 mean = 7.356 As the number of samples increases, the mean of the sampling distribution gets closer and closer to the mean of the population.

Sampling Distribution 100 Samples of Size 10 mean = 7.67 square units
Statistics Standards for Algebra II/Math III Understand and Evaluate Random Processes Sampling Distribution 100 Samples of Size 10 mean = 7.67 square units Sample Distribution 500 Samples of Size 10 mean = square units S.IC.1 provides an introduction to the Central Limit Theorem. The two main ideas that students should understand is that when the population is not too unreasonably skewed, as more and more samples are taken from the population, the mean of the sample distribution approaches the mean of the population. As the sample size increases, the spread of the sampling distribution decreases. By looking at the mean of the sample distribution, we can make an inference (draw a conclusion) about the mean of the population. The mean of the sampling distribution of 100 samples of size 10 is Even with only 100 samples, we can get a pretty good approximation of the mean of the population. The actual mean area of the population is 7.42 square units. The standard deviation of the population is 5.22 square units. As the sample size increases, the spread of the sampling distribution decreases. (The standard deviation gets smaller.)

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. Check for student understanding on these additional concepts: Students should be able to define and identify in context the population, the sample, the population parameter, and the sample statistic. Students should be able to explain how to take a simple random sample. Students, when reading statistical studies, should identify how the sample was selected. Students should understand the relationship between a population, random samples taken from the population, and the sampling distribution.

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 .) Beth Chance and Allan Rossman, leading statistics educators and authors of Workshop Statistics, have created several applets that can be used to illustrate simulations and sampling distributions. (Beth and Allan have given their permission for these applets to be shared with Illinois math teachers.) When using simulation to make inferences about a population, the recommended method is to first guide students through a physical (hands-on) simulation. Technology can then be used to illustrate what happens in the long run. Beth Chance and Allan Rossman’s Reese’s Pieces applet simulates taking random samples of Reese’s Pieces from a population of Reese’s Pieces and creating a sample distribution of the proportion of orange candies. Before going to the applet, start by giving each student a random sample of size 25 from a large bag of Reese’s Pieces. Have them calculate the proportion of orange and plot this on a class dot plot on the chalkboard. If the bag of Reese’s Pieces is representative of the population of all Reese’s Pieces, looking at the class sampling distribution (dot plot) what can students infer about the proportion of orange candies in the population of all Reese’s Pieces? (Answer: the center of distribution should be close to 0.5.) Next go to and select the Reese’s Pieces applet. Suppose the true population proportion of orange Reese’s Pieces is 0.5 ( 50% of the candies are orange.) If we repeatedly took random samples from the population, calculated the proportion of orange, and created a dot plot of all of the sample proportions, what would be the shape of the distribution? What would be the center of the distribution? If we increased the sample size, how would this change the sampling distribution?

Statistics Standards for Algebra II/Math III Making Inferences & Justifying Conclusions
PBA EOY S.IC.3 X S.IC.6 The PARCC Evidence Tables indicate that all statistics standards for Algebra II and Math III will be assessed on both the Performance Based Assessment and the End-Of-Year Assessment. (The exception is the conditional probability standards which will be assessed on the End-Of-Year Assessment in Algebra II and Math II.) S-IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. S-IC.6 Evaluate reports based on data.

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

More on Observational Studies
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 long- term 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-and- effect relationship from an observational study.* * There is a notable exception to this rule. The Comprehensive Smoking Education Act of 1984 (Public Law 98–474) requires that following message appear on all cigarette packaging and advertising: SURGEON GENERAL’S WARNING: Smoking Causes Lung Cancer, Heart Disease, Emphysema, and May Complicate Pregnancy. While no experiment was conducted to allow this cause-and-effect conclusion, many observational studies were conducted. After years of study, with the some results being observed, it was determined that in the case of smoking, a causal relationship could be claimed.

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 No Random Sampling How were subjects assigned to treatment groups? Random Assignment May infer cause and effect AND May generalize findings from sample to the population May infer cause and effect, but Cannot generalize findings from sample to the population. (We can conclude that the treatment caused a response for this sample only.) No Random Assignment May generalize findings from sample to population, Cannot infer cause and effect. Cannot generalize findings from sample to population AND Cannot infer cause and effect In statistical studies, random is used in two different ways. We randomly select the participants for our study, and, in an experiment, we randomly assign participants to treatment groups. The type of inferences (conclusions) that we can make and for what particular group are determined by whether or not participants were selected randomly and assigned to treatment groups randomly. (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/ # ) 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. TV news programs may provide examples of statistical studies that make for good classroom discussion. The Today Show, for example, features several segments by Jeff Rossen in which he investigates topics such as “Chocolate Taste Test – Expensive vs. Cheap: Can You Tell The Difference?” or “Underage Alcohol Buys.” While these news segments are investigations, not necessarily statistical investigations, they make for great class discussion. After showing the video footage, ask students: What was the population of interest? Is this study a survey, an observational study, or an experiment How was the sample of participants selected? If this was an experiment, what are the treatment groups? How were participants assigned to the treatment groups? What conclusions did the authors of this article draw? 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 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

Statistics Standards for Algebra II/Math III Making Inferences & Justifying Conclusions
PBA EOY S.IC.5 X The PARCC Evidence Tables indicate that all statistics standards for Algebra II and Math III will be assessed on both the Performance Based Assessment and the End-Of-Year Assessment. (The exception is the conditional probability standards which will be assessed on the End-Of-Year Assessment in Algebra II and Math II.) 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
S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Activity: Sleep Deprivation
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. This activity is based on activity found in Investigating Statistical Concepts, Applications and Methods by Beth Chance and Allan Rossman. Beth Chance and Alan Rossman have given permission for the activity and the corresponding applet at to be shared with Illinois math teachers. Sleep Deprivation is based on a study by Stickgold, R., James, L. & Hobson, J. (2000) titled Visual discrimination learning requires sleep after training. This original study can be found at What is the difference between the means for the two groups? 15.92

Primary Question of Inference:
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 -10.7 4.5 2.2 21.3 -14.7 9.6 2.4 21.8 7.2 10.0 Unrestricted Sleep Group 25.2 14.5 -7.0 12.6 34.5 45.6 11.6 18.6 12.1 30.5

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. To help students understand the process of simulation, it is important to first do the physical simulation. Students then can be shown the same simulation using an applet, Fathom or some other tool.

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) Used with permission from Beth Chance and Allan Rossman.

Computer Simulation of 1000 Trials
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 or higher by chance (random assignment)? Notice the sampling distribution is approximately normal and centered close to 0. When the improvement scores are randomly assigned to the two treatment groups, the average difference between the means of the two treatment groups is close to 0. The standard deviation for the sampling distribution is Anything out beyond two standard deviations would be considered an unusually large value (outlier). Two standard deviations above the mean would be 2(6.76) = In the experiment, the difference between the means of the two treatment groups was This is more than two standard deviations above the mean. It is not very likely that we would get a difference of means as high as just by random sampling. 10 out of 1000 samples (1.0%) were 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
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. Can we conclude cause and effect? Yes. The subjects were assigned randomly to the two treatment groups, so we can conclude that there is a cause-and-effect relationship between sleep deprivation and improvement on the visual discrimination task. However, we cannot generalize these findings from this sample to the population. The subjects were volunteers and were not selected randomly from a population. Probably the population of interest is all people, but we do not know if the sample of 21 volunteers was representative of the population of all people. Given that the subjects were between the ages of 18 and 25, it is unlikely that the sample is representative of all people. We also cannot assume that they are representative of the population of 18 to 25 year olds!

Activity: Distracted Driver
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

Activity: Distracted Driver
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

Primary Question of Inference:
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)? Key Ideas to discuss with students: Did the two treatments (talking on a cell phone and talking to a passenger) have the same effect on whether the driver missed the exit or a different effect? Researchers believe there is a difference and must provide compelling evidence to support this claim. Is it possible that talking on a cell phone and talking to a passenger had the same effect on the driver’s distraction? Or could this big a difference between the treatment groups have occurred simply by chance? To answer these questions, we use simulation to reassign the subjects randomly to the two treatment groups and count how many times drivers miss the exit in each group. We repeat this random reassignment many times. How often do we get a difference as large as (or larger) between the two treatment groups? If it rarely happens, we conclude that results from the experiment are not likely to have happened “just by chance.” If our simulation results in us frequently getting a difference as large or larger than 0.209, then we conclude that it is possible that the results from the experiment could have happened “just by chance” and are not necessarily due to a difference in the effect of the two treatments. 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? Be sure to read through the entire Teacher Notes provided with this activity. These notes provide excellent guidance as to how to explain the simulation and its relationship to the original experiment. Students may look at 6.8% and think this means that it is not very likely to have happened by chance. In more formal statistical reasoning (hypothesis test), we look for probability that is less than .05, or sometimes smaller! For now, we want students to recognize that while 6.8% is small, it indicates that having 7 or more by chance is a possibility. 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), After physical (hands-on) simulation, use applet titled Randomization Test for quantitative response (two groups) available from 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) Distracted Driving is an activity created by Roxy Peck, Daren Starnes and Celia Rowland. The Teacher Notes for this activity provides excellent guidance on how to do a physical (hands-on) simulation using a deck of playing cards.

The New Illinois Learning Standards
Algebra I & Math I

Possibly quadratic functions (See EOY Evidence Table – S-ID.Int.2)

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 linear, quadratic, and exponential models.* 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
PBA EOY S.ID.6 X S.ID.7 S.ID.8 S.ID.9  X The PARCC Evidence Tables indicate that all statistics standards for Algebra I and Math I will be assessed on the End-Of-Year Assessment. 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 linear, quadratic, and exponential models.* 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.* 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.*

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. Answer: D As math teachers, we tend to interpret scatterplots by saying, “As x increases, y increases.” However, when interpreting data in context, this description does not always work. Answer A uses the phrase, “As heights go up,” to imply that we are seeing growth in height. This data is about 65 different men at a given instance of time (Spring 2014) and does not show information about any of the Chicago Bears having an increase in height. Similarly, as this is a snap shot of a moment in time, we do not see any of the Bears gain (or lose) weight. Answer B is slightly better. Now we are using the phrase, “tends to increase” to describe the general trend. Answer C is indicating that a single person is growing and gaining weight. Even a graph of the height and weight of a single person might find that, as this individual grows, his weight might not always increase. Answer D is the best answer. It describes the overall tendency accurately. Taller football players tend to be heavier and shorter football players tend to be lighter.

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. When r is close to 0, this does not mean that there is not relationship. There may well be a curved relationship or some other relationship between x and y. An r value close to 0 tell us that the relationship is not linear.

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. If r is calculated for the two examples shown, it may appear to indicate that there is a strong relationship, but the graph tells us the relationship is not 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? The graph shows a linear relationship between the amount of fresh lemons imported to the USA from Mexico and the total US highway fatality rate. In years when more metric tons of fresh lemons are imported, the US highway fatality rate is lower, and in years when fewer metric tons of fresh lemons are imported, the US highway fatality rate is higher. The r value is indicating there is a strong, negative linear relationship between metric tons of fresh lemons imported to USA from Mexico and the total US highway fatality rate. However, this does not mean that an increase in the amount of imported fresh lemons causes a decrease in the total US highway fatality rate. Association does not allow us to infer a cause-and-effect relationship! This example is shared with permission from John Oakes.

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. In the residual plot on the left, the residuals appear to be scattered with no discernable pattern. The residuals help us to see that this data set has a possible outlier in the y-direction (an unusually large residual). In the middle residual plot, we can see a curved pattern. A linear function is not the best choice to model this data. The residual plot indicates that another function, perhaps a quadratic function, would be a better fit to the data. In the residual plot on the right, the residuals are larger for larger values of x. There is probably some other relationship between x and y other than linear. 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.

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? Residuals are the vertical deviations from the linear regression line. What appears to be very small deviations in the scatterplot on the left, are more pronounced in the residual plot on the right.

Altitude vs. Temperature
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?

Altitude vs. Temperature
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 Use residuals as a means to classify snakes. Graph shows length of snake and head length Let’s look at Snake 1. Snake 1 appears to lie just above the pattern of the points in the Species A graph. If we fit a best fit line to Species A, Snake 1’s values would place it above this regression line. In the Species B graph, Snake 1 appears to lie in the pattern. In fact, it falls on the exact same coordinates as another snake. If we fit a best fit line to the Species B data, Snake 1’s point would lie very close or possibly on the regression line. Snake 1 would have a greater residual in the Species A graph, and a smaller residual in the Species B graph. Snake 1 is likely to be a Species B snake. 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 Additional Activities: Altitude vs. Temperature Adapted from an activity featured at Snakes – A residual activity from Inside Mathematics (http://www.insidemathematics.org/common-core-math-tasks/high-school/HS-S- 2003%20Snakes.pdf) Engage NY features modules by grade or course designed to engage students in the Common Core State Standards. Roxy Peck, one of the leading statistics educators in the country, and a team of other top educators have created an entire unit for the statistics standards in Algebra I that is featured on the Engage NY website. The unit includes activities with explanations that can be used for class, homework and assessments. For more information about Roxy Peck and her contributions to statistics education, see the article Interview With Roxy Peck by Allan Rossman that appeared in Journal of Statistics Education (Volume 20, November 2012). Retrieved from Census at School (http://www.amstat.org/censusatschool/) allows students to complete an online survey, take random samples of students in the United States, and compare their class’ responses with the responses from their random samples.

The New Illinois Learning Standards
Algebra II and Math II The Conditional Probability standards are allocated to Algebra II and Math II.

Statistics Standards for Algebra II/Math II Making Inferences & Justifying Conclusions
PBA EOY S.CP.1 X S.CP.2 S.CP.3 S.CP.4 S.CP.5 S.CP.6 S.CP.7 The PARCC Evidence Tables indicate that the Conditional Probability (S.CP) standards for Algebra II and Math II will be assessed on the End-Of-Year Assessment. S-CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S-CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S-CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

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:
𝑃 𝐸 𝐹 =𝑃(𝐸) If E and F are not independent, they are said to be dependent events. If 𝑃 𝐸 𝐹 =𝑃(𝐸), it is also true that 𝑃 𝐹 𝐸 =𝑃(𝐹), and vice versa. If two events E and F are independent then the Multiplication Rule becomes: 𝑃 𝐸∩𝐹 =𝑃(𝐸)∙𝑃(𝐹) 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.

Activity: Conditional Probability Four Square
Statistics Standards for Algebra II/Math II Conditional Probability and the Rules of Probability Activity: Conditional Probability Four Square

Acknowledgements and Resources

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 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 content/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 Peck, R., Gould, R., & Miller, S. (2013). Developing Essential Understand of Statistics for Teaching Mathematics in Grades 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 Rossen, J. (2014, February 26). Underage Alcohol Buys. Retrieved from Rossman, A. (2012). Interview With Roxy Peck. Journal of Statistics Education, 20(2). pp. 1 – 14. Retrieved from 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 Retrieved from Strayer, D. and Johnston, W. (2001, November 6) 12(6). Driven to Distraction: Dual- Task Studies of Simulated Driving and Conversing on a Cellular Telephone. Pp Retrieved from Reprint.pdf. The Institute for Preventative Sports Medicine. (2001) Study finds break-away bases effective in professional baseball. Retrieved from

Online Resources Census at School. Consortium for the Advancement of Undergraduate Statistics Education. Engage NY. Illustrative Mathematics. Inside Mathematics. Mathematics Assessment Project. Math Vision Project. NCSSM Statistics Institutes. NCTM Core Math Tools – Data Sets

Online Resources PARCC Model Content Frameworks. er2012V3_FINAL.pdf PARCC Mathematics Evidence Tables. https://www.parcconline.org/assessment- blueprints-test-specs Smarter Balanced Assessment Consortium. Statistics Education Web (STEW). The Data and Story Library (DASL). The High School Flip Book Common Core State Standards for Mathematics. book-usd pdf

Thank you for joining us!
The New Illinois Learning Standards for Algebra I / Math I Statistics and Probability Thank you for joining us! Dana Cartier Julia Brenson Tina Dunn