What is statistics? Statistics is about using data to answer questions.

1. What is statistics? Statistics is about using data to answer questions. • Why do we study statistics? We gather, summarize, and analyze data to search for answers to our questions.

2. Question 1: • LeBron James is an NBA basketball super star who, in 2008, played for the Cleveland Cavaliers. His team made it to the playoffs that year and faced off against the Boston Celtics. However, sports announcers and fans alike claimed that he did not perform up to his abilities in the playoffs, claiming that he “choked”. Imagine you want to determine if this crisis is justified. You know that data involves facts, observations, and measurements about a particular topic or idea. If you wanted to gather data bout LeBron’s performance, what information might you look for? How could you use data to answer this question? • What was his shooting percentage in the regular season? • What was it in the playoffs? • How different are they?

3. Statistical analysis is the process of looking at data to learn about something bigger. We can think of the statistical analysis process in 4 steps. Steps in Statistical Analysis • Ask a question that can be answered by collecting data. • Decide what to measure and then collect data. • Summarize and analyze the data. • Draw a conclusion and communicate the results.

4. Did LeBron James choke in the 2008 playoffs? We will now do an activity that will help us learn about the statistical analysis process. In this activity we use the statistical analysis process to investigate a question about whether LeBron choked in the 2008 playoffs.

5. As you discovered in answering question number 1 above, there are many ways to judge the performance of a basketball player. We will choose to look at his 3-point percentage and use this to see if his ability to make 3-point shots decreased in the playoffs. Note, we are using his 3-point shooting percentage to answer the same question we saw above: Did LeBron James choke in the playoffs? In order to answer our questions, we need to collect the necessary data. Specifically we ned to know what LeBron’s 3-point shooting percentages were in the regular season of 2007-2008 and in the 2008 playoffs. The data shown below is from www.baskeballreference.com.

6. Reminder about class format One important part of the pedagogy is to do problems that are going to stretch you beyond what you know. Although this may be uncomfortable sometimes, we do this on purpose so you can learn. It’s important to know that when you feel stretched, it’s not a sign that you can’t be successful. Usually it’s a sign that you’re learning.

7. In the past, when some people get a problem wrong, they might have thought that they just don’t have the ability to study math – that they’re not math people. But when you talk to professionals mathematicians, the people who are best at math, it turns out that they work a long time on the same problem- and they only spend their time on problems that they struggle with the most. And even though you might think they make up answers on their own, almost always mathematicians have to ask people for help.

8. Like a professional mathematician, the goal of this course is to help you learn things that you can actually use in life – not so you can memorize it for a test and then forget it. And so in this course, sometimes we will work on problems that either don’t have a correct answer or that, for most, the answer isn’t obvious right away. At those times, it’s important to realize that the way that you thought about math in the past – like getting right means you are smart and getting in wrong means that you aren't – that type of thinking just isn’t true in this course. Instead we will work on problems that take three things: effort, a good strategy, and help from others. If you get it right, it’s because of one of those three things, and if you get it wrong, then you know what to do: change your effort, change your strategy, or ask for more help.

9. Answer each of the following questions about the four steps of the statistical analysis process. Step 1: Ask a question that can be answered with data. • What question are we trying to answer in our investigation of LeBron’s 3-point shooting percentages in the regular season and in the playoffs of 2007-2008? Did LeBron choke in the 2008 playoffs? Specifically, did his ability to make 3-point shots decrease in the playoffs?

10. Step 2: Decide what to measure and the collect data. • What information did we get from the website? Are the data related to the question we are trying to answer? We got his percentage of 3 points shots made during the regular season and during the playoffs. Yes, we will use this data to answer the question posed.

11. Step 3: Summarize and Analyze Data • Use the data shown above to summarize the necessary data in the table below. How might we use this data to decide if his ability to make 3-point shots in the playoffs decreased? Record the appropriate data below: We see that it is less in the playoffs than in the regular season, but we would expect it to be different just cause of chance (or luck). What we need to decide is if his decreased percentage in the playoffs Is more than what we could expect just because of chance? So we need to know what range of percentages, base on his regular season 3-point percentage , would we expect just because of chance.

12. Step 4: Draw a conclusion and communicate the results. • Once the data are summarized and analyzed, how can we use this to answer the research question? If LeBron’s 3-point percentage in the playoffs is not less than what we would expect just due to random variation, then we might conclude his poor performance was just due to chance. It if is less than what we would expect because of random variation, then we have evidence that his ability to make 3-point shots in the playoff did decrease, and perhaps conclude that he did choke.

13. Wrap-Up Were going to need percentages of made shots. What about random variation in shooting ability? How could we possibly rule out chance as a possibility for deviations from expected results. Simulation of his shootings!

14. Wrap-Up Consider his “normal” 3-pont percentage in the regular season, what range of 3-point percentages for LeBron would we expect in the playoffs, just because of chance/ random variation/ luck? What percentages are most likely? If you can rule out chance as an explanation for his performance in the playoffs, you have evidence to support the assertion that he choked.

15. Next Steps As we proceed, we will address the question of whether LeBron choked in the playoffs. We have completed the data collecting and summarizing process by researching on the internet and then recoding the data we found in a table. We now prepare to analyze our data. To do this, we need to think about how the data can be used to answer our question. We will use simulation to help in this. As a group, lets answer the following questions.

16. Suppose LeBron did not choke in the playoffs. Would you expect his 3 – point percentage to be different than what it was in the regular season? Explain why you think this. Yes, his shooting percentage will vary from game to game just due to random variation/chance/luck. If LeBron did not choke in the playoffs, about what percentage of 3-point shots would you expect him to make? Somewhere “close” to his regular season percentage of 31.5%

17. 4 Suppose LeBron did choke in the playoffs. Would you expect the percentage of 3 – point shots he made in the playoffs to be greater or less than the percentage you wrote above? Less Than How poorly would LeBron had to have performed in order to convince you that he choked in the playoffs, that is, that his 3 - point shooting ability decreased in the playoffs? Give a specific percentage. Tell why you think this. The percentage should be enough less than 31.5% that we can be convinced – but not 0!

18. Imagine that LeBron’s 3-point percentage in the playoffs was 22%. Does this prove without a doubt that LeBron’s performance in the playoffs decreased in the playoffs? If not, give another explanation for why his percentage might have been so different than his percentage in the regular season. It is not a guarantee. Maybe he shot that poorly just by chance.

19. Using simulation to discover what may happen by chance If LeBron did not choke in the playoffs, we would expect him to perform at close to his regular season percentage, 31.5%. But how far below 31.5% would convince us that he did choke? We will answer this by simulating LeBron’s 3-point shooting ability in the regular season. From that, we’ll see the typical range of percentages to expect from LeBron when he’s performing at his usual level of ability, in other words, when he’s not choking. This range of percentages is what would likely occur just due to random (or chance) variation. We’ll also be able to see what percentages are most likely.

20. Discussion Question A: Why do you expect the percentage to be around 31.5%? Note: This is what we would expect if LeBron was performing as he had over the regular season. Discussion Question B: Why is it “around 31.5%" and not “exactly 31.5%"? Note: You do not expect him to perform at exactly what his season percentage was. He would make a few more or a few less than 31.5% by chance. Another way to say this would be PERFROMANCE = ABILITY ± RANDOM VARIATION We might expect his shooting percentage to be some less than 31.5%, even if he hadn’t choked, just because of chance. In order to be convinced that he choked we need to see a percentage far enough below 31.5% to convince us that it wasn’t just due to chance. How far below 31.5% convinces us that he choked? How do you figure out that our results are not “by chance”? To be sure that “chance” does not explain our results, you need to rule out chance as a possible explanation. To rule out chance as a possible explanation, you need to have an idea of what range of 3 point percentages to expect just by chance and which are most likely, assuming he was shooting at his normal rate of 31.5%.

21. Spinner Activity • Is your percentage of “Made Its” equal to 31.5%? If not, is it greater than or less than 31.5%? Report out results, answers will vary. • Do you think everyone in the class got the same percentage of “Made Its”? Why or why not? Spinner are random so not everyone will get exactly the same percentage.

22. Looking at your one percentage, is that enough to determine what shooting percentages we would expect to observe just by chance? This just tells us one value, but what we really need to see is a pattern of values. This could have been an unusual outcome. We want to look at many outcomes to see what values come up most often. Knowing this one percentage is not enough. What additional information would help us get a better understanding of the range of shooting percentages that we would expect from LeBron just due to chance? Make sure that the discussion ends up with “we need to try this more times

23. First, write the class shooting percentages in the following table. Add to the table as necessary to accommodate the number of trials your class produced. Each Trial represents 70 shots or 70 spins. Observed Percentage means the percentage of “Made Its” in each round of 70 spins produced. (Also fill out dotplot)

24. Answer question 10. What was the smallest shooting percentage observed? What was the largest shooting percentage observed? Did the shooting percentage differ much from trial to trial? “To really understand how the observed proportion varies from one trial to another, we need to look at many more trials. This process can take a long time, so we will turn to technology to perform more trials. Use the dotplot produced in the movie to answer the following questions. 11 What does each dot represent in the dotplot? Each dot represents the percentage of “Made Its” in 70 spins or each dot represents the percentage of 70 3-point shots Lebron made in the playoffs if he shot at the same percentage that he did in the regular season.

25. 12 Where is the dotplot centered? Why do you think this is? This dotplot should be centered around 0.315 or 31.5%. We expect most proportions to be fairly close to 31.5%, LeBron’s regular season 3-point shooting percentage. 13 What range of percentages is most likely? Since this is a random process answers will vary, but the most likely are the ones near the center, say between 21% and 43%, even though we see percentages as low as 19% and as high as 45%. Random (or chance) variation Reminder.

26. Look at the dotplot. How can this graph help you decide whether LeBron choked or not? This graph shows the range of shooting percentages we would expect if LeBron shot at the same ability in the playoffs as he did in the regular season. If the percentage LeBron actually shot in the playoffs is lower than what we’re seeing in the dotplot, then we might say we have enough evidence to conclude that LeBron made a lower percentage of his 3-point shots in the playoffs than what we would expect just due to chance, if he was shooting with the same percentage as the regular season. In other words, he choked. 15 Recall that LeBron made 25.7% of his 3-point shots in the 2008 playoffs. Use the dotplot to answer the following question. If LeBron was shooting in the playoffs with the same percentage as he did in the regular season, would it be unusual for him to have made 25.7% of his 3-point shots? Hint: Look at your answer to number 13 It would not be unlikely to see this percentage if he was shooting in the playoffs at the same percentage as he was in the regular season.

27. Draw a Conclusion and Communicate the Results. • So, is there enough evidence for us to conclude that LeBron choked? That is, is there enough evidence for us to conclude that if LeBron was shooting in the playoffs at the same percentage as he was in the regular season, his shooting percentage in the playoffs was sufficiently low enough for us to conclude that his lower percentage was not just due to chance? No there is not enough evidence for us to conclude that he choked. While 25.7% is lower than his regular season percentage it would not be unlikely for him to shoot 25.7% in the playoffs just due to random variation in his performance. Note: From the dotplot, 25.7% is closer to the middle of the distribution rather than way out in the tail.

28. WRAP-UP for our spinner activity —state a theory, collect data to test the theory, and consider what kind of evidence provides support for this theory. Two possible explanations • the theory is correct • the theory is not correct and the observed result is due to chance variability. To conclude that there is evidence in support of the theory, they need to rule out chance as a plausible explanation for what was observed. Accomplished by generating results assuming the theory is false and what they are seeing is only due to chance variability. they then compare the observed results to see whether they can be considered consistent with chance.

29. Next Steps. The task you have just completed with the LeBron example illustrates the statistical analysis process, which we have described in four steps. These are given again below. Steps in a Statistical Investigation • Ask a question that can be answered by collecting data. • Decide what to measure and then collect data. • Summarize and analyze the data. • Draw a conclusion and communicate the results.

30. 17. Identify each step of the Statistical Analysis Process for the LeBron investigation below. • Ask a question that can be answered by collecting data. Did LeBron James choke in the playoffs? • Decide what to measure and then collect data. We decided to use 3 – point percentages as a measure of LeBron’s performance in the playoffs. We researched and found this information on the internet. • Summarize and analyze the data. We recorded the number of 3-point shots LeBron made in the regular season and in the playoffs and calculated the percentage of 3 – point shots he made in each and compared it to what might happen by random variation in his performance • Draw a conclusion and communicate the results. Assuming his regular season 3- point percentage of 31.5% indicates his ability, his playoff percentage of 25.7% was not lower than what we would expect just due to random variation in his performance. Therefore, we cannot conclude that he choked in the 2008 playoffs

31. Take it Home Activity in Binder Book. Leave in Binder.