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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 Lyons Township High School The New Illinois Learning Standards Agenda Resources Available Through ISBE Algebra II/Math III – Normal Distribution, Random Sampling, Experimental Design, and Comparing Two Treatments Algebra I/Math I (and above) – Assessing the Fit of a Function to Data Algebra II/Math II – Conditional Probability The New Illinois Learning Standards ILStats http://ilstats.weebly.com/ All materials from this session are available at this website. This website is currently under construction, but 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 Algebra II & Math III Standard PBA EOY S.ID.4 X X Statistics Standards for Algebra II/Math III S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use 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 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 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. 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. MATHPRB 5: randInt(1, 100, 5) ENTER. Statistics Standards for Algebra II/Math III Understand and Evaluate Random Processes Sampling Distribution 100 Samples of Size 5 mean = 7.762 Dot Plot Measures from Sample of Rectangles 2 4 mean = 7.762 count = 100 stdDev = 2.45438 6 Sample Distribution 500 Samples of Size 5 mean = 7.356 8 10 Sam pleMean 12 14 Dot Plot Measures from Sample of Rectangles 2 16 4 mean = 7.3564 count = 500 stdDev 6 8 10 Sam pleMean 12 = 2.39474 As the number of samples increases, the mean of the sampling distribution gets closer and closer to the mean of the population. 14 16 Statistics Standards for Algebra II/Math III Understand and Evaluate Random Processes Sampling Distribution 100 Samples of Size 10 mean = 7.67 square units 4 6 8 10 Sam pleMean 12 14 16 2 4 mean = 7.67 mean = 7.5814 count = 100 count = 500 stdDev = 1.84826 Dot Plot Measures from Sample of Rectangles Dot Plot Measures from Sample of Rectangles 2 Sample Distribution 500 Samples of Size 10 mean = 7.581 square units stdDev 6 8 10 Sam pleMean 12 14 = 1.71477 As the sample size increases, the spread of the sampling distribution decreases. (The standard deviation gets smaller.) 16 Statistics Standards for Algebra II/Math III Understand and Evaluate Random Processes Big Ideas: When multiple samples are taken from the population, the values of the sample statistics vary from sample to sample. This is known as sampling variability. If the population distribution is not too unreasonably skewed, as more and more samples are taken from the population, the mean (center) of the sampling distribution approaches the population parameter. As the sample size increases, the spread of the sampling distribution decreases. The shape of the sampling distribution is approximately normal. Statistics Standards for Algebra II/Math III Understand and Evaluate Random Processes Activities: Random Rectangles Reese’s Pieces What proportion of Reese’s Pieces are Orange? (http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html Permission to share with Illinois math teachers has been given by Beth Chance and Allan Rossman .) Statistics Standards for Algebra II/Math III Making Inferences & Justifying Conclusions Algebra II & Math III Standard PBA EOY S.IC.3 X X S.IC.6 X X Statistics Standards for Algebra II/Math III Making Inferences & Justifying Conclusions Three types of statistical studies are surveys, observational studies, and experiments. In a survey the researcher gathers information by asking the subjects questions. In an observational study, the researcher observes and 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.5 X X Statistics Standards for Algebra II/Math III S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Statistics Standards for Algebra II/Math III Making Inferences & Justifying Conclusions Activity: Sleep Deprivation Does the effect of sleep deprivation linger or can we “make up” for lost sleep? To test this, 21 volunteer subjects ages 18 to 25 were randomly assigned to one of two treatment groups. Both groups first received training on a visual discrimination task. One group was deprived of sleep for the first night following this training, but were allowed unlimited sleep on the next two nights. The second group was allowed unlimited sleep on all three nights. Both groups were retested on the third day. Used with permission from Beth Chance and Allan Rossman. Statistics Standards for Algebra II/Math III Making Inferences & Justifying Conclusions Based on the graph and statistics below, do you think there is evidence that sleep deprivation on the first night might have had an effect on a subject’s improvement on the visual 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) The New Illinois Learning Standards Algebra I & Math I Standard S.ID.6a S.ID.6b Additional Information provided by PBA EOY PARCC Model Content Frameworks and Blue Prints i) Tasks have a real-world context. Algebra I ii) Exponential functions are limited to those with domains in X the integers. i) Tasks have a real-world context. Algebra II ii) Tasks are limited to exponential functions with domains not X X in the integers and trigonometric functions. i) Tasks have real-world context. Math I X ii) Tasks are limited to linear functions and exponential functions with domains in the integers. i) Tasks have real-world context. Math II ii) Tasks are limited to quadratic functions and Exponential X Functions (See EOY Evidence Table S-ID.Int.2 and S-ID.6a-1) i) Tasks have a real-world context. Math III ii) Tasks are limited to exponential functions with domains not X X in the integers and trigonometric functions. Possibly quadratic functions (See EOY Evidence Table – SAlgebra I X ID.Int.2) ii) Tasks are limited to linear functions, quadratic functions, Math II and exponential functions with domains in the integers. X Course Math III i) Tasks have a real-world context. ii) Tasks are limited to exponential functions with domains not in the integers and trigonometric functions. X The New Illinois Learning Standards S-ID.6 Summarize, represent, and interpret data on two categorical and quantitative variables. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.* S-ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize 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 Algebra I & Math I Standard PBA EOY S.ID.6 X S.ID.7 X S.ID.8 X S.ID.9 X The New Illinois Learning Standards S-ID.7 Interpret linear models. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.* S-ID.8 Interpret linear models. Compute (using technology) and interpret the correlation coefficient of a linear fit.* S-ID.9 Interpret linear models. Distinguish between correlation and causation.* Statistics Standards for Algebra I/Math I Interpreting Scatterplots Below is a graph of the heights and weights of currently rostered Chicago Bears (April 2014). Which is the best interpretation of the scatterplot? A. As heights go up, weight increases. B. As heights go up, weight tends to increase. C. If you get taller, you will get heavier. D. Taller football players tend to be heavier football players. http://chicagosports.sportsdirectinc.com/football/nfl-teams.aspx?page=/data/nfl/teams/rosters/roster16.html Statistics Standards for Algebra I/Math I Assessing the Fit of a Linear Function To assess the fit of a linear function to a scatterplot consider all of the following: 1. Look at the scatterplot. Does the data appear linear? 2. Look at r-value. What is the strength of the linear relationship between x and y? 3. Look at the residuals. Are they scattered with no discernable pattern? Statistics Standards for Algebra I/Math I Correlation When interpreting the correlation coefficient there are FOUR things that should be discussed: 1. The strength of the linear relationship: strong, moderate, weak, or no linear relationship. Statistics Standards for Algebra I/Math I Correlation 2. Whether the relationship between x and y is positive or negative. If the slope of the best fit line is positive, then the r value is also positive. If the slope of the best fit line is negative, then the r value is negative. Statistics Standards for Algebra I/Math I Correlation 3. The relationship we are evaluating is a linear relationship between x and y. For example, the two graphs below show a relationship between x and y that is something other than linear. Statistics Standards for Algebra I/Math I Correlation 4. CONTEXT! Be sure to interpret the correlation coefficient in the context of the problem. The linear association is between what two variables? Interpretation: r = 0.7 There is a moderate, positive linear relationship between the height (in inches) and the weight (in pounds) for the currently rostered Chicago Bears. Statistics Standards for Algebra I/Math I Correlation vs. Causation Example: Do fresh lemons cause a lower highway fatality rate? www.grossmont.edu/johnoakes/s110online/Causation%20versus%20Correlation.pdf Statistics Standards for Algebra I/Math I Residuals The residual is the vertical distance from the data point to the line of best fit. (5, 28) actual (5, 21) predicted Residual = Actual – Predicted Residual = 28 – 21 Residual = 7 Statistics Standards for Algebra I/Math I Residuals To create a residual plot, graph each x-coordinate with its corresponding residual. (x-value, residual) (x-value, residual) (5, 7) Residual = 7 This horizontal axis corresponds to the regression line. Statistics Standards for Algebra I/Math I Residuals Patterns of a residual plot are used to assess the fit of a function to the data. Patterns should appear scattered with no discernible pattern. Outlier Curved pattern Larger residuals for larger values of x Statistics Standards for Algebra I/Math I Residuals Why do we need to look at residual plots? Here is an example from Engage NY: The temperature (in degrees Fahrenheit) was measured at various altitudes (in thousands of feet) above Los Angeles. The scatter plot (next slide) seems to show a linear (straight line) relationship between these two quantities. http://www.engageny.org/sites/default/files/resource/attachments/algebra_i-m2-teacher-materials.pdf Statistics Standards for Algebra I/Math I Assessing the Fit of a Function to Data Activity: Altitude vs. Temperature The outside air temperature (in degrees Fahrenheit) for various altitudes (in thousands of feet) was measured for a plane flying above Los Angeles. Is there a linear relationship between altitude and temperature? http://www.engageny.org/sites/default/files/resource/attachments/algebra_i-m2-teacher-materials.pdf Statistics Standards for Algebra I/Math I Assessing the Fit of a Function to Data Altitude vs. Temperature Part I: Scatterplot. Does the data appear linear? Part II: Correlation Coefficient What is the strength of the linear relationship between x and y? Statistics Standards for Algebra I/Math I Assessing the Fit of a Function to Data Altitude vs. Temperature Part III: Residuals Are they scattered with no discernable pattern? Statistics Standards for Algebra I/Math I Residuals Example – Snakes! (www.insidemathematics.org) Snake 1 Snake 1 Rita catches 5 more snakes. She wants to know whether they belong to species A or to species B. The measurements of these snakes are shown in the table below. Statistics Standards for Algebra I/Math I Fit a Function to Data and Interpret Linear Models Additional Activities: Altitude vs. Temperature Adapted from an activity featured at www.engageny.org. Snakes – A residual activity from Inside Mathematics (http://www.insidemathematics.org/common-core-math-tasks/high-school/HS-S2003%20Snakes.pdf) The New Illinois Learning Standards Algebra II and Math II Statistics Standards for Algebra II/Math II Making Inferences & Justifying Conclusions Algebra II & Math II Standard PBA EOY S.CP.1 X S.CP.2 X S.CP.3 X S.CP.4 X S.CP.5 X S.CP.6 X S.CP.7 X Statistics Standards for Algebra II/Math II Conditional Probability and the Rules of Probability Big Ideas Provide opportunities for students to independently determine if a problem is asking for an AND, OR, or conditional probability. Provide data from real life problems. Ask students to calculate probabilities and interpret probabilities in the context of the problem. Students should be able to create and interpret two-way frequency tables, diagrams, probability trees, and other graphical displays. Determine if two events are independent or dependent, and interpret what this means in the context of the problem. Independence Two events E and F are said to be independent if: = () If E and F are not independent, they are said to be dependent events. If = (), it is also true that = (), and vice versa. If two events E and F are independent then the Multiplication Rule becomes: ∩ = () ∙ () Examples of events which are likely to be dependent: The event that a student spends more time on homework and studying and the event that the student’s test average goes up. The event that the amount of snow is above average in a given winter and the event that the number of show shovel sales is also above average for that winter. The event that a randomly selected passenger on the Titanic was a woman and the event that the passenger survived. Example of events which are independent The event that a 2 is rolled on the first roll of a dice and the event that a 2 is rolled on a second roll of a dice. The event that a couple’s first child is a boy and the event that a couple’s second child is a girl. Statistics Standards for Algebra II/Math II Conditional Probability and the Rules of Probability Activity: Conditional Probability Four Square Acknowledgements and Resources Chance, B. & Rossman, A. (Preliminary Edition). Investigating Statistical Concepts, Application and Methods. Duxbury Press. 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. 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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 NCTM Core Math Tools – Data Sets http://www.nctm.org/resources/content.aspx?id=32705 Online Resources PARCC Model Content Frameworks. http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovemb er2012V3_FINAL.pdf PARCC Mathematics Evidence Tables. https://www.parcconline.org/assessmentblueprints-test-specs Smarter Balanced Assessment Consortium. http://www.smarterbalanced.org/ Statistics Education Web (STEW). http://www.amstat.org/education/STEW/ The Data and Story Library (DASL). http://lib.stat.cmu.edu/DASL/ The High School Flip Book Common Core State Standards for Mathematics. http://www.azed.gov/azcommoncore/files/2012/11/high-school-ccss-flipbook-usd-259-2012.pdf The New Illinois Learning Standards for Algebra I / Math I Statistics and Probability Thank you for joining us! Dana Cartier Julia Brenson Tina Dunn [email protected] [email protected] [email protected]