SACE Stage 1 Mathematics STATISTICS

1. SACE Stage 1 Mathematics STATISTICS LINKING MATHEMATICS WITH RELEVANT TEACHING AND LEARNING PRACTICES IN THE SENIOR YEARS Session 1

2. Statistical thinking will one day be as necessary a qualification for efficient citizenship as the ability to read and write. --H.G. Wells

3. What is Statistics? Statistics is the art of solving problems or answering questions that require the collection and analysis of data.

4. What are these workshops? • These workshops present a small number of investigations and activities using statistical methods prescribed in the syllabus. • The purpose is to illustrate the motivation and development of statistical reasoning through its application in problem solving.

5. What they are not • These workshops are not a refresher course on elementary statistics. • It is assumed that teachers can access basic knowledge of the necessary techniques. • There is no discussion of formulas. • There are few example calculations. • There are no step by step computer or calculator instruction.

6. What they are not • However, the data are available separately and participants are encouraged to reproduce the calculations between the sessions and experiment with their own analysis. • Some source material is also provided.

7. What they are not • They do not exhaustively cover the syllabus. • It is again assumed that the participants will be able make themselves familiar with this material.

8. What they are not • They are not a practice run for classroom teaching. • They are not detailed lesson plans. • In contrast to the classroom, the routine technical aspects of the subject are not addressed. • Those aspects will, of course, occupy a significant amount of classroom time.

9. Outcomes • At the end of the sessions, it is anticipated the participants will come away with: • An appreciation of the elegance and power of statistical ideas. • An appreciation of the role of data analytic investigation as a vehicle for the development of statistical reasoning and methods.

10. Outcomes • At the end of the sessions, it is anticipated the participants will come away with: • A better understanding of what makes an appropriate investigation. • The confidence to implement teaching programs based on the problem solving approach.

11. Problemsto investigate • The Road Accident problem. • The Titanic question.

12. Entering the problem zone • Improving our accident record. • Focus on the recognition of students prior knowledge. • Video presentation and so on.

13. The road traffic problem • It is widely known that stopping distance of a vehicle increases dramatically with speed. • This can be checked using high school level physics and has been proved experimentally. • For this reason we can be sure that excessive speed increases the risk of accident. • The question of interest is: By how much?

14. The road traffic problem • If speed is an important factor in a significant number of accidents then there is justification for increased spending on: • Driver education. • Advertising campaigns. • Policing speed laws. • It also justifies the use of speed cameras as life savers rather than revenue raisers.

15. The road traffic problem If speed is not an important factor then spending could be directed to: • Better roads. • Combating drink driving. • Vehicle inspections.

16. The Research Question Were cars involved in serious crashes travelling faster than other cars?

17. What might we do to answer this question? Groups suggest an appropriate way to investigate the road accident problem.

18. The Study • Serious crashes that occurred on rural roads in a 150 km radius of Adelaide were studied. • Alcohol was determined not to be a factor. • The vehicles were travelling at ‘free speed’.

19. Free Speed • Free speed means that the vehicles were travelling without obstruction: • They were not trying to overtake another vehicle. • They were not trying to enter a road or merge with traffic. • If they were at an intersection, they had right of way.

20. Accident Vehicle Speeds • For each ‘free speed’ crash the researchers: • Attended the accident scene. • Obtained measurements of tyre marks and point of impact etc. • Used computerized accident reconstruction techniques to estimate the speed of the accident vehicle at the time of the crash.

21. Control Vehicles • For each ‘free speed’ crash the researchers: • Returned to the crash scene some days after the accident. • Chose the same day of week as the actual accident. • Chose the same hour of the day. • Chose a day with similar weather and lighting conditions.

22. Control Vehicles • Identified 10 vehicles travelling at the free speed and measured their speeds using a hand-held radar. • Were careful to conceal themselves from the view of the motorists they were observing. • These vehicles are called the control vehicles.

23. The Accident Data For 83 accidents the following data was recorded: • The speed of the accident vehicle in km/h. • The speeds of each of the 10 control vehicles in km/h. • The speed limit on the section of road.

24. How might we use this data to answer the question? Groups consider appropriate ways to investigate the road accident problem.

25. A First Look at the Data • We want to compare the speeds of the 83 accident vehicles to the 830 controls. • Since speed is a quantitative variable, it is appropriate to consider histograms. • We use separate histograms with the same horizontal scale.

26. A First Look at the Data

27. Summary Statistics

28. First Conclusions • On average the accident vehicles were travelling 13.1kmk/h faster than the control vehicles. • A small number of the accidents were travelling very fast. (five were going above 140km/hour) • No control vehicles were recorded above 140km/hour. • The accident vehicles had also a peak at 110-119 km/hour. • No such peak was present in the control speeds.

29. The 110-119 km/h Peak • Explanation 1 The researchers were biased toward guessing near the speed limit when the reconstruction was difficult. The researchers were sure that this was not the case. • Explanation 2 The drivers misunderstood the “no limit” sign to mean 110km/h instead of 100km/h.

30. Investigating the Peak • Explanation 2 can be investigated by tabulating the speed limits for these vehicles. The majority of these where recorded in 100 km/hour zones • We must compare this to the overall distribution of speed limits • This shows roughly the same percentage amongst all cases, so there is no real evidence for this explanation

31. Conclusion Apart from a small number of very high speeds and the unexplained peak between 110-119km/h there appears to be no major differences between the distributions of speed for accidents and controls .

32. Reflections • Have we answered the original research question? • Are we satisfied with this answer? Groups to consider these points and offer suggestions for further analysis.

33. Entering a new problem zone • An historic adventure. • Focus on the recognition of students prior knowledge. • Video presentation and so on.

34. Titanic Study • The tragic maiden voyage of the Titanic has captured the interest of many people and it is now our turn to investigate some of the issues related to the voyage. • The question of interest is: Did all passengers on the Titanic have an equal chance of surviving?

35. Consider the question Groups to discuss how they might investigate the question.

36. The Data • Data collected from range of websites: • The OzDASL site at http://www.maths.uq.edu.au/~gks/data/index.html • An Excel file of ‘Titanic’ data can be found through this site but is incomplete. The file supplied is as complete as is possible. • The OzDASL site is an Australian version of the DASL (Data and Story Library) site at http://dasl.datadesk.com/ • Both sites contain data files to suit many areas of interest.

37. Passenger information • There were 1313 passengers on board the Titanic. • For each passenger the following is recorded: • age • gender • class of travel (1st, 2nd or 3rd) • whether or not they survived the sinking. • Some data values are missing.

38. The data

39. How might the data be analysed? Groups to discuss how they might proceed in order to reach an answer to the question posed.

40. A first look at the Data • If we want to examine any relationships amongst survivors, then we need to consider the population of passengers on board the Titanic. • Gender, class of travel and survival are categorical variables and the relationship between them can be examined by charts and tables.

41. A summary of the data Class count % 1st 322 24.5 2nd 280 21.3 3rd 711 54.2 Gender count % female 462 35.2 male 851 64.8 Survival Status count % yes 450 34.3 no 863 65.7

42. Comments • From the tabular information we can see that • There were more male passengers than female. • The majority of passengers travelled third class. • About one third of the passengers survived.

43. Comments While this summary provided a good description of our population, and is an important step, it does not answer our question, more information can be obtained if cross tabulation is used.

44. Survival by class

45. Survival by gender Survival by gender for the Titanic passengers 90.00 80.00 70.00 60.00 50.00 No percentage Yes 40.00 30.00 20.00 10.00 0.00 female male gender

46. Observations when two variables are considered • The class with the largest percentage of survivors is first class whereas third class has the smallest percentage of survivors. • The percentage of females surviving is much larger than the percentage of males.

47. Reflections on the observations • Do these observations answer our question? • Are we satisfied with this answer? Groups reflect and offer suggestions.

48. Gender by class • Offers a way to investigate the gender balance in classes. • Offers a pathway for further analysis.

49. Gender by class

50. Comments when two variables are considered. • The number of males in third class is more than double the number of females. • Any more comments ?