The New Illinois Learning Standards for Algebra II / Math III Statistics and Probability

1. The New Illinois Learning Standards for Algebra II / Math IIIStatistics and Probability Julia Brenson

2. The Four Components of a Statistical Investigation* 1) Formulate a question 2) Design and implement a plan to collect data 3) Analyze the data by measures and graphs 4) Interpret the results in the context of the original question *Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report American Statistical Association http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf

3. The New Illinois Learning Standards Algebra II and Math III

4. Statistics Standards for Algebra II/Math IIINormal Distribution

5. Statistics Standards for Algebra II/Math IIINormal 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

6. Statistics Standards for Algebra II/Math IIINormal Distribution The Empirical Rule 68% 95% 97.5%

7. Statistics Standards for Algebra II/Math IIINormal 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? .0235 + .0015 = .0250 6 11 16 21 26 31 36

8. Statistics Standards for Algebra II/Math IIINormal 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?

9. Statistics Standards for Algebra II/Math IIINormal Distribution Animal Cracker Lab Mean = 20.04 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?

10. Statistics Standards for Algebra II/Math IIINormal Distribution Activities: • Animal Cracker Lab • See Illustrative Mathematics Activities: • SAT Scores • Should We Send Out a Certificate? • Do You Fit In This Car?

11. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes

12. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes A statistic is a numerical summary computed from a sample. A parameter is a numerical summary computed from a population. Astatistic will vary depending on the sample from which it was calculated, but a population parameter is a constant value that does not change.

13. Statistics Standards for Algebra II/Math IIIUnderstand 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.

14. Statistics Standards for Algebra II/Math IIIUnderstand 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

15. Statistics Standards for Algebra II/Math IIIUnderstand 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.

16. Statistics Standards for Algebra II/Math IIIUnderstand 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.

17. Statistics Standards for Algebra II/Math IIIUnderstand 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?

18. Statistics Standards for Algebra II/Math IIIUnderstand 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.

19. Statistics Standards for Algebra II/Math IIIUnderstand 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.

20. Statistics Standards for Algebra II/Math IIIUnderstand 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

21. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes • Calculator • Reseed: Enter a four digit number of your choice into your TI-84 then • STO • MATH  PRB • 1: rand • ENTER. • Generate five random numbers from 1 to 100 inclusive. • MATHPRB • 5: randInt(1, 100, 5) • ENTER.

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

23. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes Sample Distribution 500 Samples of Size 10 mean = 7.581 square units Sampling Distribution 100 Samples of Size 10 mean = 7.67 square units As the sample size increases, the spread of the sampling distribution decreases. (The standard deviation gets smaller.)

24. Statistics Standards for Algebra II/Math IIIUnderstand 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.

25. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes 100 Rectangles 5.6 6.4 9.6 7.7 6.4 7.7 5.6 9.6 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Simulation Process Model shared with permission from Sharon Lane-Getaz

26. Statistics Standards for Algebra II/Math IIIUnderstand 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 .)

27. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes

28. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes Big Idea • Use simulation to determine probabilities or verify a probability model. Example: Spinning Pennies • Stanford professor Persi Diaconis’ research indicates a spinning penny lands tails up approximately 80% of the time. Can this be true? See “What are the odds? New study shows how guessing heads or tails isn’t really a 50-50 game,” (Daily Mail http://www.dailymail.co.uk/news/article-2241854/What-odds-New-study-shows-guessing-heads-tails-isnt-really-50-50-game.html ) • Questions: • What is the probability of a spinning penny landing tails up? • What is the probability that a spinning penny lands tails up at least 80% of the time?

29. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes Number of Tails in 40 Spins ? Spins

30. Statistics Standards for Algebra II/Math IIIUnderstand and Evaluate Random Processes Activities: • Penny Spinning • Sarah the Chimpanzee (www.illustrativemathematics.org) • Block Scheduling (www.illustrativemathematics.org)

31. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions

32. Statistics Standards for Algebra II/Math IIIMaking 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

33. Statistics Standards for Algebra II/Math IIIMaking 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.*

34. Statistics Standards for Algebra II/Math IIIMaking 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.

35. What conclusions may we draw from statistical studies? (adapted from Ramsey and Schafer’s The Statistical Sleuth)

36. Statistics Standards for Algebra II/Math IIIMaking 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.

37. Statistics Standards for Algebra II/Math IIIMaking 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.

38. Statistics Standards for Algebra II/Math IIIMaking 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)

39. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions

40. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions In our work with S.IC.2 (Random Rectangles), we found that the center of the sampling distribution provides a reasonable estimate of the population parameter. In S.IC.4, we use a sample statistic +/- a margin of error to make inferences about the population parameter. Here are the key ideas: • We take a random sample from the population. The sample statistic will be our estimate of the population parameter. • We know that our sample proportion is likely to differ from the population proportion. How much do we expect it to differ? The margin of error is the anticipated difference between the sample proportion and the true population proportion. • S.IC.4 states that students should be able to find a margin of error using simulation.

41. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Activity: Margin of Error Part I Do you Tweet? The Pew Internet Research Project, in an internet article titled The Demographics of Social Media Users – 2012, reported the results from a landline and cellphone survey of internet users. The table below is taken from the online article.

42. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Part I Do You Tweet? Questions: 1. What was the population of interest? 2. How many people were in the sample? 3. What percent of all internet users use Twitter? 4. What margin of error is reported? 5. What do you think this margin of error means?

43. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Activity: Margin of Error Part II Paper Bag Population - Exploring the Margin of Error I have an entire population of colored beads in my paper bag. What questions might we ask about my population?

44. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Activity: Margin of Error (Paper Bag Population) Let’s conduct a statistical investigation. Formulate a question What proportion of the beads in the bag is blue? Design and implement a plan to collect data Take random samples of size 25 from the paper bag population. The sample proportion of blue beads will be our estimate of the proportion of blue beads in the paper bag population. We use the symbol (p-hat) to represent our estimate of the population proportion.

45. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Activity: Margin of Error (Paper Bag Population) Analyze the data by measures and graphs A possible sampling distribution: Mean =0.61 Std Dev = 0.10 n = 28

46. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Margin of Error One method of constructing an interval is to use a sample statistic ± a margin of error. The margin of error is 2 times the standard deviation of the sampling distribution.

47. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions We use the reasoning that if the sample statistic is likely to be within 2 standard deviations of the center of the sampling distribution, then the center of the distribution is also likely to be within 2 standard deviations of the sample statistic. Remember that the population parameter is approximately equal to the mean of the sampling distribution, so the population parameter is likely to be within two standard deviations of our sample statistic. In other words, we conclude that the population parameter is likely to be in this interval. (Or at least we expect 95% of the intervals created by this method to include the population parameter.)  

48. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Let’s look at this mathematically. We anticipate that 95% of our sample proportion, will fall within 2 standard deviation of the population parameter, .

49. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Back to our Paper Bag Population! Interpret the results in the context of the original question • Find the margin of error Margin of error = = 0.10 = 0.20 • Report a margin of error   Suppose my sample proportion was 0.52, then 0.52 0.20 • Interpret the margin of error The true population proportion of blue beads is estimated to be 0.52 0.20. I anticipate that the proportion of blue beads in the paper bag population is between 0.32 and 0.72.

50. Statistics Standards for Algebra II/Math IIIMaking Inferences & Justifying Conclusions Back to Twitter! • Interpret the margin of error Based on the Pew Research survey, the true percentage of internet users who use Twitter is estimated to be 16% 2.6%. We anticipate that the percentage of internet users that use Twitter is between 13.4% and 18.6%.